Research ArticleUniform Treatment of Jensenrsquos Inequality byMontgomery Identity

Tahir Rasheed1 Saad Ihsan Butt 1 ETHilda Pecaric2 Josip Pecaric 3

and Ahmet Ocak Akdemir 4

1COMSATS University Islamabad Lahore Campus Lahore Pakistan2Department of Media and Communication University North Trg dr Zarka Dolinara 1 Koprivnica Croatia3Peoples Friendship University of Russia (RUDN University) 6 Miklukho-Maklaya St Moscow 117198 Russia4Department of Mathematics Faculty of Arts and Sciences A grı Ibrahim Ccedileccedilen University Agrı Turkey

Correspondence should be addressed to Ahmet Ocak Akdemir aocakakdemirgmailcom

Received 26 February 2021 Revised 2 April 2021 Accepted 12 April 2021 Published 17 May 2021

Academic Editor Xiaolong Qin

Copyright copy 2021 Tahir Rasheed et alis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We generalize Jensenrsquos integral inequality for real Stieltjes measure by using Montgomery identity under the effect of nminusconvexfunctions also we give different versions of Jensenrsquos discrete inequality along with its converses for real weights As an applicationwe give generalized variants of HermitendashHadamard inequality Montgomery identity has a great importance as many inequalitiescan be obtained fromMontgomery identity in qminuscalculus and fractional integrals Also we give applications in information theoryfor our obtained results especially for Zipf and Hybrid ZipfndashMandelbrot entropies

1 Introduction

Convex functions have a great importance in mathematicalinequalities and the well-known Jensenrsquos inequality is thecharacterization of convex functions Jensenrsquos inequality fordifferentiable convex functions plays a significant role in thefield of inequalities as several other inequalities can be seenas special cases of it One can find the application of Jensenrsquosdiscrete inequality in discrete-time delay systems in [1]

Taking into consideration the tremendous applicationsof Jensenrsquos inequality in various fields of mathematics andother applied sciences the generalizations and improve-ments of Jensenrsquos inequality have been a topic of supremeinterest for the researchers during the last few decades asevident from a large number of publications on the topic (see[2ndash4] and the references therein)

e well-known Jensenrsquos inequality asserts that for thefunction Γ it holds that

Ψ1

Pm

1113944

m

J1pJxJ

⎛⎝ ⎞⎠le1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 (1)

if Ψ is a convex function on interval I sub R where pJ arepositive real numbers and xJ isin I(J 1 m) whilePm 1113936

mJ1 pJ

However the well-known integral analogue of Jensenrsquosinequality is as follows

Theorem 1 Let Z [a b]⟶ [α β] be a continuous func-tion and λ [a b]⟶ R be an increasing and boundedfunction with λ(a)ne λ(b) =en for every continuous convexfunction Ψ [α β]⟶ R the following inequality holds

Ψ(1113957Z)le1113938

b

aΨ(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

(2)

where

1113957Z 1113938

b

aZ(ζ)dλ(ζ)

1113938b

adλ(ζ)

isin [α β] (3)

ere are several inequalities coming from Jensenrsquosinequality both in integral and discrete cases which can be

HindawiJournal of MathematicsVolume 2021 Article ID 5564647 17 pageshttpsdoiorg10115520215564647

obtained by varying conditions on the function Z andmeasure λ defined in eorem 1

Montgomery identity is used in quantum calculus orqminuscalculus ere are different identities of Montgomeryand several inequalities of Ostrowski type were formulatedby using these identities Budak and Sarikaya established thegeneralized Montgomery-type identities for differentialmappings in [5] Applications of Montgomery identity canbe found in fractional integrals as well as in quantum integraloperators Here we utilize Montgomeryrsquos identity for thegeneralization of Jensenrsquos inequality In [6] Cerone andDragomir developed a systematic study which producedsome novel inequalities Several interesting results related toinequalities and different types of convexity can be found in[7ndash21] e class of convex functions is a very useful conceptthat has become a focus of interest for researchers in sta-tistics convex programming and many other applied dis-ciplines as well as in inequality theory e readers can find

some motivated findings related to convex functions andsome new integral inequalities in [22ndash27]

In [28] Khan et al have mentioned about n-convexfunctions as follows

Definition 1 A function f I⟶ R is called convex oforder n or n-convex if for all choices of (n + 1) distinctpoints xi xi+n we have Δ(n)f(xi)ge 0

If n-th order derivative f(n) exists then f is nminusconvex ifand only if f(n) ge 0 For 1le kle (n minus 2) a function f isnminusconvex if and only if f(k) exists and is (n minus k)minusconvex

In the present paper we will use Montgomery identitythat is presented as following

Theorem 2 Let n isin N Ψ I⟶ R be such that Ψ(nminus1) isabsolutely continuous I sub R is an open interval and α β isin Iαlt β =en the following identity holds

Ψ(x) 1

β minus α1113946β

αΨ(t)dt + 1113944

nminus2

ℓ0

Ψ(ℓ+1)(α)

ℓ(ℓ + 2)

(x minus α)ℓ+2

β minus αminus 1113944

nminus2

ℓ0

Ψ(ℓ+1)(β)

ℓ(ℓ + 2)

(x minus β)ℓ+2

β minus α+

1(n minus 1)

1113946β

αRn(x s)Ψ(n)

(s)ds (4)

where

Rn(x s)

minus(x minus s)

n

n(β minus α)+

x minus αβ minus α

(x minus s)nminus 1

αle slex

minus(x minus s)

n

n(β minus α)+

x minus ββ minus α

(x minus s)nminus 1

xlt sle β

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(5)

2 Generalization of Jensenrsquos IntegralInequality by Using Montgomery Identity

Before giving our main results we consider the followingassumptions that we use throughout our paper

A1 Let Z [a b]⟶ R be continuous functionA2 Let λ [a b]⟶ R be a continuous function or thefunctions of bounded variation such that λ(a)ne λ(b)

21 New Generalization of Jensenrsquos Integral Inequality Inour first main result we employ Montgomery identity toobtain the following real Stieltjes measurersquos theoreticalrepresentations of Jensenrsquos inequality

Theorem 3 Let g λ be as defined in A1 A2 such thatZ([a b]) sub [α β] Also let Ψ [α β]⟶ R be such that fornge 1 Ψ(nminus1) is absolutely continuous If Ψ is nminusconvex suchthat

Rn(1113957Z s)le1113938

b

aRn(Z(ζ) s)dλ(ζ)

1113938b

adλ(ζ)

s isin [α β] (6)

with 1113957Z and Rn(x s) as defined in (3) and (5) respectivelythen we have

Ψ(1113957Z) minus1113938

b

aΨ(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

le 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

times

Ψ(ℓ+1)(α) (1113957Z minus α)

(ℓ+2)minus

1113938b

a(Z(ζ) minus α)

(ℓ+2)dλ(ζ)

1113938b

adλ(ζ)

⎛⎝ ⎞⎠ minus Ψ(ℓ+1)(β) (1113957Z minus β)

(ℓ+2)+

1113938b

a(Z(ζ) minus β)

(ℓ+2)dλ(ζ)

1113938b

adλ(ζ)

⎛⎝ ⎞⎠⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

(7)

2 Journal of Mathematics

Proof AsΨ(nminus1) is absolutely continuous for (nge 1) we canuse the representation of Ψ using Montgomery identity (4)and can calculate

Ψ(1113957Z) 1

β minus α1113946β

αΨ(ζ)d(ζ) + 1113944

nminus2

ℓ0

Ψ(ℓ+1)(α)

ℓ(ℓ + 2)

(1113957Z minus α)ℓ+2

β minus αminus 1113944

nminus2

ℓ0

Ψ(ℓ+1)(β)

ℓ(ℓ + 2)

(1113957Z minus β)ℓ+2

β minus α+

1(n minus 1)

1113946β

αRn(1113957Z s)Ψ(n)

(s)ds (8)

e integration of the composition of functionsΨ ∘ Z forthe real measure λ on [a b] gives

1113938b

aΨ(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

1

β minus α1113946β

αΨ(ζ)d(ζ) + 1113944

nminus2

ℓ0

Ψ(ℓ+1)(α)

ℓ(ℓ + 2)

1113938b

a(Z(ζ) minus α)

(ℓ+2)dλ(ζ)

(β minus α) 1113938b

adλ(ζ)

minus 1113944nminus2

ℓ0

Ψ(ℓ+1)(β)

ℓ(ℓ + 2)

1113938b

a(Z(ζ) minus β)

(ℓ+2)dλ(ζ)

(β minus α) 1113938b

adλ(ζ)

+1

(n minus 1)1113946β

αRn(Z(ζ) s)Ψ(n)

(s)ds

(9)

Now computing the difference Ψ(1113957Z) minus 1113938b

aΨ(Z(ζ))

dλ(ζ)1113938b

adλ(ζ) we get the following generalized identity

involving real Stieltjes measure

Ψ(1113957Z) minus1113938

b

aΨ(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

times Ψ(ℓ+1)(α) (1113957Z minus α)

(ℓ+2)minus

1113938b

a(Z(ζ) minus α)

(ℓ+2)dλ(ζ)

1113938b

adλ(ζ)

⎛⎝ ⎞⎠ minus Ψ(ℓ+1)(β) (1113957Z minus β)

(ℓ+2)+

1113938b

a(Z(ζ) minus β)

(ℓ+2)dλ(ζ)

1113938b

adλ(ζ)

⎛⎝ ⎞⎠⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

+1

(n minus 1)1113946β

αRn(1113957Z s) minus

1113938b

aRn(Z(ζ) s)dλ(ζ)

1113938b

adλ(ζ)

⎛⎝ ⎞⎠Ψ(n)(s)ds(GI1)

(10)

Finally by our assumption Ψ(nminus1) is absolutely con-tinuous on [α β] as a result Ψ(n) exists almost everywhereMoreover Ψ is supposed to be nminusconvex so we haveΨ(n)(x)ge 0 almost everywhere on [α β] erefore bytaking into account the last term in generalized identity(GI1) and integral analogue of Jensenrsquos inequality that isgiven in (6) we get (7)

In the later part of this section we will vary our con-ditions on functions g and Stieltjes measure dλ to obtaingeneralized variants of JensenndashSteffensen JensenndashBoasJensenndashBrunk and Jensen-type inequalities We start with

the following generalization of JensenndashSteffensen inequalityfor nminusconvex functions

Theorem 4 Let Ψ defined in =eorem 3 be nminusconvex and Z

defined in M1 be monotonic =en the following results hold

(i) If λ defined in M2 satisfies

λ(a)le λ(x)le λ(b) forallx isin [a b] λ(b)gt λ(a) (11)

then for even nge 3 (6) is valid(ii) Moreover if (6) is valid and the function

H(x) ≔ 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)(x minus α)

ℓ+2minus Ψ(ℓ+1)

(β)(x minus β)ℓ+2

1113872 1113873 (12)

Journal of Mathematics 3

is convex then we get inequality (2) which is calledgeneralized JensenndashSteffensen inequality fornminusconvex function

Proof (i) By applying second derivative test we can showthat the function

Rn(x s) is convex for even ngt 3 Now using theassumed conditions one can employ JensenndashSteffensen inequality given by Boas (see [29] or [30]p 59) for convex function Rn(x s) to obtain (6)

(ii) Since we can rewrite the RHS of (7) in thedifference

H(1113957Z) minus1113938

b

aH(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

(13)

for convex function H and by our assumed conditions onfunctions Z and λ this difference is non-positive by usingJensenndashSteffensen inequality difference [29] As a result theRHS of inequality (7) is non-positive and we get gener-alized JensenndashSteffensen inequality (2) for nminusconvexfunction

Now we give similar results related to JensenndashBoasinequality [30] p 59] which is a generalization of Jen-senndashSteffensen inequality

Corollary 1 Let Ψ defined in =eorem 3 be nminusconvexfunction Also let Z be as defined in M1 witha y0 lty1 lt ltyk lt ltymminus1 ltym b and Z bemonotonic in each of the m intervals ((ykminus1 yk)) =en thefollowing results hold

(i) If λ as defined in M2 satisfies

λ(a)le λ x1( 1113857le λ y1( 1113857le λ x2( 1113857le λ y2( 1113857le le λ ymminus1( 1113857le λ xm( 1113857le λ(b) (14)

forallxk isin (ykminus1 yk) and λ(b)gt λ(a) then for even nge 3(6) is valid

(ii) Moreover if (6) is valid and the function H(middot) definedin (18) is convex then again inequality (2) holds andis called JensenndashBoas inequality for nminusconvexfunction

Proof We follow the similar argument as in the proof ofeorem 4 but under the conditions of this corollary weutilize JensenndashBoas inequality (see [29] or [24] p 59) insteadof JensenndashSteffensen inequality

Next we give results for JensenndashBrunk inequality

Corollary 2 Let Ψ defined in =eorem 3 be nminusconvex and Z

defined in M1 be an increasing function =en the followingresults hold

(i) If λ defined in M2 with λ(b)gt λ(a) and

1113946x

a(Z(x) minus Z(ζ))dλ(ζ)ge 0 (15)

and

1113946b

x(Z(x) minus Z(ζ))dλ(ζ)le 0 (16)

forallx isin [a b] holds then for even nge 3 (6) is valid(ii) Moreover if (6) is valid and the function H(middot) defined

in (18) is convex then again inequality (2) holds andis called JensenndashBrunk inequality for nminusconvexfunction

Proof We proceed with the similar idea as in the proof ofeorem 4 but under the conditions of this corollary weemploy JensenndashBrunk inequality (see [31] or [30] p 59])instead of JensenndashSteffensen inequality

Remark 1 e similar result in Corollary 2 is also validprovided that the function Z is decreasing Also assumingthat the function Z is monotonic one can replace theconditions in Corollary 2(i) by

0le 1113946x

a|Z(x) minus Z(ζ)|dλ(ζ)le 1113946

b

x|g(x) minus Z(ζ)|dλ(ζ)

(17)

Remark 2 It is interesting to see that by employing similarmethod as in eorem 4 we can also get the generalizationof classical Jensenrsquos inequality (2) for nminusconvex functions byassuming the functions Z and λ along with the respectiveconditions in eorem 1

Another important consequence of eorem 3 can begiven by setting the function Z as Z(ζ) ζ is form is thegeneralized version of LHS inequality of the Hermite-Hadamard inequality

Corollary 3 Let λ [a b]⟶ R be a function of boundedvariation such that λ(a)ne λ(b) with [a b] sub [α β] and1113957ζ 1113938

b

aζdλ(ζ) 1113938

b

adλ(ζ) isin [α β] Under the assumptions of

=eorem 3 if Ψ is nminusconvex such that

Rn(1113957Z s)le1113938

b

aRn(Z(ζ) s)dλ(ζ)

1113938b

adλ(ζ)

s isin [α β] (18)

then we have

4 Journal of Mathematics

Ψ(1113957ζ)le1113938

b

aΨ(ζ)dλ(ζ)

1113938b

adλ(ζ)

+ 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

times Ψ(ℓ+1)(α) (1113957ζ minus α)

(ℓ+2)minus

1113938b

a(ζ minus α)

(ℓ+2)dλ(ζ)

1113938b

adλ(ζ)

⎛⎝ ⎞⎠ minus Ψ(ℓ+1)(β) (1113957ζ minus β)

(ℓ+2)+

1113938b

a(ζ minus β)

(ℓ+2)dλ(ζ)

1113938b

adλ(ζ)

⎛⎝ ⎞⎠⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

(19)

If the inequality (18) holds in reverse direction then (19)also holds reversely

e special case of above corollary can be given in theform of following remark

Remark 3 It is interesting to see that substituting λ(ζ) ζgives 1113938

b

adλ(ζ) b minus a and 1113957ζ a + b2 Using these substi-

tutions in (2) and by following remark (20) we get the LHSinequality of renowned HermitendashHadamard inequality fornminusconvex functions

22 New Generalization of Converse of Jensenrsquos IntegralInequality In this section we give the results for the

converse of Jensenrsquos inequality to hold giving the conditionson the real Stieltjes measure dλ such that λ(a)ne λ(b)allowing that the measure can also be negative butemploying Montgomery identity

To start with we need the following assumption for theresults of this section

A3 Let mM isin [α β](mneM) be such thatmle Z(ζ)leM for all ζ isin [a b] where Z is defined in A1

For a given function Ψ [α β]⟶ R we consider thedifference

CJ Ψ Z mM λ1113872 1113873 1113938

b

aΨ(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

minusM minus 1113957Z

M minus mΨ(m) minus

1113957Z minus mM minus mΨ(M) (20)

where 1113957Z is defined in (3)Using Montgomery identity we obtain the following

representation of the converse of Jensenrsquos inequality

Theorem 5 Let Z λ be as defined in A1 A2 and letΨ [α β]⟶ R be such that for nge 1 Ψ(nminus1) is absolutelycontinuous If Ψ is nminusconvex such that

CJ Rn(x s) Z mM λ1113872 1113873le 0 s isin [α β] (21)

or

1113938b

aRn(Z(ζ) s)dλ(ζ)

1113938b

adλ(ζ)

leM minus 1113957Z

M minus mRn(m s)( 1113857 +

1113957Z minus mM minus m

Rn(M s)( 1113857 s isin [α β] (22)

then we get the following extension of the converse of Jensenrsquosdifference

1113938b

aΨ(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

leM minus 1113957Z

M minus mΨ(m) +

1113957Z minus mM minus mΨ(M) + 1113944

nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889

times Ψ(ℓ+1)(α)CJ (x minus α)

ℓ+2 Z mM λ1113872 1113873 minus Ψ(ℓ+1)

(β)CJ (x minus β)ℓ+2

Z mM λ1113872 11138731113872 1113873

(23)

where Rn(middot s) is defined in (5)

Journal of Mathematics 5

Proof AsΨ(nminus1) is absolutely continuous for (nge 1) we canuse the representation of Ψ using Montgomery identity (4)in the difference CJ(Ψ Z mM λ)

CJ Ψ Z mM λ1113872 1113873 CJ1

β minus α1113946β

αΨ(ζ)dζ Z mM λ1113888 1113889

+ 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889Ψ(ℓ+1)(α)CJ (x minus α)

ℓ+2 Z mM λ1113872 1113873

minus 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889Ψ(ℓ+1)(β)CJ (x minus β)

ℓ+2 Z mM λ1113872 1113873 +

1(n minus 1)

1113946β

αCJ Rn(x s) Z mM λ1113872 1113873Ψ(n)

(s)ds

(24)

After simplification and following the fact thatCJ(Ψ Z mM λ) is zero for Ψ to be constant or linear we getthe following generalized identity

CJ Ψ Z mM λ1113872 1113873j 1113944nminus2

ℓ0

ℓ(ℓ + 2)(β minus α)1113888 1113889

times Ψ(ℓ+1)(α)CJ (x minus α)

ℓ+2 Z mM λ1113872 1113873 minus Ψ(ℓ+1)

(β)CJ (x minus β)ℓ+2

Z mM λ1113872 11138731113872 1113873

+1

(n minus 1)1113946β

αCJ Rn(x s) Z mM λ1113872 1113873Ψ(n)

(s)ds(CGI1)

(25)

Now using characterizations of nminusconvex functions likein the proof of eorem 3 we get (23)

e next result gives converse of Jensenrsquos inequality forhigher-order convex functions

Theorem 6 Let Ψ defined in =eorem 5 be nminusconvex and Z

be as defined in A3 =en the following results hold

(i) If λ is non-negative measure on [a b] then for evennge 3 (22) is valid

(ii) Moreover if (22) is valid and the function H(middot)

defined in (12) is convex then we get the followinginequality for nminusconvex function to be valid

1113938b

aΨ(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

leM minus 1113957Z

M minus mΨ(m) minus

1113957Z minus mM minus mΨ(M) (26)

Proof e idea of the proof is similar to that of (6) but we useconverse of Jensenrsquos inequality (see [32] or [30] p 98)

23 Applications of Jensenrsquos Integral Inequality In this sec-tion we give applications of Jensenrsquos integral inequality

Another important consequence of eorem 3 is bysetting the function Z as Z(ζ) ζ gives generalized version ofL H S inequality of the HermitendashHadamard inequality

Corollary 4 Let λ [a b]⟶ R be a function of boundedvariation such that λ(a)ne λ(b) with [a b] sub [α β] and1113957ζ 1113938

b

aζdλ(ζ) 1113938

b

adλ(ζ) isin [α β] Under the assumptions of

=eorem 5 if Ψ is nminusconvex such that

1113938b

aRn(ζ s)dλ(ζ)

1113938b

adλ(ζ)

leb minus 1113957ζb minus a

Rn(a s)( 1113857 +1113957ζ minus a

b minus aRn(b s)( 1113857 s isin [α β]

(27)

then we have

6 Journal of Mathematics

1113938b

aΨ(ζ)dλ(ζ)

1113938b

adλ(ζ)

leb minus 1113957ζb minus aΨ(a) +

1113957ζ minus a

b minus aΨ(b) + 1113944

nminus2

ℓ0

ℓ(ℓ + 2)(β minus α)1113888 1113889times

Ψ(ℓ+1)(α)CJ (x minus α)

ℓ+2 id ab λ1113872 1113873 minus Ψ(ℓ+1)

(β)CJ (x minus β)ℓ+2

id ab λ1113872 11138731113872 1113873

(28)

If the inequality (27) holds in reverse direction then (28)also holds reversely

e special case of above corollary can be given in theform of following remark

Remark 4 It is interesting to see that substituting λ(ζ) ζand by following eorem 6 we get the RHS inequality ofrenowned HermitendashHadamard inequality for nminusconvexfunctions

3 Generalization of Jensenrsquos DiscreteInequality by Using Montgomery Identity

In this section we give generalizations for Jensenrsquos discreteinequality by using Montgomery identity e proofs aresimilar to those of continuous case as given in previoussection therefore we give results directly

31 Generalization of Jensenrsquos Discrete Inequality for RealWeights In discrete case we have that pJ gt 0 for allJ 1 2 m Here we give generalizations of results

allowing pJ to be negative real numbers Also with usualnotations for pJxJ(J 1 2 n) we notate

x x1 x2 xm( 1113857 and p p1 p2 pm( 1113857 (29)

to be mminustuples

Pv 1113944v

J1pJ Pv Pm minus Pvminus1 (v 1 2 m) (30)

and

x 1

Pm

1113944

m

J1pJxJ (31)

Using Montgomery identity (4) we obtain the followingrepresentations of Jensenrsquos discrete inequality

Theorem 7 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also let xJ isin [a b]sube[α β]pJ isin R(J 1 m) be such that Pm ne 0 and x isin [α β]

(i) =en the following generalized identity holds

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

Ψ(ℓ+1)(α) (x minus α)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus α1113872 1113873

ℓ+2⎛⎝ ⎞⎠ minus Ψ(ℓ+1)(β) (x minus β)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus β1113872 1113873

ℓ+2⎛⎝ ⎞⎠⎧⎨

⎩

⎫⎬

⎭

+1

(n minus 1)1113946β

αRn(x s) minus

1Pm

1113944

m

J1pJRn xJ s1113872 1113873⎡⎢⎢⎣ ⎤⎥⎥⎦Ψ(n)

(s)ds (DGI1)

(32)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

Rn(x s)le1

Pm

1113944

m

J1pJRn xJ s1113872 1113873 (33)

holds then we have the following generalized inequality

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873le 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

Ψ(ℓ+1)(α) (x minus α)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus α1113872 1113873

ℓ+2⎛⎝ ⎞⎠ minus Ψ(ℓ+1)(β) (x minus β)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus β1113872 1113873

ℓ+2⎛⎝ ⎞⎠⎧⎨

⎩

⎫⎬

⎭

(34)

Journal of Mathematics 7

If inequality (33) holds in reverse direction then (34) alsoholds reversely

Proof Similar to that of eorem 3In the later part of this section we will vary our con-

ditions on pJxJ(J 1 2 n) to obtain generalizeddiscrete variants of JensenndashSteffensen Jensenrsquos and Jen-senndashPetrovic type inequalities We start with the followinggeneralization of JensenndashSteffensen discrete inequality fornminusconvex functions

Theorem 8 Let Ψ be as defined in =eorem 7 Also let x bemonotonic nminustuple xJ isin [a b]sube[α β] and p be a realnminustuple such that

0lePv lePm (v 1 2 m minus 1) Pm gt 0 (35)

is satisfied

(i) If Ψ is nminusconvex then for even nge 3 (33) is valid(ii) Moreover if (33) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized JensenndashSteffensen discrete inequality

Ψ(x)le1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 (36)

Proof It is interesting to see that under the assumed con-ditions on tuples x and p we have that x isin [a b] Forx1 gex2 ge gexm

Pm x1 minus x( 1113857 1113944m

J2pJ x1 minus xJ1113872 1113873 1113944

m

v2xvminus1 minus xv( 1113857 Pm minus Pvminus1( 1113857ge 0

(37)

is shows that x1 ge x Also xge xn since we have

Pm x minus xm( 1113857 1113944

mminus1

J1pJ xJ minus xm1113872 1113873 1113944

mminus1

v1xv minus xvminus1( 1113857Pv ge 0

(38)

For further details see the proof of JensenndashSteffensendiscrete inequality ([24] p 57) e idea of the rest of theproof is similar to that of eorem 3 but here we employeorem 7 and JensenndashSteffensen discrete inequality

Corollary 5 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube [α β] with p being a positive nminustuple

(i) If Ψ is nminusconvex then for even nge 3 (34) is valid(ii) Moreover if (33) is valid and the function H(middot) de-

fined in (12) is convex then again we get (36) which iscalled Jensenrsquos inequality for nminusconvex functions

Proof For pJ gt 0 xJ isin [a b] (J 1 2 3 m) ensuresthat x isin [a b] So by applying classical Jensenrsquos discrete

inequality (1) and idea ofeorem 8 we will get the requiredresults

Remark 5 Under the assumptions of Corollary 5 if wechoose Pm 1 then Corollary 5 (ii) gives the followinginequality for nminusconvex functions

Ψ 1113944m

J1pJxJ

⎛⎝ ⎞⎠le 1113944m

J1pJΨ xJ1113872 1113873 (39)

Nowwe give following reverses of JensenndashSteffensen andJensen-type inequalities

Corollary 6 Let Ψ be as defined in =eorem 7 Also let x bemonotonic mminustuple xJ isin [a b]sube[α β] and p be a realmminustuple such that there exist m isin 1 2 m such that

0gePv for vltm and 0gePv for vgtm (40)

where Pm gt 0 and x isin [α β]

(i) If is nminusconvex then for even nge 3 then reverse ofinequality (33) holds

(ii) Moreover if (33) holds reversely and the function H(middot)

defined in (12) is convex then we get reverse ofgeneralized JensenndashSteffensen inequality (36) fornminusconvex functions

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ reverse of JensenndashSteffensen inequality to obtain results

In the next corollary we give explicit conditions on realtuple p such that we get reverse of classical Jensen inequality

Corollary 7 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube[α β] such that x isin [α β] Let p be a real nminustuplesuch that

0ltp1 0gep2 p3 pm 0ltPm (41)

is satisfied

(i) If Ψ is nminusconvex then for even nge 3 the reverse ofinequality (33) is valid

(ii) Also if reverse of (33) is valid and the function H(middot)

defined in (12) is convex then we get reverse of (36)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ reverse of Jensen in-equality to obtain results

In [33] (see also [30]) one can find the result which isequivalent to the JensenndashSteffensen and the reverse Jen-senndashSteffensen inequality together It is the so-calledJensenndashPetrovic inequality Here without the proof we givethe adequate corollary which uses that result e proof goesthe same way as in the previous corollaries

8 Journal of Mathematics

Corollary 8 Let Ψ be as defined in =eorem 7 and letxi isin [a b]sube[α β] be such that xm ge xmminus1 x2 lex1 Let pbe a real mminustuple with Pm 1 such that

0lePv for 1le vltm minus 1 and 0lePv for 2le vltm

(42)

is satisfied =en we get the equivalent results given in=eorem 8 (i) and (ii) respectively

Remark 6 Under the assumptions of Corollary 8 if thereexist m isin 1 2 n such that

0gePv for vltm and 0gePv for vgtm (43)

and x isin [α β] then we get the equivalent results for reverseJensenndashSteffensen inequality given in Corollary 6 (i) and(ii) respectively

Remark 7 It is interesting to see that the conditions onpJJ 1 2 m given in Corollary 8 and Remark 6 arecoming from JensenndashPetrovic inequality which becomeequivalent to conditions for pJJ 1 2 m for Jen-senndashSteffensen results given in eorem 8 and Corollary 6respectively when Pm 1

Now we give results for Jensen and its reverses fornminustuples x and p when n is an odd number

Corollary 9 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube[α β] for J 1 2 m be such that x p berealmminustuplesm 2m + 1 m isin N and 1113954x 111139362k+1

J1 pJ11139362k+1J1

pJxJ isin [α β] for all k 1 2 m If for everyk 1 2 m we have

(ilowast) p1 gt 0 p2k le 0 p2k + p2k+1 le 0 11139362kJ1pJ ge 0

11139362k+1J1 pJ gt 0

(iilowast) x2k lex2k+1 11139362k+1J1 pJ (xJ minus x2k+1)ge 0

then we have the following statements to be valid(i) If Ψ is nminusconvex then for even nge 3 the inequality

Rn(1113954x s)ge1

P2m+11113944

2m+1

J1pJRn xJ s1113872 1113873 (44)

(ii) Also if (44) is valid and the function H(middot) defined in(12) is convex then we get the following generalizedinequality

Ψ(1113954x)ge1

P2m+11113944

2m+1

J1pJΨ xJ1113872 1113873 (45)

Proof We employ the idea of the proofs of eorems 7 and8 for n odd along with inequality of Vasic and Janic[34]

Remark 8 We can also discuss the following importantcases by considering the explicit conditions given in [34]

We conclude this section by giving the following im-portant cases

(Case 1)Let the condition (ilowast) hold and the reverse inequalitiesin condition (iilowast) hold en again we can give in-equalities (44) and (45) respectively given in Corollary9(Case 2)If in case of conditions (ilowast) and (iilowast) the following arevalid(iiilowast) p1 gt 0 p2k+1 ge 0 p2k + p 2k+1 ge 0 1113936

2kJ1pJ ge 0

11139362k+1J1 pJ gt 0

(ivlowast) x2k lex2k+1 11139362kminus1J1 pJ(xJ minus x2k)le 0

then we can give reverses of inequalities (44) and (45)respectively given in Corollary 9(Case 3)Finally we can also give reverses of inequalities (44)and (45) respectively given in Corollary 9 providedthat the condition (iiilowast) holds and the reverse in-equalities in condition (ivlowast) hold

e result given in (Case 3) is type of generalization ofinequality by Szego [35]

32 Generalization of Converse JensenrsquosDiscrete Inequality forRealWeights In this section we give the results for converseof Jensenrsquos inequality in discrete case by using the Mont-gomery identity

Let xJ isin [a b]sube[α β] ane b pJ isin R(J 1 n) besuch that Pm ne 0 en we have the following difference ofconverse of Jensenrsquos inequality for Ψ [α β]⟶ R

CJdis(Ψ) 1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 minus

b minus x

b minus aΨ(a) minus

x minus a

b minus aΨ(b)

(46)

Similarly we assume the Giaccardi difference [36] givenas

Gcardi(Ψ) 1113944m

J1pJΨ xJ1113872 1113873 minus AΨ 1113944

m

J1pJxJ

⎛⎝ ⎞⎠ minus B 1113944m

J1pJ minus 1⎛⎝ ⎞⎠Ψ x0( 1113857 (47)

where

Journal of Mathematics 9

A 1113936

mJ1 pJ xJ minus x01113872 11138731113872 1113873

1113936mJ1 pJxJ minus x01113872 1113873

B 1113936

mJ1 pJxJ

1113936mJ1 pJxJ minus x01113872 1113873

and 1113944m

J1pJxJ ne x0 (48)

Theorem 9 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also letx0 xJ isin [a b]sube[α β] pJ isin R(J 1 m) be such that1113936

mJ1 pJxJ ne x0

(i) =en the following generalized identity holds

CJdis(Ψ) 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)CJdis xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)CJdis xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875

+1

(n minus 1)1113946β

αCJdis Rn xJ s1113872 11138731113872 1113873Ψ(n)

(s)ds (DCGI)

(49)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

CJdis Rn xJ s1113872 11138731113872 1113873le 0 (50)

holds then we have the following generalized inequality

CJdis(Ψ)le 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)CJdis xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)CJdis xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875 (51)

If inequality (50) holds in reverse direction then (51) alsoholds reversely

Theorem 10 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also let xJ isin [a b]sube[α β]pJ isin R(J 1 m) be such that Pm ne 0 and x isin [α β]

(i) =en the following generalized Giaccardi identityholds

Gcardi(Ψ) 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)Gcardi xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)Gcardi xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875

+1

(n minus 1)1113946β

αGcardi Rn xJ s1113872 11138731113872 1113873Ψ(n)

(s)ds (GIAGI)

(52)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

Gcardi Rn xJ s1113872 11138731113872 1113873le 0 (53)

holds then we have the following generalized Giaccardiinequality

Gcardi(Ψ)le 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)Gcardi xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)Gcardi xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875 (54)

If inequality (53) holds in reverse direction then (54) alsoholds reversely

In the later part of this section we will vary our con-ditions on pJxJ (J 1 2 m) to obtain generalized

10 Journal of Mathematics

converse discrete variants of Jensenrsquos inequality and Giac-cardi inequality for nminusconvex functions

Theorem 11 Let Ψ be as defined in =eorem 9 Also letxJ isin [a b]sube [α β] and p be a positive mminustuple

(i) If Ψ is nminusconvex then for even nge 3 (50) is valid(ii) Moreover if (50) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized converse of Jensenrsquos inequality

1Pm

1113944

m

J1pJΨ xJ1113872 1113873le

b minus x

b minus aΨ(a) +

x minus a

b minus aΨ(b) (55)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ converse of Jensenrsquosinequality (see [32] or [30] p 98) to obtain results

Finally in this section we give Giaccardi inequality forhigher-order convex functions

Theorem 12 Let Ψ be as defined in =eorem 9 Also letx0 xJ isin [a b]sube[α β] and p be a positive mminustuple such that

1113944

m

J1pJxJ nex0 and xv minus x0( 1113857 1113944

m

J1pJxJ minus xv

⎛⎝ ⎞⎠ge 0 (v 1 m) (56)

(i) If Ψ is nminusconvex then for even nge 3 (53) is valid(ii) Moreover if (53) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized Giaccardi inequality

1113944

m

J1pJΨ xJ1113872 1113873leAΨ 1113944

m

J1pJxJ

⎛⎝ ⎞⎠ + B 1113944m

J1pJ minus 1⎛⎝ ⎞⎠Ψ x0( 1113857

(57)

where A and B are defined in (47)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ Giaccardi inequality(see [36] or [37] p 11) to obtain results

33 Applications in Information =eory for Jensenrsquos DiscreteInequality Jensenrsquos inequality plays a key role in infor-mation theory to construct lower bounds for some notableinequalities but here we will use it to make connectionsbetween inequalities in information theory

Let Ψ R+⟶ R+ be a convex function and letp ≔ (p1 pm) and q ≔ (q1 qm) be positive proba-bility distributions then Ψ-divergence functional is defined(in [38]) as follows

IΨ(p q) 1113944m

J1qJΨ

pJ

qJ1113888 1113889 (58)

Horvath et al in [39] defined the generalized Csiszardivergence functional as follows

Definition 2 Let I be an interval in R and Ψ I⟶ R be afunction Also let p ≔ (p1 pm) isin Rm andq ≔ (q1 qm) isin (0infin)m such that

pJ

qJisin I J 1 m (59)

en let

1113957IΨ(p q) 1113944m

J1qJΨ

pJ

qJ1113888 1113889 (60)

In this section we write Jensenrsquos difference here that weuse in upcoming results

F p xJΨ1113872 1113873 Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 (61)

Theorem 13 Under the assumptions of =eorem 9 (ii) let(51) hold and Ψ be nminusconvex Also letp ≔ (p1 pm) inRm and q ≔ (q1 qm) isin (0infin)mthen we have the following results

Journal of Mathematics 11

1113957IΨ(p q)gePmΨ(1) minus Pm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F q

pJ

qJ (x minus α)

ℓ+21113888 11138891113890 1113891 minus Ψ(ℓ+1)

(β) F qpJ

qJ (x minus β)

ℓ+21113888 11138891113890 11138911113896 1113897

(62)

Proof From eorem 9 by following Jensenrsquos difference(61) we can rearrange (34) as

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873le 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F p xJ (x minus α)

ℓ+21113872 11138731113872 1113873 minus Ψ(ℓ+1)

(β) F p xJ (x minus β)ℓ+2

1113872 11138731113872 11138731113966 1113967

(63)

Now replace pJ with qJ and xJ with pJqJ and we get(62)

For positive n-tuple q (q1 qm) such that1113936

mJ1 qJ 1 the Shannon entropy is defined by

S(q) minus 1113944m

J1qJ ln qJ (64)

Corollary 10 Under the assumptions of =eorem 9 (ii) let(51) hold and Ψ be nminusconvex

(i) If q ≔ (q1 qm) isin (0infin)m then

1113944

m

J1qJ ln qJ lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (65)

(ii) We can get bounds for the Shannon entropy of q if wechoose q ≔ (q1 qn) to be a positive probabilitydistribution

S(q)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (66)

Proof (i) Substituting Ψ(x) ≔ minus lnx and usingp ≔ (1 1 1) in eorem 13 we get (65)

(ii) Since we have 1113936mJ1 qJ 1 by multiplying minus1 on

both sides of (65) and taking into account (64) weget (66)

e KullbackndashLeibler distance [40] between the positiveprobability distributions p (p1 pm) andq (q1 qm) is defined by

D(q p) 1113944

m

J1qJ ln

qJ

pJ

1113888 1113889 (67)

Corollary 11 Under the assumptions of Corollary 10

(i) If q ≔ (q1 qm) p ≔ (p1 pm) isin (0infin)mthen

1113944

m

J1qJ ln

qJ

pJ

1113888 1113889lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

pJ

qJ minusln(middot)1113888 1113889

⎧⎨

⎩

⎫⎬

⎭ (68)

(ii) If q ≔ (q1 qm) p ≔ (p1 pm) are positiveprobability distributions then we have

12 Journal of Mathematics

D(q p)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

βℓ+1F q

pJ

qJ minusln(middot)1113888 11138891113896 1113897 (69)

Proof

(i) UsingΨ(x) ≔ minus lnx (which isn-convex for evenn)in eorem 13 we get (68) after simplification

(ii) It is a special case of (i)

34 Results for Zipf and Hybrid ZipfndashMandelbrot EntropyOne of the basic laws in information science is Zipfrsquos law[4142] which is highly applied in linguistics Let cge 0 dgt 0and N isin 1 2 ZipfndashMandelbrot entropy can be given as

ZM(H c d) d

HNcd

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (70)

where

HNcd 1113944

N

σ1

1(σ + c)

d (71)

Consider

qJ Ψ(J N c d) 1

(J + c)dH

Ncd1113872 1113873

(72)

where Ψ(J m c d) is discrete probability distributionknown as ZipfndashMandelbrot law ZipfndashMandelbrot law hasmany application in linguistics and information sciencesSome of the recent study about ZipfndashMandelbrot law can beseen in the listed references (see [39 43]) Now we state ourresults involving entropy introduced by Mandelbrot law byestablishing the relationship with Shannon and relativeentropies

Theorem 14 Let q be ZipfndashMandelbrot law as defined in(72) with parameters cge 0 dgt 0 and N isin 1 2 and wehave

ZM(H c d) S(q)leN times 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873

⎧⎨

⎩

⎫⎬

⎭

(73)

Proof It is interesting to see that for qJ defined in (72)1113936

NJ1 qJ 1 erefore using above qJ in Shannon entropy

(64) we get Mandelbrot entropy (70)

S(q) minus 1113944

N

J1qJ ln qJ minus 1113944

N

J1

1(J + c)

dH

Ncd1113872 1113873

ln1

(J + c)dH

Ncd1113872 1113873

d

HNcd1113872 1113873

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (74)

Finally substituting this qJ 1((J + c)dHNcd) in

Corollary 10 (ii) we get the desired result Corollary 12 Let q and p be ZipfndashMandelbrot law withparameters c1 c2 isin [0infin) d1 d2 gt 0 and let

Journal of Mathematics 13

HNc1 d1

1113936Nσ1 1(σ + c1)

d1 and HNc2 d2

1113936Nσ1 1(σ + c2)

d2 Now using qJ 1(J + c1)

d1HNc1 d1

andpJ 1(J + c2)

d2HNc2 d2

in Corollary 11 (ii) the followingholds

D(q p) 1113944N

J1

1J + c1( 1113857

d1HNc1d1

lnJ + c2( 1113857

d2HNc2 d2

J + c1( 1113857d1H

mc1 d1

⎛⎝ ⎞⎠

minusZ H c1 d1( 1113857 +d2

HNc1 d1

1113944

N

J1

ln J + c2( 1113857

J + c1( 1113857d1

+ ln Hmc2d2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qJ + c1( 1113857

d1HNc1 d1

J + c2( 1113857d2H

Nc2 d2

minusln(middot)⎛⎝ ⎞⎠ minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qJ + c1( 1113857

d1HNc1d1

J + c2( 1113857d2H

Nc2d2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(75)

=e Next Result for Hybrid ZipfndashMandelbrot Entropy Fur-ther generalization of ZipfndashMandelbrot entropy is Hybrid

ZipfndashMandelbrot entropy Let N isin 1 2 cge 0ωgt 0then Hybrid ZipfndashMandelbrot entropy can be given as

1113954ZM Hlowast c dω( 1113857

1Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873 (76)

where

Hlowastcdω 1113944

N

J1

ωJ

(J + c)d (77)

Consider

qJ Ψ(J N c dω) ωJ

(J + c)dHlowastcdω

(78)

which is called Hybrid ZipfndashMandelbrot law =ere is aunified approach maximization of Shannon entropy [44]

that naturally follows the path of generalization from Zipfrsquos toHybrid Zipfrsquos law Extending this idea Jaksetic et al in [45]presented a transition from ZipfndashMandelbrot to HybridZipfndashMandelbrot law by employing maximum entropytechnique with one additional constraint It is interesting thatexamination of its densities provides some new insights ofLerchrsquos transcendent

Theorem 15 Let q be Hybrid ZipfndashMandelbrot law asdefined in (78) with parameters cge 0 dωgt 0 andN isin 1 2 and we have

1113954ZM Hlowast c dω( 1113857 S(q)leN 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠ minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠

⎧⎨

⎩

⎫⎬

⎭

(79)

14 Journal of Mathematics

Proof It is interesting to see that for qJ defined in (78)1113936

NJ1 qJ 1 erefore using above qJ in Shannon en-

tropy (64) we get Hybrid ZipfndashMandelbrot entropy (76)

S(q) minus 1113944N

J1qJ ln qJ minus 1113944

N

J1

ωJ

(J + c)dHlowastcdω

lnωJ

(J + c)dHlowastcdω

minus1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

lnωJ

(J + c)d

1113888 1113889 + ln1

Hlowastcdω

1113888 11138891113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

ln(J + c)

d

ωJ1113888 1113889 + ln H

lowastcdω1113872 11138731113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873

(80)

Finally substituting this qJ ωJ(J + c)dHlowastcdω inCorollary 10 (ii) we get the desired result

Corollary 13 Let q and p be Hybrid ZipfndashMandelbrotlaw with parameters c1 c2 isin [0infin)ω1ω2 d1 d2 gt 0 Nowusing qJ ωJ

1 (J + c1)d1Hlowastc1 d1 ω1

and pJ ωJ2

(J + c2)d2Hlowastc2 d2 ω2

in Corollary 11 (ii) the following holds

D(q p) 1113944N

J1

ωJ1

J + c1( 1113857d1Hlowastc1 d1 ω1

lnωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

ωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

⎛⎝ ⎞⎠

minus1113954ZM Hlowast c1 d1ω1( 1113857 +

1Hlowastc1 d1 ω1

1113944

N

J1

ωJ1

J + c1( 1113857d1ln

J + c2( 1113857d2

ωJ2

⎛⎝ ⎞⎠ + ln Hlowastc2 d2 ω2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎫⎪⎬

⎪⎭

(81)

Remark 9 Similarly we can give results for Shannonentropy KullbackndashLeibler distance ZipfndashMandelbrotentropy and Hybrid ZipfndashMandelbrot entropy by usinggeneralized Giaccardi inequality defined in (54) on thesame steps

4 Concluding Remarks

In this paper we gave generalization of Jensenrsquos inequality aswell as converse of Jensenrsquos inequality by using Montgomeryidentity We also formulate results for other inequalities like

JensenndashSteffensen inequality JensenndashBoas inequality andJensenndashBrunk inequality We can obtain JensenndashSteffenseninequality JensenndashBoas inequality and JensenndashBrunk in-equality by changing the assumption of Jensenrsquos inequalityAt the end we gave applications in information theory forour obtained results especially we gave results for HybridZipfndashMandelbrot entropy for our obtained results [46]

Data Availability

No data were used to support this study

Journal of Mathematics 15

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is research is supported by HEC Pakistan under NRPUProject 7906 and RUDN University Strategic AcademicLeadership Program

References

[1] X L Zhu and G H Yang ldquoJensen inequality approach tostability analysis of discrete-time systems with time-varyingdelayrdquo in Proceedings of the American Control Conferencepp 1644ndash1649 Seattle WA USA June 2008

[2] S I Butt M Klaricic Bakula ETH Pecaric and J PecaricldquoJensen-gruss inequality and its applications for the zipf-mandelbrot lawrdquo Mathematical Methods in the Applied Sci-ences vol 44 no 2 pp 1664ndash1673 2021

[3] M A Khan D Pecaric and J Pecaric ldquoNew refinement of theJensen inequality associated to certain functions with appli-cationsrdquo Journal of Inequalities and Applications vol 762020

[4] S Khan M Adil Khan S I Butt and Y-M Chu ldquoA newbound for the Jensen gap pertaining twice differentiablefunctions with applicationsrdquo Advances in Difference Equa-tions vol 1 2020

[5] H Budak andM Z Sarikaya ldquoOn generalized Ostrowski-typeinequalities for functions whose first derivatives absolutevalues are convexrdquo Turkish Journal of Mathematics vol 40pp 1193ndash1210 2016

[6] P Cerone and S S Dragomir ldquoOn some inequalities arisingfrom Montgomeryrsquos identityrdquo Journal of ComputationalAnalysis and Applications vol 5 no 4 pp 341ndash367 2003

[7] S Bernstein ldquoSur les fonctions absolument monotonesrdquo ActaMathematica vol 52 pp 1ndash66 1929

[8] V Culjak and J Pecaric ldquoInterpolation polynomials andinequalities for convex functions of higher orderrdquo ActaMathematica vol 5 pp 369ndash386 2012

[9] A M Fink ldquoBounds of the deviation of a function from itsaveragesrdquo Czechoslovak Mathematical Journal vol 42no 117 pp 289ndash310 1992

[10] J Pecaric and V Culjak ldquoInterpolation polinomials and in-equalities for convex functions of higher orderrdquo Mathe-matical Inequalities amp Applications vol 5 no 3 pp 369ndash3862002

[11] J Pecaric V Culjak and A M Fink ldquoOn some inequalitiesfor convex function of higher orderrdquo Advanced NonlinearStudies vol 6 no 2 pp 131ndash140 1999

[12] J Pecaric V Culjak andM Rogina ldquoOn some inequalities forconvex function of higher order IIrdquo Advanced NonlinearStudies vol 45 pp 281ndash294 2001

[13] J Pecaric I Peric and M Rodic-Lipanovic ldquoUniformtreatment of Jensen type inequalitiesrdquo Mathematical Reportsvol 16 no 66 pp 183ndash205 2014

[14] J Pecaric and J Peric ldquoImprovement of the Giaccardi and thepetrovic inequality and related stolarsky type meansrdquo Annalsof the University of Craiova - Mathematics and ComputerScience vol 39 no 1 pp 65ndash75 2012

[15] J Pecaric M Praljak and A Witkowski ldquoLinear operatorinequality for nminusconvex functions at a pointrdquo MathematicalInequalities amp Applications Appl vol 18 pp 1201ndash1217 2015

[16] Z Pavic ldquoRefinements of Jensenrsquos inequality for infiniteconvex combinationsrdquo Turkish Journal of Inequalities vol 2no 2 pp 44ndash53 2018

[17] R Bibi A Nosheen and J Pecaric ldquoExtended Jensenrsquos typeinequalities for diamond integrals via Taylors formulardquoTurkish Journal of Inequalities vol 3 no 1 pp 7ndash18 2019

[18] T Niaz K A Khan and J Pecaric ldquoOn refinement of Jensenrsquosinequality for 3minusconvex function at a pointrdquo Turkish Journalof Inequalities vol 4 no 1 pp 70ndash80 2020

[19] A Ekinci A O Akdemir and M E Ozdemir ldquoIntegralinequalities for different kinds of convexity via classical in-equalitiesrdquo Turkish Journal of Science vol 5 no 3 pp 305ndash313 2020

[20] Y M Chu S Talib E Set M U Awan and M A Noorldquo(p q)minus Analysis of Montgomery identity and estimates of(p q)minusbounds with applicationsrdquo Journal of Inequalities andApplications vol 2021 9 pages 2021

[21] M Z Sarikaya H Yaldrz and E Set ldquoOn fractional in-equalities via Montgomery identitiesrdquo International Journalof Open Problems in Complex Analysis vol 6 no 2 pp 36ndash432014

[22] A Bnouhachem ldquoA descent SQP alternating directionmethod for minimizing the sum of three convex functionsrdquoJournal of Nonlinear Functional Analysis vol 4 pp 469ndash4822020

[23] A O Akdemir S I Butt M Nadeem and M A RagusaldquoNew general variants of Chebyshev type inequalities viageneralized fractional integral operatorsrdquo Mathematicsvol 92 2021

[24] S I Butt M Umar S Rashid A O Akdemir and Y MingChu ldquoNew Hermite Jensen Mercer type inequalities viakminusfractional integralsrdquo Advances in Difference Equationsvol 635 2020

[25] S I Butt M Nadeem S Qaisar A O Akdemir andT Abdeljawad ldquoHermite Jensen Mercer type inequalities forconformable integrals and related resultsrdquo Advances in Dif-ference Equations vol 501 2020

[26] A Bnouhachem and X Qin ldquoAn inertial proximal Peaceman-Rachford splittingmethod with SQP regularization for convexprogrammingrdquo Journal of Nonlinear Functional Analysisvol 50 2020

[27] S Hamann ldquoMinimality conditions for convex compositefunctions and efficiency conditions in vector optimizationrdquoApplied Set-Valued Analysis and Optimization vol 1pp 221ndash229 2019

[28] A R Khan J Pecaric and M Praljak ldquoPopoviciu type in-equalities for n-convex functions via extension of Mont-gomery identitynminusconvex functions via extension ofMontgomery identityrdquo Analele Universitatii ldquoOvidiusrdquo Con-stanta-Seria Matematica vol 24 no 3 pp 161ndash188 2016

[29] R P Boas ldquoe Jensen-Steffensen inequalityrdquo ElektrotehnFak Ser Mat Fiz vol 302ndash319 pp 1ndash8 1970

[30] J Pecaric F Proschan and Y L Tong Convex FunctionsPartial Orderings and Statistical Applications Academic PressNew York NY USA 1992

[31] H D Brunk ldquoOn an inequality for convex functionsrdquo Pro-ceedings of the American Mathematical Society vol 7 no 5p 817 1956

[32] P R Beesack and J E Pecaric ldquoOn Jessenrsquos inequality forconvex functionsrdquo Journal of Mathematical Analysis andApplications vol 110 no 2 pp 536ndash552 1985

[33] R Barlow A Marshall and F Proschan ldquoSome inequalitiesfor starshaped and convex functionsrdquo Pacific Journal ofMathematics vol 29 no 1 pp 19ndash42 1969

16 Journal of Mathematics

[34] J E Pecaric ldquoOn an inequality of PMV Janicrdquo Publications delrsquoInstitut Mathematique vol 28 no 42 pp 145ndash149 1980

[35] G Szego ldquoUber eine Verallgemeinerung des DirichletschenIntegralsrdquo Mathematische Zeitschrift vol 52 no 1pp 676ndash685 1950

[36] F Giaccardi ldquoSu alcune disuguaglianierdquo Journal of Researchof the National Bureau of Standards vol 1 no 4 pp 139ndash1531955

[37] D S Mitrinovic J E Pecaric and A M Fink Classical andnew inequalities in analysis Mathematics and its Applications(East European Series) Vol 61 Kluwer Academic PublishersGroup Dordrecht Netherlands 1993

[38] I Csiszar ldquoInformation-type measures of diference ofprobability distributions and indirect observationsrdquo StudiaScience Math Hungar vol 2 pp 299ndash318 1967

[39] L Horvath D Pecaric and J Pecaric ldquoEstimations of f - andrenyi divergences by using a cyclic refinement of the Jensenrsquosinequalityrdquo Bulletin of the Malaysian Mathematical SciencesSociety vol 42 2017

[40] S Kullback Information =eory and Statistics Wiley NewYork NY USA 1959

[41] S T Piantadosi ldquoZipfrsquos word frequency law in natural lan-guage a critical review and future directionsrdquo PsychonomicBulletin and Review vol 21 no 5 pp 1112ndash1130 2014

[42] Z K Silagadze ldquoCitations and the zipfndashmandelbrot lawrdquoComplex Systems no 11 pp 487ndash499 1997

[43] J Jaksetic D Pecaric and J Pecaric ldquoSome properties of zipf-mandelbrot law and hurwitz ζ-functionrdquo Mathematical In-equalities amp Applications vol 21 no 2 pp 575ndash584 2018

[44] M Visser ldquoZipfrsquos law power laws and maximum entropyrdquoMathematical Inequalities amp Applications vol 15 Article ID043021 2013

[45] J Jaksetic D Pecaric and J Pecaric ldquoHybrid zipf-mandelbrotlawrdquo Mathematical Inequalities amp Applications vol 13 no 1pp 275ndash286 2019

[46] D V Widder ldquoCompletely convex functions and Lidstoneseriesrdquo Transactions of the American Mathematical Societyvol 51 p 387 1942

Journal of Mathematics 17

Proof AsΨ(nminus1) is absolutely continuous for (nge 1) we canuse the representation of Ψ using Montgomery identity (4)and can calculate

Ψ(1113957Z) 1

β minus α1113946β

αΨ(ζ)d(ζ) + 1113944

nminus2

ℓ0

Ψ(ℓ+1)(α)

ℓ(ℓ + 2)

(1113957Z minus α)ℓ+2

β minus αminus 1113944

nminus2

ℓ0

Ψ(ℓ+1)(β)

ℓ(ℓ + 2)

(1113957Z minus β)ℓ+2

β minus α+

1(n minus 1)

1113946β

αRn(1113957Z s)Ψ(n)

(s)ds (8)

e integration of the composition of functionsΨ ∘ Z forthe real measure λ on [a b] gives

1113938b

aΨ(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

1

β minus α1113946β

αΨ(ζ)d(ζ) + 1113944

nminus2

ℓ0

Ψ(ℓ+1)(α)

ℓ(ℓ + 2)

1113938b

a(Z(ζ) minus α)

(ℓ+2)dλ(ζ)

(β minus α) 1113938b

adλ(ζ)

minus 1113944nminus2

ℓ0

Ψ(ℓ+1)(β)

ℓ(ℓ + 2)

1113938b

a(Z(ζ) minus β)

(ℓ+2)dλ(ζ)

(β minus α) 1113938b

adλ(ζ)

+1

(n minus 1)1113946β

αRn(Z(ζ) s)Ψ(n)

(s)ds

(9)

Now computing the difference Ψ(1113957Z) minus 1113938b

aΨ(Z(ζ))

dλ(ζ)1113938b

adλ(ζ) we get the following generalized identity

involving real Stieltjes measure

Ψ(1113957Z) minus1113938

b

aΨ(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

times Ψ(ℓ+1)(α) (1113957Z minus α)

(ℓ+2)minus

1113938b

a(Z(ζ) minus α)

(ℓ+2)dλ(ζ)

1113938b

adλ(ζ)

⎛⎝ ⎞⎠ minus Ψ(ℓ+1)(β) (1113957Z minus β)

(ℓ+2)+

1113938b

a(Z(ζ) minus β)

(ℓ+2)dλ(ζ)

1113938b

adλ(ζ)

⎛⎝ ⎞⎠⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

+1

(n minus 1)1113946β

αRn(1113957Z s) minus

1113938b

aRn(Z(ζ) s)dλ(ζ)

1113938b

adλ(ζ)

⎛⎝ ⎞⎠Ψ(n)(s)ds(GI1)

(10)

Finally by our assumption Ψ(nminus1) is absolutely con-tinuous on [α β] as a result Ψ(n) exists almost everywhereMoreover Ψ is supposed to be nminusconvex so we haveΨ(n)(x)ge 0 almost everywhere on [α β] erefore bytaking into account the last term in generalized identity(GI1) and integral analogue of Jensenrsquos inequality that isgiven in (6) we get (7)

In the later part of this section we will vary our con-ditions on functions g and Stieltjes measure dλ to obtaingeneralized variants of JensenndashSteffensen JensenndashBoasJensenndashBrunk and Jensen-type inequalities We start with

the following generalization of JensenndashSteffensen inequalityfor nminusconvex functions

Theorem 4 Let Ψ defined in =eorem 3 be nminusconvex and Z

defined in M1 be monotonic =en the following results hold

(i) If λ defined in M2 satisfies

λ(a)le λ(x)le λ(b) forallx isin [a b] λ(b)gt λ(a) (11)

then for even nge 3 (6) is valid(ii) Moreover if (6) is valid and the function

H(x) ≔ 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)(x minus α)

ℓ+2minus Ψ(ℓ+1)

(β)(x minus β)ℓ+2

1113872 1113873 (12)

Journal of Mathematics 3

is convex then we get inequality (2) which is calledgeneralized JensenndashSteffensen inequality fornminusconvex function

Proof (i) By applying second derivative test we can showthat the function

Rn(x s) is convex for even ngt 3 Now using theassumed conditions one can employ JensenndashSteffensen inequality given by Boas (see [29] or [30]p 59) for convex function Rn(x s) to obtain (6)

(ii) Since we can rewrite the RHS of (7) in thedifference

H(1113957Z) minus1113938

b

aH(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

(13)

for convex function H and by our assumed conditions onfunctions Z and λ this difference is non-positive by usingJensenndashSteffensen inequality difference [29] As a result theRHS of inequality (7) is non-positive and we get gener-alized JensenndashSteffensen inequality (2) for nminusconvexfunction

Now we give similar results related to JensenndashBoasinequality [30] p 59] which is a generalization of Jen-senndashSteffensen inequality

Corollary 1 Let Ψ defined in =eorem 3 be nminusconvexfunction Also let Z be as defined in M1 witha y0 lty1 lt ltyk lt ltymminus1 ltym b and Z bemonotonic in each of the m intervals ((ykminus1 yk)) =en thefollowing results hold

(i) If λ as defined in M2 satisfies

λ(a)le λ x1( 1113857le λ y1( 1113857le λ x2( 1113857le λ y2( 1113857le le λ ymminus1( 1113857le λ xm( 1113857le λ(b) (14)

forallxk isin (ykminus1 yk) and λ(b)gt λ(a) then for even nge 3(6) is valid

(ii) Moreover if (6) is valid and the function H(middot) definedin (18) is convex then again inequality (2) holds andis called JensenndashBoas inequality for nminusconvexfunction

Proof We follow the similar argument as in the proof ofeorem 4 but under the conditions of this corollary weutilize JensenndashBoas inequality (see [29] or [24] p 59) insteadof JensenndashSteffensen inequality

Next we give results for JensenndashBrunk inequality

Corollary 2 Let Ψ defined in =eorem 3 be nminusconvex and Z

defined in M1 be an increasing function =en the followingresults hold

(i) If λ defined in M2 with λ(b)gt λ(a) and

1113946x

a(Z(x) minus Z(ζ))dλ(ζ)ge 0 (15)

and

1113946b

x(Z(x) minus Z(ζ))dλ(ζ)le 0 (16)

forallx isin [a b] holds then for even nge 3 (6) is valid(ii) Moreover if (6) is valid and the function H(middot) defined

in (18) is convex then again inequality (2) holds andis called JensenndashBrunk inequality for nminusconvexfunction

Proof We proceed with the similar idea as in the proof ofeorem 4 but under the conditions of this corollary weemploy JensenndashBrunk inequality (see [31] or [30] p 59])instead of JensenndashSteffensen inequality

Remark 1 e similar result in Corollary 2 is also validprovided that the function Z is decreasing Also assumingthat the function Z is monotonic one can replace theconditions in Corollary 2(i) by

0le 1113946x

a|Z(x) minus Z(ζ)|dλ(ζ)le 1113946

b

x|g(x) minus Z(ζ)|dλ(ζ)

(17)

Remark 2 It is interesting to see that by employing similarmethod as in eorem 4 we can also get the generalizationof classical Jensenrsquos inequality (2) for nminusconvex functions byassuming the functions Z and λ along with the respectiveconditions in eorem 1

Another important consequence of eorem 3 can begiven by setting the function Z as Z(ζ) ζ is form is thegeneralized version of LHS inequality of the Hermite-Hadamard inequality

Corollary 3 Let λ [a b]⟶ R be a function of boundedvariation such that λ(a)ne λ(b) with [a b] sub [α β] and1113957ζ 1113938

b

aζdλ(ζ) 1113938

b

adλ(ζ) isin [α β] Under the assumptions of

=eorem 3 if Ψ is nminusconvex such that

Rn(1113957Z s)le1113938

b

aRn(Z(ζ) s)dλ(ζ)

1113938b

adλ(ζ)

s isin [α β] (18)

then we have

4 Journal of Mathematics

Ψ(1113957ζ)le1113938

b

aΨ(ζ)dλ(ζ)

1113938b

adλ(ζ)

+ 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

times Ψ(ℓ+1)(α) (1113957ζ minus α)

(ℓ+2)minus

1113938b

a(ζ minus α)

(ℓ+2)dλ(ζ)

1113938b

adλ(ζ)

⎛⎝ ⎞⎠ minus Ψ(ℓ+1)(β) (1113957ζ minus β)

(ℓ+2)+

1113938b

a(ζ minus β)

(ℓ+2)dλ(ζ)

1113938b

adλ(ζ)

⎛⎝ ⎞⎠⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

(19)

If the inequality (18) holds in reverse direction then (19)also holds reversely

e special case of above corollary can be given in theform of following remark

Remark 3 It is interesting to see that substituting λ(ζ) ζgives 1113938

b

adλ(ζ) b minus a and 1113957ζ a + b2 Using these substi-

tutions in (2) and by following remark (20) we get the LHSinequality of renowned HermitendashHadamard inequality fornminusconvex functions

22 New Generalization of Converse of Jensenrsquos IntegralInequality In this section we give the results for the

converse of Jensenrsquos inequality to hold giving the conditionson the real Stieltjes measure dλ such that λ(a)ne λ(b)allowing that the measure can also be negative butemploying Montgomery identity

To start with we need the following assumption for theresults of this section

A3 Let mM isin [α β](mneM) be such thatmle Z(ζ)leM for all ζ isin [a b] where Z is defined in A1

For a given function Ψ [α β]⟶ R we consider thedifference

CJ Ψ Z mM λ1113872 1113873 1113938

b

aΨ(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

minusM minus 1113957Z

M minus mΨ(m) minus

1113957Z minus mM minus mΨ(M) (20)

where 1113957Z is defined in (3)Using Montgomery identity we obtain the following

representation of the converse of Jensenrsquos inequality

Theorem 5 Let Z λ be as defined in A1 A2 and letΨ [α β]⟶ R be such that for nge 1 Ψ(nminus1) is absolutelycontinuous If Ψ is nminusconvex such that

CJ Rn(x s) Z mM λ1113872 1113873le 0 s isin [α β] (21)

or

1113938b

aRn(Z(ζ) s)dλ(ζ)

1113938b

adλ(ζ)

leM minus 1113957Z

M minus mRn(m s)( 1113857 +

1113957Z minus mM minus m

Rn(M s)( 1113857 s isin [α β] (22)

then we get the following extension of the converse of Jensenrsquosdifference

1113938b

aΨ(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

leM minus 1113957Z

M minus mΨ(m) +

1113957Z minus mM minus mΨ(M) + 1113944

nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889

times Ψ(ℓ+1)(α)CJ (x minus α)

ℓ+2 Z mM λ1113872 1113873 minus Ψ(ℓ+1)

(β)CJ (x minus β)ℓ+2

Z mM λ1113872 11138731113872 1113873

(23)

where Rn(middot s) is defined in (5)

Journal of Mathematics 5

Proof AsΨ(nminus1) is absolutely continuous for (nge 1) we canuse the representation of Ψ using Montgomery identity (4)in the difference CJ(Ψ Z mM λ)

CJ Ψ Z mM λ1113872 1113873 CJ1

β minus α1113946β

αΨ(ζ)dζ Z mM λ1113888 1113889

+ 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889Ψ(ℓ+1)(α)CJ (x minus α)

ℓ+2 Z mM λ1113872 1113873

minus 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889Ψ(ℓ+1)(β)CJ (x minus β)

ℓ+2 Z mM λ1113872 1113873 +

1(n minus 1)

1113946β

αCJ Rn(x s) Z mM λ1113872 1113873Ψ(n)

(s)ds

(24)

After simplification and following the fact thatCJ(Ψ Z mM λ) is zero for Ψ to be constant or linear we getthe following generalized identity

CJ Ψ Z mM λ1113872 1113873j 1113944nminus2

ℓ0

ℓ(ℓ + 2)(β minus α)1113888 1113889

times Ψ(ℓ+1)(α)CJ (x minus α)

ℓ+2 Z mM λ1113872 1113873 minus Ψ(ℓ+1)

(β)CJ (x minus β)ℓ+2

Z mM λ1113872 11138731113872 1113873

+1

(n minus 1)1113946β

αCJ Rn(x s) Z mM λ1113872 1113873Ψ(n)

(s)ds(CGI1)

(25)

Now using characterizations of nminusconvex functions likein the proof of eorem 3 we get (23)

e next result gives converse of Jensenrsquos inequality forhigher-order convex functions

Theorem 6 Let Ψ defined in =eorem 5 be nminusconvex and Z

be as defined in A3 =en the following results hold

(i) If λ is non-negative measure on [a b] then for evennge 3 (22) is valid

(ii) Moreover if (22) is valid and the function H(middot)

defined in (12) is convex then we get the followinginequality for nminusconvex function to be valid

1113938b

aΨ(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

leM minus 1113957Z

M minus mΨ(m) minus

1113957Z minus mM minus mΨ(M) (26)

Proof e idea of the proof is similar to that of (6) but we useconverse of Jensenrsquos inequality (see [32] or [30] p 98)

23 Applications of Jensenrsquos Integral Inequality In this sec-tion we give applications of Jensenrsquos integral inequality

Another important consequence of eorem 3 is bysetting the function Z as Z(ζ) ζ gives generalized version ofL H S inequality of the HermitendashHadamard inequality

Corollary 4 Let λ [a b]⟶ R be a function of boundedvariation such that λ(a)ne λ(b) with [a b] sub [α β] and1113957ζ 1113938

b

aζdλ(ζ) 1113938

b

adλ(ζ) isin [α β] Under the assumptions of

=eorem 5 if Ψ is nminusconvex such that

1113938b

aRn(ζ s)dλ(ζ)

1113938b

adλ(ζ)

leb minus 1113957ζb minus a

Rn(a s)( 1113857 +1113957ζ minus a

b minus aRn(b s)( 1113857 s isin [α β]

(27)

then we have

6 Journal of Mathematics

1113938b

aΨ(ζ)dλ(ζ)

1113938b

adλ(ζ)

leb minus 1113957ζb minus aΨ(a) +

1113957ζ minus a

b minus aΨ(b) + 1113944

nminus2

ℓ0

ℓ(ℓ + 2)(β minus α)1113888 1113889times

Ψ(ℓ+1)(α)CJ (x minus α)

ℓ+2 id ab λ1113872 1113873 minus Ψ(ℓ+1)

(β)CJ (x minus β)ℓ+2

id ab λ1113872 11138731113872 1113873

(28)

If the inequality (27) holds in reverse direction then (28)also holds reversely

e special case of above corollary can be given in theform of following remark

Remark 4 It is interesting to see that substituting λ(ζ) ζand by following eorem 6 we get the RHS inequality ofrenowned HermitendashHadamard inequality for nminusconvexfunctions

3 Generalization of Jensenrsquos DiscreteInequality by Using Montgomery Identity

In this section we give generalizations for Jensenrsquos discreteinequality by using Montgomery identity e proofs aresimilar to those of continuous case as given in previoussection therefore we give results directly

31 Generalization of Jensenrsquos Discrete Inequality for RealWeights In discrete case we have that pJ gt 0 for allJ 1 2 m Here we give generalizations of results

allowing pJ to be negative real numbers Also with usualnotations for pJxJ(J 1 2 n) we notate

x x1 x2 xm( 1113857 and p p1 p2 pm( 1113857 (29)

to be mminustuples

Pv 1113944v

J1pJ Pv Pm minus Pvminus1 (v 1 2 m) (30)

and

x 1

Pm

1113944

m

J1pJxJ (31)

Using Montgomery identity (4) we obtain the followingrepresentations of Jensenrsquos discrete inequality

Theorem 7 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also let xJ isin [a b]sube[α β]pJ isin R(J 1 m) be such that Pm ne 0 and x isin [α β]

(i) =en the following generalized identity holds

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

Ψ(ℓ+1)(α) (x minus α)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus α1113872 1113873

ℓ+2⎛⎝ ⎞⎠ minus Ψ(ℓ+1)(β) (x minus β)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus β1113872 1113873

ℓ+2⎛⎝ ⎞⎠⎧⎨

⎩

⎫⎬

⎭

+1

(n minus 1)1113946β

αRn(x s) minus

1Pm

1113944

m

J1pJRn xJ s1113872 1113873⎡⎢⎢⎣ ⎤⎥⎥⎦Ψ(n)

(s)ds (DGI1)

(32)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

Rn(x s)le1

Pm

1113944

m

J1pJRn xJ s1113872 1113873 (33)

holds then we have the following generalized inequality

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873le 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

Ψ(ℓ+1)(α) (x minus α)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus α1113872 1113873

ℓ+2⎛⎝ ⎞⎠ minus Ψ(ℓ+1)(β) (x minus β)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus β1113872 1113873

ℓ+2⎛⎝ ⎞⎠⎧⎨

⎩

⎫⎬

⎭

(34)

Journal of Mathematics 7

If inequality (33) holds in reverse direction then (34) alsoholds reversely

Proof Similar to that of eorem 3In the later part of this section we will vary our con-

ditions on pJxJ(J 1 2 n) to obtain generalizeddiscrete variants of JensenndashSteffensen Jensenrsquos and Jen-senndashPetrovic type inequalities We start with the followinggeneralization of JensenndashSteffensen discrete inequality fornminusconvex functions

Theorem 8 Let Ψ be as defined in =eorem 7 Also let x bemonotonic nminustuple xJ isin [a b]sube[α β] and p be a realnminustuple such that

0lePv lePm (v 1 2 m minus 1) Pm gt 0 (35)

is satisfied

(i) If Ψ is nminusconvex then for even nge 3 (33) is valid(ii) Moreover if (33) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized JensenndashSteffensen discrete inequality

Ψ(x)le1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 (36)

Proof It is interesting to see that under the assumed con-ditions on tuples x and p we have that x isin [a b] Forx1 gex2 ge gexm

Pm x1 minus x( 1113857 1113944m

J2pJ x1 minus xJ1113872 1113873 1113944

m

v2xvminus1 minus xv( 1113857 Pm minus Pvminus1( 1113857ge 0

(37)

is shows that x1 ge x Also xge xn since we have

Pm x minus xm( 1113857 1113944

mminus1

J1pJ xJ minus xm1113872 1113873 1113944

mminus1

v1xv minus xvminus1( 1113857Pv ge 0

(38)

For further details see the proof of JensenndashSteffensendiscrete inequality ([24] p 57) e idea of the rest of theproof is similar to that of eorem 3 but here we employeorem 7 and JensenndashSteffensen discrete inequality

Corollary 5 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube [α β] with p being a positive nminustuple

(i) If Ψ is nminusconvex then for even nge 3 (34) is valid(ii) Moreover if (33) is valid and the function H(middot) de-

fined in (12) is convex then again we get (36) which iscalled Jensenrsquos inequality for nminusconvex functions

Proof For pJ gt 0 xJ isin [a b] (J 1 2 3 m) ensuresthat x isin [a b] So by applying classical Jensenrsquos discrete

inequality (1) and idea ofeorem 8 we will get the requiredresults

Remark 5 Under the assumptions of Corollary 5 if wechoose Pm 1 then Corollary 5 (ii) gives the followinginequality for nminusconvex functions

Ψ 1113944m

J1pJxJ

⎛⎝ ⎞⎠le 1113944m

J1pJΨ xJ1113872 1113873 (39)

Nowwe give following reverses of JensenndashSteffensen andJensen-type inequalities

Corollary 6 Let Ψ be as defined in =eorem 7 Also let x bemonotonic mminustuple xJ isin [a b]sube[α β] and p be a realmminustuple such that there exist m isin 1 2 m such that

0gePv for vltm and 0gePv for vgtm (40)

where Pm gt 0 and x isin [α β]

(i) If is nminusconvex then for even nge 3 then reverse ofinequality (33) holds

(ii) Moreover if (33) holds reversely and the function H(middot)

defined in (12) is convex then we get reverse ofgeneralized JensenndashSteffensen inequality (36) fornminusconvex functions

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ reverse of JensenndashSteffensen inequality to obtain results

In the next corollary we give explicit conditions on realtuple p such that we get reverse of classical Jensen inequality

Corollary 7 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube[α β] such that x isin [α β] Let p be a real nminustuplesuch that

0ltp1 0gep2 p3 pm 0ltPm (41)

is satisfied

(i) If Ψ is nminusconvex then for even nge 3 the reverse ofinequality (33) is valid

(ii) Also if reverse of (33) is valid and the function H(middot)

defined in (12) is convex then we get reverse of (36)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ reverse of Jensen in-equality to obtain results

In [33] (see also [30]) one can find the result which isequivalent to the JensenndashSteffensen and the reverse Jen-senndashSteffensen inequality together It is the so-calledJensenndashPetrovic inequality Here without the proof we givethe adequate corollary which uses that result e proof goesthe same way as in the previous corollaries

8 Journal of Mathematics

Corollary 8 Let Ψ be as defined in =eorem 7 and letxi isin [a b]sube[α β] be such that xm ge xmminus1 x2 lex1 Let pbe a real mminustuple with Pm 1 such that

0lePv for 1le vltm minus 1 and 0lePv for 2le vltm

(42)

is satisfied =en we get the equivalent results given in=eorem 8 (i) and (ii) respectively

Remark 6 Under the assumptions of Corollary 8 if thereexist m isin 1 2 n such that

0gePv for vltm and 0gePv for vgtm (43)

and x isin [α β] then we get the equivalent results for reverseJensenndashSteffensen inequality given in Corollary 6 (i) and(ii) respectively

Remark 7 It is interesting to see that the conditions onpJJ 1 2 m given in Corollary 8 and Remark 6 arecoming from JensenndashPetrovic inequality which becomeequivalent to conditions for pJJ 1 2 m for Jen-senndashSteffensen results given in eorem 8 and Corollary 6respectively when Pm 1

Now we give results for Jensen and its reverses fornminustuples x and p when n is an odd number

Corollary 9 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube[α β] for J 1 2 m be such that x p berealmminustuplesm 2m + 1 m isin N and 1113954x 111139362k+1

J1 pJ11139362k+1J1

pJxJ isin [α β] for all k 1 2 m If for everyk 1 2 m we have

(ilowast) p1 gt 0 p2k le 0 p2k + p2k+1 le 0 11139362kJ1pJ ge 0

11139362k+1J1 pJ gt 0

(iilowast) x2k lex2k+1 11139362k+1J1 pJ (xJ minus x2k+1)ge 0

then we have the following statements to be valid(i) If Ψ is nminusconvex then for even nge 3 the inequality

Rn(1113954x s)ge1

P2m+11113944

2m+1

J1pJRn xJ s1113872 1113873 (44)

(ii) Also if (44) is valid and the function H(middot) defined in(12) is convex then we get the following generalizedinequality

Ψ(1113954x)ge1

P2m+11113944

2m+1

J1pJΨ xJ1113872 1113873 (45)

Proof We employ the idea of the proofs of eorems 7 and8 for n odd along with inequality of Vasic and Janic[34]

Remark 8 We can also discuss the following importantcases by considering the explicit conditions given in [34]

We conclude this section by giving the following im-portant cases

(Case 1)Let the condition (ilowast) hold and the reverse inequalitiesin condition (iilowast) hold en again we can give in-equalities (44) and (45) respectively given in Corollary9(Case 2)If in case of conditions (ilowast) and (iilowast) the following arevalid(iiilowast) p1 gt 0 p2k+1 ge 0 p2k + p 2k+1 ge 0 1113936

2kJ1pJ ge 0

11139362k+1J1 pJ gt 0

(ivlowast) x2k lex2k+1 11139362kminus1J1 pJ(xJ minus x2k)le 0

then we can give reverses of inequalities (44) and (45)respectively given in Corollary 9(Case 3)Finally we can also give reverses of inequalities (44)and (45) respectively given in Corollary 9 providedthat the condition (iiilowast) holds and the reverse in-equalities in condition (ivlowast) hold

e result given in (Case 3) is type of generalization ofinequality by Szego [35]

32 Generalization of Converse JensenrsquosDiscrete Inequality forRealWeights In this section we give the results for converseof Jensenrsquos inequality in discrete case by using the Mont-gomery identity

Let xJ isin [a b]sube[α β] ane b pJ isin R(J 1 n) besuch that Pm ne 0 en we have the following difference ofconverse of Jensenrsquos inequality for Ψ [α β]⟶ R

CJdis(Ψ) 1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 minus

b minus x

b minus aΨ(a) minus

x minus a

b minus aΨ(b)

(46)

Similarly we assume the Giaccardi difference [36] givenas

Gcardi(Ψ) 1113944m

J1pJΨ xJ1113872 1113873 minus AΨ 1113944

m

J1pJxJ

⎛⎝ ⎞⎠ minus B 1113944m

J1pJ minus 1⎛⎝ ⎞⎠Ψ x0( 1113857 (47)

where

Journal of Mathematics 9

A 1113936

mJ1 pJ xJ minus x01113872 11138731113872 1113873

1113936mJ1 pJxJ minus x01113872 1113873

B 1113936

mJ1 pJxJ

1113936mJ1 pJxJ minus x01113872 1113873

and 1113944m

J1pJxJ ne x0 (48)

Theorem 9 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also letx0 xJ isin [a b]sube[α β] pJ isin R(J 1 m) be such that1113936

mJ1 pJxJ ne x0

(i) =en the following generalized identity holds

CJdis(Ψ) 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)CJdis xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)CJdis xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875

+1

(n minus 1)1113946β

αCJdis Rn xJ s1113872 11138731113872 1113873Ψ(n)

(s)ds (DCGI)

(49)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

CJdis Rn xJ s1113872 11138731113872 1113873le 0 (50)

holds then we have the following generalized inequality

CJdis(Ψ)le 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)CJdis xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)CJdis xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875 (51)

If inequality (50) holds in reverse direction then (51) alsoholds reversely

Theorem 10 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also let xJ isin [a b]sube[α β]pJ isin R(J 1 m) be such that Pm ne 0 and x isin [α β]

(i) =en the following generalized Giaccardi identityholds

Gcardi(Ψ) 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)Gcardi xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)Gcardi xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875

+1

(n minus 1)1113946β

αGcardi Rn xJ s1113872 11138731113872 1113873Ψ(n)

(s)ds (GIAGI)

(52)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

Gcardi Rn xJ s1113872 11138731113872 1113873le 0 (53)

holds then we have the following generalized Giaccardiinequality

Gcardi(Ψ)le 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)Gcardi xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)Gcardi xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875 (54)

If inequality (53) holds in reverse direction then (54) alsoholds reversely

In the later part of this section we will vary our con-ditions on pJxJ (J 1 2 m) to obtain generalized

10 Journal of Mathematics

converse discrete variants of Jensenrsquos inequality and Giac-cardi inequality for nminusconvex functions

Theorem 11 Let Ψ be as defined in =eorem 9 Also letxJ isin [a b]sube [α β] and p be a positive mminustuple

(i) If Ψ is nminusconvex then for even nge 3 (50) is valid(ii) Moreover if (50) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized converse of Jensenrsquos inequality

1Pm

1113944

m

J1pJΨ xJ1113872 1113873le

b minus x

b minus aΨ(a) +

x minus a

b minus aΨ(b) (55)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ converse of Jensenrsquosinequality (see [32] or [30] p 98) to obtain results

Finally in this section we give Giaccardi inequality forhigher-order convex functions

Theorem 12 Let Ψ be as defined in =eorem 9 Also letx0 xJ isin [a b]sube[α β] and p be a positive mminustuple such that

1113944

m

J1pJxJ nex0 and xv minus x0( 1113857 1113944

m

J1pJxJ minus xv

⎛⎝ ⎞⎠ge 0 (v 1 m) (56)

(i) If Ψ is nminusconvex then for even nge 3 (53) is valid(ii) Moreover if (53) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized Giaccardi inequality

1113944

m

J1pJΨ xJ1113872 1113873leAΨ 1113944

m

J1pJxJ

⎛⎝ ⎞⎠ + B 1113944m

J1pJ minus 1⎛⎝ ⎞⎠Ψ x0( 1113857

(57)

where A and B are defined in (47)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ Giaccardi inequality(see [36] or [37] p 11) to obtain results

33 Applications in Information =eory for Jensenrsquos DiscreteInequality Jensenrsquos inequality plays a key role in infor-mation theory to construct lower bounds for some notableinequalities but here we will use it to make connectionsbetween inequalities in information theory

Let Ψ R+⟶ R+ be a convex function and letp ≔ (p1 pm) and q ≔ (q1 qm) be positive proba-bility distributions then Ψ-divergence functional is defined(in [38]) as follows

IΨ(p q) 1113944m

J1qJΨ

pJ

qJ1113888 1113889 (58)

Horvath et al in [39] defined the generalized Csiszardivergence functional as follows

Definition 2 Let I be an interval in R and Ψ I⟶ R be afunction Also let p ≔ (p1 pm) isin Rm andq ≔ (q1 qm) isin (0infin)m such that

pJ

qJisin I J 1 m (59)

en let

1113957IΨ(p q) 1113944m

J1qJΨ

pJ

qJ1113888 1113889 (60)

In this section we write Jensenrsquos difference here that weuse in upcoming results

F p xJΨ1113872 1113873 Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 (61)

Theorem 13 Under the assumptions of =eorem 9 (ii) let(51) hold and Ψ be nminusconvex Also letp ≔ (p1 pm) inRm and q ≔ (q1 qm) isin (0infin)mthen we have the following results

Journal of Mathematics 11

1113957IΨ(p q)gePmΨ(1) minus Pm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F q

pJ

qJ (x minus α)

ℓ+21113888 11138891113890 1113891 minus Ψ(ℓ+1)

(β) F qpJ

qJ (x minus β)

ℓ+21113888 11138891113890 11138911113896 1113897

(62)

Proof From eorem 9 by following Jensenrsquos difference(61) we can rearrange (34) as

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873le 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F p xJ (x minus α)

ℓ+21113872 11138731113872 1113873 minus Ψ(ℓ+1)

(β) F p xJ (x minus β)ℓ+2

1113872 11138731113872 11138731113966 1113967

(63)

Now replace pJ with qJ and xJ with pJqJ and we get(62)

For positive n-tuple q (q1 qm) such that1113936

mJ1 qJ 1 the Shannon entropy is defined by

S(q) minus 1113944m

J1qJ ln qJ (64)

Corollary 10 Under the assumptions of =eorem 9 (ii) let(51) hold and Ψ be nminusconvex

(i) If q ≔ (q1 qm) isin (0infin)m then

1113944

m

J1qJ ln qJ lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (65)

(ii) We can get bounds for the Shannon entropy of q if wechoose q ≔ (q1 qn) to be a positive probabilitydistribution

S(q)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (66)

Proof (i) Substituting Ψ(x) ≔ minus lnx and usingp ≔ (1 1 1) in eorem 13 we get (65)

(ii) Since we have 1113936mJ1 qJ 1 by multiplying minus1 on

both sides of (65) and taking into account (64) weget (66)

e KullbackndashLeibler distance [40] between the positiveprobability distributions p (p1 pm) andq (q1 qm) is defined by

D(q p) 1113944

m

J1qJ ln

qJ

pJ

1113888 1113889 (67)

Corollary 11 Under the assumptions of Corollary 10

(i) If q ≔ (q1 qm) p ≔ (p1 pm) isin (0infin)mthen

1113944

m

J1qJ ln

qJ

pJ

1113888 1113889lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

pJ

qJ minusln(middot)1113888 1113889

⎧⎨

⎩

⎫⎬

⎭ (68)

(ii) If q ≔ (q1 qm) p ≔ (p1 pm) are positiveprobability distributions then we have

12 Journal of Mathematics

D(q p)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

βℓ+1F q

pJ

qJ minusln(middot)1113888 11138891113896 1113897 (69)

Proof

(i) UsingΨ(x) ≔ minus lnx (which isn-convex for evenn)in eorem 13 we get (68) after simplification

(ii) It is a special case of (i)

34 Results for Zipf and Hybrid ZipfndashMandelbrot EntropyOne of the basic laws in information science is Zipfrsquos law[4142] which is highly applied in linguistics Let cge 0 dgt 0and N isin 1 2 ZipfndashMandelbrot entropy can be given as

ZM(H c d) d

HNcd

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (70)

where

HNcd 1113944

N

σ1

1(σ + c)

d (71)

Consider

qJ Ψ(J N c d) 1

(J + c)dH

Ncd1113872 1113873

(72)

where Ψ(J m c d) is discrete probability distributionknown as ZipfndashMandelbrot law ZipfndashMandelbrot law hasmany application in linguistics and information sciencesSome of the recent study about ZipfndashMandelbrot law can beseen in the listed references (see [39 43]) Now we state ourresults involving entropy introduced by Mandelbrot law byestablishing the relationship with Shannon and relativeentropies

Theorem 14 Let q be ZipfndashMandelbrot law as defined in(72) with parameters cge 0 dgt 0 and N isin 1 2 and wehave

ZM(H c d) S(q)leN times 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873

⎧⎨

⎩

⎫⎬

⎭

(73)

Proof It is interesting to see that for qJ defined in (72)1113936

NJ1 qJ 1 erefore using above qJ in Shannon entropy

(64) we get Mandelbrot entropy (70)

S(q) minus 1113944

N

J1qJ ln qJ minus 1113944

N

J1

1(J + c)

dH

Ncd1113872 1113873

ln1

(J + c)dH

Ncd1113872 1113873

d

HNcd1113872 1113873

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (74)

Finally substituting this qJ 1((J + c)dHNcd) in

Corollary 10 (ii) we get the desired result Corollary 12 Let q and p be ZipfndashMandelbrot law withparameters c1 c2 isin [0infin) d1 d2 gt 0 and let

Journal of Mathematics 13

HNc1 d1

1113936Nσ1 1(σ + c1)

d1 and HNc2 d2

1113936Nσ1 1(σ + c2)

d2 Now using qJ 1(J + c1)

d1HNc1 d1

andpJ 1(J + c2)

d2HNc2 d2

in Corollary 11 (ii) the followingholds

D(q p) 1113944N

J1

1J + c1( 1113857

d1HNc1d1

lnJ + c2( 1113857

d2HNc2 d2

J + c1( 1113857d1H

mc1 d1

⎛⎝ ⎞⎠

minusZ H c1 d1( 1113857 +d2

HNc1 d1

1113944

N

J1

ln J + c2( 1113857

J + c1( 1113857d1

+ ln Hmc2d2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qJ + c1( 1113857

d1HNc1 d1

J + c2( 1113857d2H

Nc2 d2

minusln(middot)⎛⎝ ⎞⎠ minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qJ + c1( 1113857

d1HNc1d1

J + c2( 1113857d2H

Nc2d2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(75)

=e Next Result for Hybrid ZipfndashMandelbrot Entropy Fur-ther generalization of ZipfndashMandelbrot entropy is Hybrid

ZipfndashMandelbrot entropy Let N isin 1 2 cge 0ωgt 0then Hybrid ZipfndashMandelbrot entropy can be given as

1113954ZM Hlowast c dω( 1113857

1Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873 (76)

where

Hlowastcdω 1113944

N

J1

ωJ

(J + c)d (77)

Consider

qJ Ψ(J N c dω) ωJ

(J + c)dHlowastcdω

(78)

which is called Hybrid ZipfndashMandelbrot law =ere is aunified approach maximization of Shannon entropy [44]

that naturally follows the path of generalization from Zipfrsquos toHybrid Zipfrsquos law Extending this idea Jaksetic et al in [45]presented a transition from ZipfndashMandelbrot to HybridZipfndashMandelbrot law by employing maximum entropytechnique with one additional constraint It is interesting thatexamination of its densities provides some new insights ofLerchrsquos transcendent

Theorem 15 Let q be Hybrid ZipfndashMandelbrot law asdefined in (78) with parameters cge 0 dωgt 0 andN isin 1 2 and we have

1113954ZM Hlowast c dω( 1113857 S(q)leN 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠ minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠

⎧⎨

⎩

⎫⎬

⎭

(79)

14 Journal of Mathematics

Proof It is interesting to see that for qJ defined in (78)1113936

NJ1 qJ 1 erefore using above qJ in Shannon en-

tropy (64) we get Hybrid ZipfndashMandelbrot entropy (76)

S(q) minus 1113944N

J1qJ ln qJ minus 1113944

N

J1

ωJ

(J + c)dHlowastcdω

lnωJ

(J + c)dHlowastcdω

minus1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

lnωJ

(J + c)d

1113888 1113889 + ln1

Hlowastcdω

1113888 11138891113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

ln(J + c)

d

ωJ1113888 1113889 + ln H

lowastcdω1113872 11138731113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873

(80)

Finally substituting this qJ ωJ(J + c)dHlowastcdω inCorollary 10 (ii) we get the desired result

Corollary 13 Let q and p be Hybrid ZipfndashMandelbrotlaw with parameters c1 c2 isin [0infin)ω1ω2 d1 d2 gt 0 Nowusing qJ ωJ

1 (J + c1)d1Hlowastc1 d1 ω1

and pJ ωJ2

(J + c2)d2Hlowastc2 d2 ω2

in Corollary 11 (ii) the following holds

D(q p) 1113944N

J1

ωJ1

J + c1( 1113857d1Hlowastc1 d1 ω1

lnωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

ωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

⎛⎝ ⎞⎠

minus1113954ZM Hlowast c1 d1ω1( 1113857 +

1Hlowastc1 d1 ω1

1113944

N

J1

ωJ1

J + c1( 1113857d1ln

J + c2( 1113857d2

ωJ2

⎛⎝ ⎞⎠ + ln Hlowastc2 d2 ω2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎫⎪⎬

⎪⎭

(81)

Remark 9 Similarly we can give results for Shannonentropy KullbackndashLeibler distance ZipfndashMandelbrotentropy and Hybrid ZipfndashMandelbrot entropy by usinggeneralized Giaccardi inequality defined in (54) on thesame steps

4 Concluding Remarks

In this paper we gave generalization of Jensenrsquos inequality aswell as converse of Jensenrsquos inequality by using Montgomeryidentity We also formulate results for other inequalities like

JensenndashSteffensen inequality JensenndashBoas inequality andJensenndashBrunk inequality We can obtain JensenndashSteffenseninequality JensenndashBoas inequality and JensenndashBrunk in-equality by changing the assumption of Jensenrsquos inequalityAt the end we gave applications in information theory forour obtained results especially we gave results for HybridZipfndashMandelbrot entropy for our obtained results [46]

Data Availability

No data were used to support this study

Journal of Mathematics 15

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is research is supported by HEC Pakistan under NRPUProject 7906 and RUDN University Strategic AcademicLeadership Program

References

[1] X L Zhu and G H Yang ldquoJensen inequality approach tostability analysis of discrete-time systems with time-varyingdelayrdquo in Proceedings of the American Control Conferencepp 1644ndash1649 Seattle WA USA June 2008

[2] S I Butt M Klaricic Bakula ETH Pecaric and J PecaricldquoJensen-gruss inequality and its applications for the zipf-mandelbrot lawrdquo Mathematical Methods in the Applied Sci-ences vol 44 no 2 pp 1664ndash1673 2021

[3] M A Khan D Pecaric and J Pecaric ldquoNew refinement of theJensen inequality associated to certain functions with appli-cationsrdquo Journal of Inequalities and Applications vol 762020

[4] S Khan M Adil Khan S I Butt and Y-M Chu ldquoA newbound for the Jensen gap pertaining twice differentiablefunctions with applicationsrdquo Advances in Difference Equa-tions vol 1 2020

[5] H Budak andM Z Sarikaya ldquoOn generalized Ostrowski-typeinequalities for functions whose first derivatives absolutevalues are convexrdquo Turkish Journal of Mathematics vol 40pp 1193ndash1210 2016

[6] P Cerone and S S Dragomir ldquoOn some inequalities arisingfrom Montgomeryrsquos identityrdquo Journal of ComputationalAnalysis and Applications vol 5 no 4 pp 341ndash367 2003

[7] S Bernstein ldquoSur les fonctions absolument monotonesrdquo ActaMathematica vol 52 pp 1ndash66 1929

[8] V Culjak and J Pecaric ldquoInterpolation polynomials andinequalities for convex functions of higher orderrdquo ActaMathematica vol 5 pp 369ndash386 2012

[9] A M Fink ldquoBounds of the deviation of a function from itsaveragesrdquo Czechoslovak Mathematical Journal vol 42no 117 pp 289ndash310 1992

[10] J Pecaric and V Culjak ldquoInterpolation polinomials and in-equalities for convex functions of higher orderrdquo Mathe-matical Inequalities amp Applications vol 5 no 3 pp 369ndash3862002

[11] J Pecaric V Culjak and A M Fink ldquoOn some inequalitiesfor convex function of higher orderrdquo Advanced NonlinearStudies vol 6 no 2 pp 131ndash140 1999

[12] J Pecaric V Culjak andM Rogina ldquoOn some inequalities forconvex function of higher order IIrdquo Advanced NonlinearStudies vol 45 pp 281ndash294 2001

[13] J Pecaric I Peric and M Rodic-Lipanovic ldquoUniformtreatment of Jensen type inequalitiesrdquo Mathematical Reportsvol 16 no 66 pp 183ndash205 2014

[14] J Pecaric and J Peric ldquoImprovement of the Giaccardi and thepetrovic inequality and related stolarsky type meansrdquo Annalsof the University of Craiova - Mathematics and ComputerScience vol 39 no 1 pp 65ndash75 2012

[15] J Pecaric M Praljak and A Witkowski ldquoLinear operatorinequality for nminusconvex functions at a pointrdquo MathematicalInequalities amp Applications Appl vol 18 pp 1201ndash1217 2015

[16] Z Pavic ldquoRefinements of Jensenrsquos inequality for infiniteconvex combinationsrdquo Turkish Journal of Inequalities vol 2no 2 pp 44ndash53 2018

[17] R Bibi A Nosheen and J Pecaric ldquoExtended Jensenrsquos typeinequalities for diamond integrals via Taylors formulardquoTurkish Journal of Inequalities vol 3 no 1 pp 7ndash18 2019

[18] T Niaz K A Khan and J Pecaric ldquoOn refinement of Jensenrsquosinequality for 3minusconvex function at a pointrdquo Turkish Journalof Inequalities vol 4 no 1 pp 70ndash80 2020

[19] A Ekinci A O Akdemir and M E Ozdemir ldquoIntegralinequalities for different kinds of convexity via classical in-equalitiesrdquo Turkish Journal of Science vol 5 no 3 pp 305ndash313 2020

[20] Y M Chu S Talib E Set M U Awan and M A Noorldquo(p q)minus Analysis of Montgomery identity and estimates of(p q)minusbounds with applicationsrdquo Journal of Inequalities andApplications vol 2021 9 pages 2021

[21] M Z Sarikaya H Yaldrz and E Set ldquoOn fractional in-equalities via Montgomery identitiesrdquo International Journalof Open Problems in Complex Analysis vol 6 no 2 pp 36ndash432014

[22] A Bnouhachem ldquoA descent SQP alternating directionmethod for minimizing the sum of three convex functionsrdquoJournal of Nonlinear Functional Analysis vol 4 pp 469ndash4822020

[23] A O Akdemir S I Butt M Nadeem and M A RagusaldquoNew general variants of Chebyshev type inequalities viageneralized fractional integral operatorsrdquo Mathematicsvol 92 2021

[24] S I Butt M Umar S Rashid A O Akdemir and Y MingChu ldquoNew Hermite Jensen Mercer type inequalities viakminusfractional integralsrdquo Advances in Difference Equationsvol 635 2020

[25] S I Butt M Nadeem S Qaisar A O Akdemir andT Abdeljawad ldquoHermite Jensen Mercer type inequalities forconformable integrals and related resultsrdquo Advances in Dif-ference Equations vol 501 2020

[26] A Bnouhachem and X Qin ldquoAn inertial proximal Peaceman-Rachford splittingmethod with SQP regularization for convexprogrammingrdquo Journal of Nonlinear Functional Analysisvol 50 2020

[27] S Hamann ldquoMinimality conditions for convex compositefunctions and efficiency conditions in vector optimizationrdquoApplied Set-Valued Analysis and Optimization vol 1pp 221ndash229 2019

[28] A R Khan J Pecaric and M Praljak ldquoPopoviciu type in-equalities for n-convex functions via extension of Mont-gomery identitynminusconvex functions via extension ofMontgomery identityrdquo Analele Universitatii ldquoOvidiusrdquo Con-stanta-Seria Matematica vol 24 no 3 pp 161ndash188 2016

[29] R P Boas ldquoe Jensen-Steffensen inequalityrdquo ElektrotehnFak Ser Mat Fiz vol 302ndash319 pp 1ndash8 1970

[30] J Pecaric F Proschan and Y L Tong Convex FunctionsPartial Orderings and Statistical Applications Academic PressNew York NY USA 1992

[31] H D Brunk ldquoOn an inequality for convex functionsrdquo Pro-ceedings of the American Mathematical Society vol 7 no 5p 817 1956

[32] P R Beesack and J E Pecaric ldquoOn Jessenrsquos inequality forconvex functionsrdquo Journal of Mathematical Analysis andApplications vol 110 no 2 pp 536ndash552 1985

[33] R Barlow A Marshall and F Proschan ldquoSome inequalitiesfor starshaped and convex functionsrdquo Pacific Journal ofMathematics vol 29 no 1 pp 19ndash42 1969

16 Journal of Mathematics

[34] J E Pecaric ldquoOn an inequality of PMV Janicrdquo Publications delrsquoInstitut Mathematique vol 28 no 42 pp 145ndash149 1980

[35] G Szego ldquoUber eine Verallgemeinerung des DirichletschenIntegralsrdquo Mathematische Zeitschrift vol 52 no 1pp 676ndash685 1950

[36] F Giaccardi ldquoSu alcune disuguaglianierdquo Journal of Researchof the National Bureau of Standards vol 1 no 4 pp 139ndash1531955

[37] D S Mitrinovic J E Pecaric and A M Fink Classical andnew inequalities in analysis Mathematics and its Applications(East European Series) Vol 61 Kluwer Academic PublishersGroup Dordrecht Netherlands 1993

[38] I Csiszar ldquoInformation-type measures of diference ofprobability distributions and indirect observationsrdquo StudiaScience Math Hungar vol 2 pp 299ndash318 1967

[39] L Horvath D Pecaric and J Pecaric ldquoEstimations of f - andrenyi divergences by using a cyclic refinement of the Jensenrsquosinequalityrdquo Bulletin of the Malaysian Mathematical SciencesSociety vol 42 2017

[40] S Kullback Information =eory and Statistics Wiley NewYork NY USA 1959

[41] S T Piantadosi ldquoZipfrsquos word frequency law in natural lan-guage a critical review and future directionsrdquo PsychonomicBulletin and Review vol 21 no 5 pp 1112ndash1130 2014

[42] Z K Silagadze ldquoCitations and the zipfndashmandelbrot lawrdquoComplex Systems no 11 pp 487ndash499 1997

[43] J Jaksetic D Pecaric and J Pecaric ldquoSome properties of zipf-mandelbrot law and hurwitz ζ-functionrdquo Mathematical In-equalities amp Applications vol 21 no 2 pp 575ndash584 2018

[44] M Visser ldquoZipfrsquos law power laws and maximum entropyrdquoMathematical Inequalities amp Applications vol 15 Article ID043021 2013

[45] J Jaksetic D Pecaric and J Pecaric ldquoHybrid zipf-mandelbrotlawrdquo Mathematical Inequalities amp Applications vol 13 no 1pp 275ndash286 2019

[46] D V Widder ldquoCompletely convex functions and Lidstoneseriesrdquo Transactions of the American Mathematical Societyvol 51 p 387 1942

Journal of Mathematics 17

is convex then we get inequality (2) which is calledgeneralized JensenndashSteffensen inequality fornminusconvex function

Proof (i) By applying second derivative test we can showthat the function

Rn(x s) is convex for even ngt 3 Now using theassumed conditions one can employ JensenndashSteffensen inequality given by Boas (see [29] or [30]p 59) for convex function Rn(x s) to obtain (6)

(ii) Since we can rewrite the RHS of (7) in thedifference

H(1113957Z) minus1113938

b

aH(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

(13)

for convex function H and by our assumed conditions onfunctions Z and λ this difference is non-positive by usingJensenndashSteffensen inequality difference [29] As a result theRHS of inequality (7) is non-positive and we get gener-alized JensenndashSteffensen inequality (2) for nminusconvexfunction

Now we give similar results related to JensenndashBoasinequality [30] p 59] which is a generalization of Jen-senndashSteffensen inequality

Corollary 1 Let Ψ defined in =eorem 3 be nminusconvexfunction Also let Z be as defined in M1 witha y0 lty1 lt ltyk lt ltymminus1 ltym b and Z bemonotonic in each of the m intervals ((ykminus1 yk)) =en thefollowing results hold

(i) If λ as defined in M2 satisfies

λ(a)le λ x1( 1113857le λ y1( 1113857le λ x2( 1113857le λ y2( 1113857le le λ ymminus1( 1113857le λ xm( 1113857le λ(b) (14)

forallxk isin (ykminus1 yk) and λ(b)gt λ(a) then for even nge 3(6) is valid

(ii) Moreover if (6) is valid and the function H(middot) definedin (18) is convex then again inequality (2) holds andis called JensenndashBoas inequality for nminusconvexfunction

Proof We follow the similar argument as in the proof ofeorem 4 but under the conditions of this corollary weutilize JensenndashBoas inequality (see [29] or [24] p 59) insteadof JensenndashSteffensen inequality

Next we give results for JensenndashBrunk inequality

Corollary 2 Let Ψ defined in =eorem 3 be nminusconvex and Z

defined in M1 be an increasing function =en the followingresults hold

(i) If λ defined in M2 with λ(b)gt λ(a) and

1113946x

a(Z(x) minus Z(ζ))dλ(ζ)ge 0 (15)

and

1113946b

x(Z(x) minus Z(ζ))dλ(ζ)le 0 (16)

forallx isin [a b] holds then for even nge 3 (6) is valid(ii) Moreover if (6) is valid and the function H(middot) defined

in (18) is convex then again inequality (2) holds andis called JensenndashBrunk inequality for nminusconvexfunction

Proof We proceed with the similar idea as in the proof ofeorem 4 but under the conditions of this corollary weemploy JensenndashBrunk inequality (see [31] or [30] p 59])instead of JensenndashSteffensen inequality

Remark 1 e similar result in Corollary 2 is also validprovided that the function Z is decreasing Also assumingthat the function Z is monotonic one can replace theconditions in Corollary 2(i) by

0le 1113946x

a|Z(x) minus Z(ζ)|dλ(ζ)le 1113946

b

x|g(x) minus Z(ζ)|dλ(ζ)

(17)

Remark 2 It is interesting to see that by employing similarmethod as in eorem 4 we can also get the generalizationof classical Jensenrsquos inequality (2) for nminusconvex functions byassuming the functions Z and λ along with the respectiveconditions in eorem 1

Another important consequence of eorem 3 can begiven by setting the function Z as Z(ζ) ζ is form is thegeneralized version of LHS inequality of the Hermite-Hadamard inequality

Corollary 3 Let λ [a b]⟶ R be a function of boundedvariation such that λ(a)ne λ(b) with [a b] sub [α β] and1113957ζ 1113938

b

aζdλ(ζ) 1113938

b

adλ(ζ) isin [α β] Under the assumptions of

=eorem 3 if Ψ is nminusconvex such that

Rn(1113957Z s)le1113938

b

aRn(Z(ζ) s)dλ(ζ)

1113938b

adλ(ζ)

s isin [α β] (18)

then we have

4 Journal of Mathematics

Ψ(1113957ζ)le1113938

b

aΨ(ζ)dλ(ζ)

1113938b

adλ(ζ)

+ 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

times Ψ(ℓ+1)(α) (1113957ζ minus α)

(ℓ+2)minus

1113938b

a(ζ minus α)

(ℓ+2)dλ(ζ)

1113938b

adλ(ζ)

⎛⎝ ⎞⎠ minus Ψ(ℓ+1)(β) (1113957ζ minus β)

(ℓ+2)+

1113938b

a(ζ minus β)

(ℓ+2)dλ(ζ)

1113938b

adλ(ζ)

⎛⎝ ⎞⎠⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

(19)

If the inequality (18) holds in reverse direction then (19)also holds reversely

e special case of above corollary can be given in theform of following remark

Remark 3 It is interesting to see that substituting λ(ζ) ζgives 1113938

b

adλ(ζ) b minus a and 1113957ζ a + b2 Using these substi-

tutions in (2) and by following remark (20) we get the LHSinequality of renowned HermitendashHadamard inequality fornminusconvex functions

22 New Generalization of Converse of Jensenrsquos IntegralInequality In this section we give the results for the

converse of Jensenrsquos inequality to hold giving the conditionson the real Stieltjes measure dλ such that λ(a)ne λ(b)allowing that the measure can also be negative butemploying Montgomery identity

To start with we need the following assumption for theresults of this section

A3 Let mM isin [α β](mneM) be such thatmle Z(ζ)leM for all ζ isin [a b] where Z is defined in A1

For a given function Ψ [α β]⟶ R we consider thedifference

CJ Ψ Z mM λ1113872 1113873 1113938

b

aΨ(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

minusM minus 1113957Z

M minus mΨ(m) minus

1113957Z minus mM minus mΨ(M) (20)

where 1113957Z is defined in (3)Using Montgomery identity we obtain the following

representation of the converse of Jensenrsquos inequality

Theorem 5 Let Z λ be as defined in A1 A2 and letΨ [α β]⟶ R be such that for nge 1 Ψ(nminus1) is absolutelycontinuous If Ψ is nminusconvex such that

CJ Rn(x s) Z mM λ1113872 1113873le 0 s isin [α β] (21)

or

1113938b

aRn(Z(ζ) s)dλ(ζ)

1113938b

adλ(ζ)

leM minus 1113957Z

M minus mRn(m s)( 1113857 +

1113957Z minus mM minus m

Rn(M s)( 1113857 s isin [α β] (22)

then we get the following extension of the converse of Jensenrsquosdifference

1113938b

aΨ(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

leM minus 1113957Z

M minus mΨ(m) +

1113957Z minus mM minus mΨ(M) + 1113944

nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889

times Ψ(ℓ+1)(α)CJ (x minus α)

ℓ+2 Z mM λ1113872 1113873 minus Ψ(ℓ+1)

(β)CJ (x minus β)ℓ+2

Z mM λ1113872 11138731113872 1113873

(23)

where Rn(middot s) is defined in (5)

Journal of Mathematics 5

Proof AsΨ(nminus1) is absolutely continuous for (nge 1) we canuse the representation of Ψ using Montgomery identity (4)in the difference CJ(Ψ Z mM λ)

CJ Ψ Z mM λ1113872 1113873 CJ1

β minus α1113946β

αΨ(ζ)dζ Z mM λ1113888 1113889

+ 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889Ψ(ℓ+1)(α)CJ (x minus α)

ℓ+2 Z mM λ1113872 1113873

minus 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889Ψ(ℓ+1)(β)CJ (x minus β)

ℓ+2 Z mM λ1113872 1113873 +

1(n minus 1)

1113946β

αCJ Rn(x s) Z mM λ1113872 1113873Ψ(n)

(s)ds

(24)

After simplification and following the fact thatCJ(Ψ Z mM λ) is zero for Ψ to be constant or linear we getthe following generalized identity

CJ Ψ Z mM λ1113872 1113873j 1113944nminus2

ℓ0

ℓ(ℓ + 2)(β minus α)1113888 1113889

times Ψ(ℓ+1)(α)CJ (x minus α)

ℓ+2 Z mM λ1113872 1113873 minus Ψ(ℓ+1)

(β)CJ (x minus β)ℓ+2

Z mM λ1113872 11138731113872 1113873

+1

(n minus 1)1113946β

αCJ Rn(x s) Z mM λ1113872 1113873Ψ(n)

(s)ds(CGI1)

(25)

Now using characterizations of nminusconvex functions likein the proof of eorem 3 we get (23)

e next result gives converse of Jensenrsquos inequality forhigher-order convex functions

Theorem 6 Let Ψ defined in =eorem 5 be nminusconvex and Z

be as defined in A3 =en the following results hold

(i) If λ is non-negative measure on [a b] then for evennge 3 (22) is valid

(ii) Moreover if (22) is valid and the function H(middot)

defined in (12) is convex then we get the followinginequality for nminusconvex function to be valid

1113938b

aΨ(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

leM minus 1113957Z

M minus mΨ(m) minus

1113957Z minus mM minus mΨ(M) (26)

Proof e idea of the proof is similar to that of (6) but we useconverse of Jensenrsquos inequality (see [32] or [30] p 98)

23 Applications of Jensenrsquos Integral Inequality In this sec-tion we give applications of Jensenrsquos integral inequality

Another important consequence of eorem 3 is bysetting the function Z as Z(ζ) ζ gives generalized version ofL H S inequality of the HermitendashHadamard inequality

Corollary 4 Let λ [a b]⟶ R be a function of boundedvariation such that λ(a)ne λ(b) with [a b] sub [α β] and1113957ζ 1113938

b

aζdλ(ζ) 1113938

b

adλ(ζ) isin [α β] Under the assumptions of

=eorem 5 if Ψ is nminusconvex such that

1113938b

aRn(ζ s)dλ(ζ)

1113938b

adλ(ζ)

leb minus 1113957ζb minus a

Rn(a s)( 1113857 +1113957ζ minus a

b minus aRn(b s)( 1113857 s isin [α β]

(27)

then we have

6 Journal of Mathematics

1113938b

aΨ(ζ)dλ(ζ)

1113938b

adλ(ζ)

leb minus 1113957ζb minus aΨ(a) +

1113957ζ minus a

b minus aΨ(b) + 1113944

nminus2

ℓ0

ℓ(ℓ + 2)(β minus α)1113888 1113889times

Ψ(ℓ+1)(α)CJ (x minus α)

ℓ+2 id ab λ1113872 1113873 minus Ψ(ℓ+1)

(β)CJ (x minus β)ℓ+2

id ab λ1113872 11138731113872 1113873

(28)

If the inequality (27) holds in reverse direction then (28)also holds reversely

e special case of above corollary can be given in theform of following remark

Remark 4 It is interesting to see that substituting λ(ζ) ζand by following eorem 6 we get the RHS inequality ofrenowned HermitendashHadamard inequality for nminusconvexfunctions

3 Generalization of Jensenrsquos DiscreteInequality by Using Montgomery Identity

In this section we give generalizations for Jensenrsquos discreteinequality by using Montgomery identity e proofs aresimilar to those of continuous case as given in previoussection therefore we give results directly

31 Generalization of Jensenrsquos Discrete Inequality for RealWeights In discrete case we have that pJ gt 0 for allJ 1 2 m Here we give generalizations of results

allowing pJ to be negative real numbers Also with usualnotations for pJxJ(J 1 2 n) we notate

x x1 x2 xm( 1113857 and p p1 p2 pm( 1113857 (29)

to be mminustuples

Pv 1113944v

J1pJ Pv Pm minus Pvminus1 (v 1 2 m) (30)

and

x 1

Pm

1113944

m

J1pJxJ (31)

Using Montgomery identity (4) we obtain the followingrepresentations of Jensenrsquos discrete inequality

Theorem 7 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also let xJ isin [a b]sube[α β]pJ isin R(J 1 m) be such that Pm ne 0 and x isin [α β]

(i) =en the following generalized identity holds

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

Ψ(ℓ+1)(α) (x minus α)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus α1113872 1113873

ℓ+2⎛⎝ ⎞⎠ minus Ψ(ℓ+1)(β) (x minus β)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus β1113872 1113873

ℓ+2⎛⎝ ⎞⎠⎧⎨

⎩

⎫⎬

⎭

+1

(n minus 1)1113946β

αRn(x s) minus

1Pm

1113944

m

J1pJRn xJ s1113872 1113873⎡⎢⎢⎣ ⎤⎥⎥⎦Ψ(n)

(s)ds (DGI1)

(32)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

Rn(x s)le1

Pm

1113944

m

J1pJRn xJ s1113872 1113873 (33)

holds then we have the following generalized inequality

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873le 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

Ψ(ℓ+1)(α) (x minus α)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus α1113872 1113873

ℓ+2⎛⎝ ⎞⎠ minus Ψ(ℓ+1)(β) (x minus β)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus β1113872 1113873

ℓ+2⎛⎝ ⎞⎠⎧⎨

⎩

⎫⎬

⎭

(34)

Journal of Mathematics 7

If inequality (33) holds in reverse direction then (34) alsoholds reversely

Proof Similar to that of eorem 3In the later part of this section we will vary our con-

ditions on pJxJ(J 1 2 n) to obtain generalizeddiscrete variants of JensenndashSteffensen Jensenrsquos and Jen-senndashPetrovic type inequalities We start with the followinggeneralization of JensenndashSteffensen discrete inequality fornminusconvex functions

Theorem 8 Let Ψ be as defined in =eorem 7 Also let x bemonotonic nminustuple xJ isin [a b]sube[α β] and p be a realnminustuple such that

0lePv lePm (v 1 2 m minus 1) Pm gt 0 (35)

is satisfied

(i) If Ψ is nminusconvex then for even nge 3 (33) is valid(ii) Moreover if (33) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized JensenndashSteffensen discrete inequality

Ψ(x)le1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 (36)

Proof It is interesting to see that under the assumed con-ditions on tuples x and p we have that x isin [a b] Forx1 gex2 ge gexm

Pm x1 minus x( 1113857 1113944m

J2pJ x1 minus xJ1113872 1113873 1113944

m

v2xvminus1 minus xv( 1113857 Pm minus Pvminus1( 1113857ge 0

(37)

is shows that x1 ge x Also xge xn since we have

Pm x minus xm( 1113857 1113944

mminus1

J1pJ xJ minus xm1113872 1113873 1113944

mminus1

v1xv minus xvminus1( 1113857Pv ge 0

(38)

For further details see the proof of JensenndashSteffensendiscrete inequality ([24] p 57) e idea of the rest of theproof is similar to that of eorem 3 but here we employeorem 7 and JensenndashSteffensen discrete inequality

Corollary 5 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube [α β] with p being a positive nminustuple

(i) If Ψ is nminusconvex then for even nge 3 (34) is valid(ii) Moreover if (33) is valid and the function H(middot) de-

fined in (12) is convex then again we get (36) which iscalled Jensenrsquos inequality for nminusconvex functions

Proof For pJ gt 0 xJ isin [a b] (J 1 2 3 m) ensuresthat x isin [a b] So by applying classical Jensenrsquos discrete

inequality (1) and idea ofeorem 8 we will get the requiredresults

Remark 5 Under the assumptions of Corollary 5 if wechoose Pm 1 then Corollary 5 (ii) gives the followinginequality for nminusconvex functions

Ψ 1113944m

J1pJxJ

⎛⎝ ⎞⎠le 1113944m

J1pJΨ xJ1113872 1113873 (39)

Nowwe give following reverses of JensenndashSteffensen andJensen-type inequalities

Corollary 6 Let Ψ be as defined in =eorem 7 Also let x bemonotonic mminustuple xJ isin [a b]sube[α β] and p be a realmminustuple such that there exist m isin 1 2 m such that

0gePv for vltm and 0gePv for vgtm (40)

where Pm gt 0 and x isin [α β]

(i) If is nminusconvex then for even nge 3 then reverse ofinequality (33) holds

(ii) Moreover if (33) holds reversely and the function H(middot)

defined in (12) is convex then we get reverse ofgeneralized JensenndashSteffensen inequality (36) fornminusconvex functions

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ reverse of JensenndashSteffensen inequality to obtain results

In the next corollary we give explicit conditions on realtuple p such that we get reverse of classical Jensen inequality

Corollary 7 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube[α β] such that x isin [α β] Let p be a real nminustuplesuch that

0ltp1 0gep2 p3 pm 0ltPm (41)

is satisfied

(i) If Ψ is nminusconvex then for even nge 3 the reverse ofinequality (33) is valid

(ii) Also if reverse of (33) is valid and the function H(middot)

defined in (12) is convex then we get reverse of (36)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ reverse of Jensen in-equality to obtain results

In [33] (see also [30]) one can find the result which isequivalent to the JensenndashSteffensen and the reverse Jen-senndashSteffensen inequality together It is the so-calledJensenndashPetrovic inequality Here without the proof we givethe adequate corollary which uses that result e proof goesthe same way as in the previous corollaries

8 Journal of Mathematics

Corollary 8 Let Ψ be as defined in =eorem 7 and letxi isin [a b]sube[α β] be such that xm ge xmminus1 x2 lex1 Let pbe a real mminustuple with Pm 1 such that

0lePv for 1le vltm minus 1 and 0lePv for 2le vltm

(42)

is satisfied =en we get the equivalent results given in=eorem 8 (i) and (ii) respectively

Remark 6 Under the assumptions of Corollary 8 if thereexist m isin 1 2 n such that

0gePv for vltm and 0gePv for vgtm (43)

and x isin [α β] then we get the equivalent results for reverseJensenndashSteffensen inequality given in Corollary 6 (i) and(ii) respectively

Remark 7 It is interesting to see that the conditions onpJJ 1 2 m given in Corollary 8 and Remark 6 arecoming from JensenndashPetrovic inequality which becomeequivalent to conditions for pJJ 1 2 m for Jen-senndashSteffensen results given in eorem 8 and Corollary 6respectively when Pm 1

Now we give results for Jensen and its reverses fornminustuples x and p when n is an odd number

Corollary 9 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube[α β] for J 1 2 m be such that x p berealmminustuplesm 2m + 1 m isin N and 1113954x 111139362k+1

J1 pJ11139362k+1J1

pJxJ isin [α β] for all k 1 2 m If for everyk 1 2 m we have

(ilowast) p1 gt 0 p2k le 0 p2k + p2k+1 le 0 11139362kJ1pJ ge 0

11139362k+1J1 pJ gt 0

(iilowast) x2k lex2k+1 11139362k+1J1 pJ (xJ minus x2k+1)ge 0

then we have the following statements to be valid(i) If Ψ is nminusconvex then for even nge 3 the inequality

Rn(1113954x s)ge1

P2m+11113944

2m+1

J1pJRn xJ s1113872 1113873 (44)

(ii) Also if (44) is valid and the function H(middot) defined in(12) is convex then we get the following generalizedinequality

Ψ(1113954x)ge1

P2m+11113944

2m+1

J1pJΨ xJ1113872 1113873 (45)

Proof We employ the idea of the proofs of eorems 7 and8 for n odd along with inequality of Vasic and Janic[34]

Remark 8 We can also discuss the following importantcases by considering the explicit conditions given in [34]

We conclude this section by giving the following im-portant cases

(Case 1)Let the condition (ilowast) hold and the reverse inequalitiesin condition (iilowast) hold en again we can give in-equalities (44) and (45) respectively given in Corollary9(Case 2)If in case of conditions (ilowast) and (iilowast) the following arevalid(iiilowast) p1 gt 0 p2k+1 ge 0 p2k + p 2k+1 ge 0 1113936

2kJ1pJ ge 0

11139362k+1J1 pJ gt 0

(ivlowast) x2k lex2k+1 11139362kminus1J1 pJ(xJ minus x2k)le 0

then we can give reverses of inequalities (44) and (45)respectively given in Corollary 9(Case 3)Finally we can also give reverses of inequalities (44)and (45) respectively given in Corollary 9 providedthat the condition (iiilowast) holds and the reverse in-equalities in condition (ivlowast) hold

e result given in (Case 3) is type of generalization ofinequality by Szego [35]

32 Generalization of Converse JensenrsquosDiscrete Inequality forRealWeights In this section we give the results for converseof Jensenrsquos inequality in discrete case by using the Mont-gomery identity

Let xJ isin [a b]sube[α β] ane b pJ isin R(J 1 n) besuch that Pm ne 0 en we have the following difference ofconverse of Jensenrsquos inequality for Ψ [α β]⟶ R

CJdis(Ψ) 1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 minus

b minus x

b minus aΨ(a) minus

x minus a

b minus aΨ(b)

(46)

Similarly we assume the Giaccardi difference [36] givenas

Gcardi(Ψ) 1113944m

J1pJΨ xJ1113872 1113873 minus AΨ 1113944

m

J1pJxJ

⎛⎝ ⎞⎠ minus B 1113944m

J1pJ minus 1⎛⎝ ⎞⎠Ψ x0( 1113857 (47)

where

Journal of Mathematics 9

A 1113936

mJ1 pJ xJ minus x01113872 11138731113872 1113873

1113936mJ1 pJxJ minus x01113872 1113873

B 1113936

mJ1 pJxJ

1113936mJ1 pJxJ minus x01113872 1113873

and 1113944m

J1pJxJ ne x0 (48)

Theorem 9 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also letx0 xJ isin [a b]sube[α β] pJ isin R(J 1 m) be such that1113936

mJ1 pJxJ ne x0

(i) =en the following generalized identity holds

CJdis(Ψ) 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)CJdis xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)CJdis xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875

+1

(n minus 1)1113946β

αCJdis Rn xJ s1113872 11138731113872 1113873Ψ(n)

(s)ds (DCGI)

(49)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

CJdis Rn xJ s1113872 11138731113872 1113873le 0 (50)

holds then we have the following generalized inequality

CJdis(Ψ)le 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)CJdis xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)CJdis xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875 (51)

If inequality (50) holds in reverse direction then (51) alsoholds reversely

Theorem 10 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also let xJ isin [a b]sube[α β]pJ isin R(J 1 m) be such that Pm ne 0 and x isin [α β]

(i) =en the following generalized Giaccardi identityholds

Gcardi(Ψ) 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)Gcardi xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)Gcardi xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875

+1

(n minus 1)1113946β

αGcardi Rn xJ s1113872 11138731113872 1113873Ψ(n)

(s)ds (GIAGI)

(52)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

Gcardi Rn xJ s1113872 11138731113872 1113873le 0 (53)

holds then we have the following generalized Giaccardiinequality

Gcardi(Ψ)le 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)Gcardi xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)Gcardi xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875 (54)

If inequality (53) holds in reverse direction then (54) alsoholds reversely

In the later part of this section we will vary our con-ditions on pJxJ (J 1 2 m) to obtain generalized

10 Journal of Mathematics

converse discrete variants of Jensenrsquos inequality and Giac-cardi inequality for nminusconvex functions

Theorem 11 Let Ψ be as defined in =eorem 9 Also letxJ isin [a b]sube [α β] and p be a positive mminustuple

(i) If Ψ is nminusconvex then for even nge 3 (50) is valid(ii) Moreover if (50) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized converse of Jensenrsquos inequality

1Pm

1113944

m

J1pJΨ xJ1113872 1113873le

b minus x

b minus aΨ(a) +

x minus a

b minus aΨ(b) (55)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ converse of Jensenrsquosinequality (see [32] or [30] p 98) to obtain results

Finally in this section we give Giaccardi inequality forhigher-order convex functions

Theorem 12 Let Ψ be as defined in =eorem 9 Also letx0 xJ isin [a b]sube[α β] and p be a positive mminustuple such that

1113944

m

J1pJxJ nex0 and xv minus x0( 1113857 1113944

m

J1pJxJ minus xv

⎛⎝ ⎞⎠ge 0 (v 1 m) (56)

(i) If Ψ is nminusconvex then for even nge 3 (53) is valid(ii) Moreover if (53) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized Giaccardi inequality

1113944

m

J1pJΨ xJ1113872 1113873leAΨ 1113944

m

J1pJxJ

⎛⎝ ⎞⎠ + B 1113944m

J1pJ minus 1⎛⎝ ⎞⎠Ψ x0( 1113857

(57)

where A and B are defined in (47)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ Giaccardi inequality(see [36] or [37] p 11) to obtain results

33 Applications in Information =eory for Jensenrsquos DiscreteInequality Jensenrsquos inequality plays a key role in infor-mation theory to construct lower bounds for some notableinequalities but here we will use it to make connectionsbetween inequalities in information theory

Let Ψ R+⟶ R+ be a convex function and letp ≔ (p1 pm) and q ≔ (q1 qm) be positive proba-bility distributions then Ψ-divergence functional is defined(in [38]) as follows

IΨ(p q) 1113944m

J1qJΨ

pJ

qJ1113888 1113889 (58)

Horvath et al in [39] defined the generalized Csiszardivergence functional as follows

Definition 2 Let I be an interval in R and Ψ I⟶ R be afunction Also let p ≔ (p1 pm) isin Rm andq ≔ (q1 qm) isin (0infin)m such that

pJ

qJisin I J 1 m (59)

en let

1113957IΨ(p q) 1113944m

J1qJΨ

pJ

qJ1113888 1113889 (60)

In this section we write Jensenrsquos difference here that weuse in upcoming results

F p xJΨ1113872 1113873 Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 (61)

Theorem 13 Under the assumptions of =eorem 9 (ii) let(51) hold and Ψ be nminusconvex Also letp ≔ (p1 pm) inRm and q ≔ (q1 qm) isin (0infin)mthen we have the following results

Journal of Mathematics 11

1113957IΨ(p q)gePmΨ(1) minus Pm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F q

pJ

qJ (x minus α)

ℓ+21113888 11138891113890 1113891 minus Ψ(ℓ+1)

(β) F qpJ

qJ (x minus β)

ℓ+21113888 11138891113890 11138911113896 1113897

(62)

Proof From eorem 9 by following Jensenrsquos difference(61) we can rearrange (34) as

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873le 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F p xJ (x minus α)

ℓ+21113872 11138731113872 1113873 minus Ψ(ℓ+1)

(β) F p xJ (x minus β)ℓ+2

1113872 11138731113872 11138731113966 1113967

(63)

Now replace pJ with qJ and xJ with pJqJ and we get(62)

For positive n-tuple q (q1 qm) such that1113936

mJ1 qJ 1 the Shannon entropy is defined by

S(q) minus 1113944m

J1qJ ln qJ (64)

Corollary 10 Under the assumptions of =eorem 9 (ii) let(51) hold and Ψ be nminusconvex

(i) If q ≔ (q1 qm) isin (0infin)m then

1113944

m

J1qJ ln qJ lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (65)

(ii) We can get bounds for the Shannon entropy of q if wechoose q ≔ (q1 qn) to be a positive probabilitydistribution

S(q)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (66)

Proof (i) Substituting Ψ(x) ≔ minus lnx and usingp ≔ (1 1 1) in eorem 13 we get (65)

(ii) Since we have 1113936mJ1 qJ 1 by multiplying minus1 on

both sides of (65) and taking into account (64) weget (66)

e KullbackndashLeibler distance [40] between the positiveprobability distributions p (p1 pm) andq (q1 qm) is defined by

D(q p) 1113944

m

J1qJ ln

qJ

pJ

1113888 1113889 (67)

Corollary 11 Under the assumptions of Corollary 10

(i) If q ≔ (q1 qm) p ≔ (p1 pm) isin (0infin)mthen

1113944

m

J1qJ ln

qJ

pJ

1113888 1113889lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

pJ

qJ minusln(middot)1113888 1113889

⎧⎨

⎩

⎫⎬

⎭ (68)

(ii) If q ≔ (q1 qm) p ≔ (p1 pm) are positiveprobability distributions then we have

12 Journal of Mathematics

D(q p)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

βℓ+1F q

pJ

qJ minusln(middot)1113888 11138891113896 1113897 (69)

Proof

(i) UsingΨ(x) ≔ minus lnx (which isn-convex for evenn)in eorem 13 we get (68) after simplification

(ii) It is a special case of (i)

34 Results for Zipf and Hybrid ZipfndashMandelbrot EntropyOne of the basic laws in information science is Zipfrsquos law[4142] which is highly applied in linguistics Let cge 0 dgt 0and N isin 1 2 ZipfndashMandelbrot entropy can be given as

ZM(H c d) d

HNcd

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (70)

where

HNcd 1113944

N

σ1

1(σ + c)

d (71)

Consider

qJ Ψ(J N c d) 1

(J + c)dH

Ncd1113872 1113873

(72)

where Ψ(J m c d) is discrete probability distributionknown as ZipfndashMandelbrot law ZipfndashMandelbrot law hasmany application in linguistics and information sciencesSome of the recent study about ZipfndashMandelbrot law can beseen in the listed references (see [39 43]) Now we state ourresults involving entropy introduced by Mandelbrot law byestablishing the relationship with Shannon and relativeentropies

Theorem 14 Let q be ZipfndashMandelbrot law as defined in(72) with parameters cge 0 dgt 0 and N isin 1 2 and wehave

ZM(H c d) S(q)leN times 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873

⎧⎨

⎩

⎫⎬

⎭

(73)

Proof It is interesting to see that for qJ defined in (72)1113936

NJ1 qJ 1 erefore using above qJ in Shannon entropy

(64) we get Mandelbrot entropy (70)

S(q) minus 1113944

N

J1qJ ln qJ minus 1113944

N

J1

1(J + c)

dH

Ncd1113872 1113873

ln1

(J + c)dH

Ncd1113872 1113873

d

HNcd1113872 1113873

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (74)

Finally substituting this qJ 1((J + c)dHNcd) in

Corollary 10 (ii) we get the desired result Corollary 12 Let q and p be ZipfndashMandelbrot law withparameters c1 c2 isin [0infin) d1 d2 gt 0 and let

Journal of Mathematics 13

HNc1 d1

1113936Nσ1 1(σ + c1)

d1 and HNc2 d2

1113936Nσ1 1(σ + c2)

d2 Now using qJ 1(J + c1)

d1HNc1 d1

andpJ 1(J + c2)

d2HNc2 d2

in Corollary 11 (ii) the followingholds

D(q p) 1113944N

J1

1J + c1( 1113857

d1HNc1d1

lnJ + c2( 1113857

d2HNc2 d2

J + c1( 1113857d1H

mc1 d1

⎛⎝ ⎞⎠

minusZ H c1 d1( 1113857 +d2

HNc1 d1

1113944

N

J1

ln J + c2( 1113857

J + c1( 1113857d1

+ ln Hmc2d2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qJ + c1( 1113857

d1HNc1 d1

J + c2( 1113857d2H

Nc2 d2

minusln(middot)⎛⎝ ⎞⎠ minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qJ + c1( 1113857

d1HNc1d1

J + c2( 1113857d2H

Nc2d2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(75)

=e Next Result for Hybrid ZipfndashMandelbrot Entropy Fur-ther generalization of ZipfndashMandelbrot entropy is Hybrid

ZipfndashMandelbrot entropy Let N isin 1 2 cge 0ωgt 0then Hybrid ZipfndashMandelbrot entropy can be given as

1113954ZM Hlowast c dω( 1113857

1Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873 (76)

where

Hlowastcdω 1113944

N

J1

ωJ

(J + c)d (77)

Consider

qJ Ψ(J N c dω) ωJ

(J + c)dHlowastcdω

(78)

which is called Hybrid ZipfndashMandelbrot law =ere is aunified approach maximization of Shannon entropy [44]

that naturally follows the path of generalization from Zipfrsquos toHybrid Zipfrsquos law Extending this idea Jaksetic et al in [45]presented a transition from ZipfndashMandelbrot to HybridZipfndashMandelbrot law by employing maximum entropytechnique with one additional constraint It is interesting thatexamination of its densities provides some new insights ofLerchrsquos transcendent

Theorem 15 Let q be Hybrid ZipfndashMandelbrot law asdefined in (78) with parameters cge 0 dωgt 0 andN isin 1 2 and we have

1113954ZM Hlowast c dω( 1113857 S(q)leN 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠ minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠

⎧⎨

⎩

⎫⎬

⎭

(79)

14 Journal of Mathematics

Proof It is interesting to see that for qJ defined in (78)1113936

NJ1 qJ 1 erefore using above qJ in Shannon en-

tropy (64) we get Hybrid ZipfndashMandelbrot entropy (76)

S(q) minus 1113944N

J1qJ ln qJ minus 1113944

N

J1

ωJ

(J + c)dHlowastcdω

lnωJ

(J + c)dHlowastcdω

minus1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

lnωJ

(J + c)d

1113888 1113889 + ln1

Hlowastcdω

1113888 11138891113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

ln(J + c)

d

ωJ1113888 1113889 + ln H

lowastcdω1113872 11138731113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873

(80)

Finally substituting this qJ ωJ(J + c)dHlowastcdω inCorollary 10 (ii) we get the desired result

Corollary 13 Let q and p be Hybrid ZipfndashMandelbrotlaw with parameters c1 c2 isin [0infin)ω1ω2 d1 d2 gt 0 Nowusing qJ ωJ

1 (J + c1)d1Hlowastc1 d1 ω1

and pJ ωJ2

(J + c2)d2Hlowastc2 d2 ω2

in Corollary 11 (ii) the following holds

D(q p) 1113944N

J1

ωJ1

J + c1( 1113857d1Hlowastc1 d1 ω1

lnωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

ωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

⎛⎝ ⎞⎠

minus1113954ZM Hlowast c1 d1ω1( 1113857 +

1Hlowastc1 d1 ω1

1113944

N

J1

ωJ1

J + c1( 1113857d1ln

J + c2( 1113857d2

ωJ2

⎛⎝ ⎞⎠ + ln Hlowastc2 d2 ω2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎫⎪⎬

⎪⎭

(81)

Remark 9 Similarly we can give results for Shannonentropy KullbackndashLeibler distance ZipfndashMandelbrotentropy and Hybrid ZipfndashMandelbrot entropy by usinggeneralized Giaccardi inequality defined in (54) on thesame steps

4 Concluding Remarks

In this paper we gave generalization of Jensenrsquos inequality aswell as converse of Jensenrsquos inequality by using Montgomeryidentity We also formulate results for other inequalities like

JensenndashSteffensen inequality JensenndashBoas inequality andJensenndashBrunk inequality We can obtain JensenndashSteffenseninequality JensenndashBoas inequality and JensenndashBrunk in-equality by changing the assumption of Jensenrsquos inequalityAt the end we gave applications in information theory forour obtained results especially we gave results for HybridZipfndashMandelbrot entropy for our obtained results [46]

Data Availability

No data were used to support this study

Journal of Mathematics 15

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is research is supported by HEC Pakistan under NRPUProject 7906 and RUDN University Strategic AcademicLeadership Program

References

[1] X L Zhu and G H Yang ldquoJensen inequality approach tostability analysis of discrete-time systems with time-varyingdelayrdquo in Proceedings of the American Control Conferencepp 1644ndash1649 Seattle WA USA June 2008

[2] S I Butt M Klaricic Bakula ETH Pecaric and J PecaricldquoJensen-gruss inequality and its applications for the zipf-mandelbrot lawrdquo Mathematical Methods in the Applied Sci-ences vol 44 no 2 pp 1664ndash1673 2021

[3] M A Khan D Pecaric and J Pecaric ldquoNew refinement of theJensen inequality associated to certain functions with appli-cationsrdquo Journal of Inequalities and Applications vol 762020

[4] S Khan M Adil Khan S I Butt and Y-M Chu ldquoA newbound for the Jensen gap pertaining twice differentiablefunctions with applicationsrdquo Advances in Difference Equa-tions vol 1 2020

[5] H Budak andM Z Sarikaya ldquoOn generalized Ostrowski-typeinequalities for functions whose first derivatives absolutevalues are convexrdquo Turkish Journal of Mathematics vol 40pp 1193ndash1210 2016

[6] P Cerone and S S Dragomir ldquoOn some inequalities arisingfrom Montgomeryrsquos identityrdquo Journal of ComputationalAnalysis and Applications vol 5 no 4 pp 341ndash367 2003

[7] S Bernstein ldquoSur les fonctions absolument monotonesrdquo ActaMathematica vol 52 pp 1ndash66 1929

[8] V Culjak and J Pecaric ldquoInterpolation polynomials andinequalities for convex functions of higher orderrdquo ActaMathematica vol 5 pp 369ndash386 2012

[9] A M Fink ldquoBounds of the deviation of a function from itsaveragesrdquo Czechoslovak Mathematical Journal vol 42no 117 pp 289ndash310 1992

[10] J Pecaric and V Culjak ldquoInterpolation polinomials and in-equalities for convex functions of higher orderrdquo Mathe-matical Inequalities amp Applications vol 5 no 3 pp 369ndash3862002

[11] J Pecaric V Culjak and A M Fink ldquoOn some inequalitiesfor convex function of higher orderrdquo Advanced NonlinearStudies vol 6 no 2 pp 131ndash140 1999

[12] J Pecaric V Culjak andM Rogina ldquoOn some inequalities forconvex function of higher order IIrdquo Advanced NonlinearStudies vol 45 pp 281ndash294 2001

[13] J Pecaric I Peric and M Rodic-Lipanovic ldquoUniformtreatment of Jensen type inequalitiesrdquo Mathematical Reportsvol 16 no 66 pp 183ndash205 2014

[14] J Pecaric and J Peric ldquoImprovement of the Giaccardi and thepetrovic inequality and related stolarsky type meansrdquo Annalsof the University of Craiova - Mathematics and ComputerScience vol 39 no 1 pp 65ndash75 2012

[15] J Pecaric M Praljak and A Witkowski ldquoLinear operatorinequality for nminusconvex functions at a pointrdquo MathematicalInequalities amp Applications Appl vol 18 pp 1201ndash1217 2015

[16] Z Pavic ldquoRefinements of Jensenrsquos inequality for infiniteconvex combinationsrdquo Turkish Journal of Inequalities vol 2no 2 pp 44ndash53 2018

[17] R Bibi A Nosheen and J Pecaric ldquoExtended Jensenrsquos typeinequalities for diamond integrals via Taylors formulardquoTurkish Journal of Inequalities vol 3 no 1 pp 7ndash18 2019

[18] T Niaz K A Khan and J Pecaric ldquoOn refinement of Jensenrsquosinequality for 3minusconvex function at a pointrdquo Turkish Journalof Inequalities vol 4 no 1 pp 70ndash80 2020

[19] A Ekinci A O Akdemir and M E Ozdemir ldquoIntegralinequalities for different kinds of convexity via classical in-equalitiesrdquo Turkish Journal of Science vol 5 no 3 pp 305ndash313 2020

[20] Y M Chu S Talib E Set M U Awan and M A Noorldquo(p q)minus Analysis of Montgomery identity and estimates of(p q)minusbounds with applicationsrdquo Journal of Inequalities andApplications vol 2021 9 pages 2021

[21] M Z Sarikaya H Yaldrz and E Set ldquoOn fractional in-equalities via Montgomery identitiesrdquo International Journalof Open Problems in Complex Analysis vol 6 no 2 pp 36ndash432014

[22] A Bnouhachem ldquoA descent SQP alternating directionmethod for minimizing the sum of three convex functionsrdquoJournal of Nonlinear Functional Analysis vol 4 pp 469ndash4822020

[23] A O Akdemir S I Butt M Nadeem and M A RagusaldquoNew general variants of Chebyshev type inequalities viageneralized fractional integral operatorsrdquo Mathematicsvol 92 2021

[24] S I Butt M Umar S Rashid A O Akdemir and Y MingChu ldquoNew Hermite Jensen Mercer type inequalities viakminusfractional integralsrdquo Advances in Difference Equationsvol 635 2020

[25] S I Butt M Nadeem S Qaisar A O Akdemir andT Abdeljawad ldquoHermite Jensen Mercer type inequalities forconformable integrals and related resultsrdquo Advances in Dif-ference Equations vol 501 2020

[26] A Bnouhachem and X Qin ldquoAn inertial proximal Peaceman-Rachford splittingmethod with SQP regularization for convexprogrammingrdquo Journal of Nonlinear Functional Analysisvol 50 2020

[27] S Hamann ldquoMinimality conditions for convex compositefunctions and efficiency conditions in vector optimizationrdquoApplied Set-Valued Analysis and Optimization vol 1pp 221ndash229 2019

[28] A R Khan J Pecaric and M Praljak ldquoPopoviciu type in-equalities for n-convex functions via extension of Mont-gomery identitynminusconvex functions via extension ofMontgomery identityrdquo Analele Universitatii ldquoOvidiusrdquo Con-stanta-Seria Matematica vol 24 no 3 pp 161ndash188 2016

[29] R P Boas ldquoe Jensen-Steffensen inequalityrdquo ElektrotehnFak Ser Mat Fiz vol 302ndash319 pp 1ndash8 1970

[30] J Pecaric F Proschan and Y L Tong Convex FunctionsPartial Orderings and Statistical Applications Academic PressNew York NY USA 1992

[31] H D Brunk ldquoOn an inequality for convex functionsrdquo Pro-ceedings of the American Mathematical Society vol 7 no 5p 817 1956

[32] P R Beesack and J E Pecaric ldquoOn Jessenrsquos inequality forconvex functionsrdquo Journal of Mathematical Analysis andApplications vol 110 no 2 pp 536ndash552 1985

[33] R Barlow A Marshall and F Proschan ldquoSome inequalitiesfor starshaped and convex functionsrdquo Pacific Journal ofMathematics vol 29 no 1 pp 19ndash42 1969

16 Journal of Mathematics

[34] J E Pecaric ldquoOn an inequality of PMV Janicrdquo Publications delrsquoInstitut Mathematique vol 28 no 42 pp 145ndash149 1980

[35] G Szego ldquoUber eine Verallgemeinerung des DirichletschenIntegralsrdquo Mathematische Zeitschrift vol 52 no 1pp 676ndash685 1950

[36] F Giaccardi ldquoSu alcune disuguaglianierdquo Journal of Researchof the National Bureau of Standards vol 1 no 4 pp 139ndash1531955

[37] D S Mitrinovic J E Pecaric and A M Fink Classical andnew inequalities in analysis Mathematics and its Applications(East European Series) Vol 61 Kluwer Academic PublishersGroup Dordrecht Netherlands 1993

[38] I Csiszar ldquoInformation-type measures of diference ofprobability distributions and indirect observationsrdquo StudiaScience Math Hungar vol 2 pp 299ndash318 1967

[39] L Horvath D Pecaric and J Pecaric ldquoEstimations of f - andrenyi divergences by using a cyclic refinement of the Jensenrsquosinequalityrdquo Bulletin of the Malaysian Mathematical SciencesSociety vol 42 2017

[40] S Kullback Information =eory and Statistics Wiley NewYork NY USA 1959

[41] S T Piantadosi ldquoZipfrsquos word frequency law in natural lan-guage a critical review and future directionsrdquo PsychonomicBulletin and Review vol 21 no 5 pp 1112ndash1130 2014

[42] Z K Silagadze ldquoCitations and the zipfndashmandelbrot lawrdquoComplex Systems no 11 pp 487ndash499 1997

[43] J Jaksetic D Pecaric and J Pecaric ldquoSome properties of zipf-mandelbrot law and hurwitz ζ-functionrdquo Mathematical In-equalities amp Applications vol 21 no 2 pp 575ndash584 2018

[44] M Visser ldquoZipfrsquos law power laws and maximum entropyrdquoMathematical Inequalities amp Applications vol 15 Article ID043021 2013

[45] J Jaksetic D Pecaric and J Pecaric ldquoHybrid zipf-mandelbrotlawrdquo Mathematical Inequalities amp Applications vol 13 no 1pp 275ndash286 2019

[46] D V Widder ldquoCompletely convex functions and Lidstoneseriesrdquo Transactions of the American Mathematical Societyvol 51 p 387 1942

Journal of Mathematics 17

Ψ(1113957ζ)le1113938

b

aΨ(ζ)dλ(ζ)

1113938b

adλ(ζ)

+ 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

times Ψ(ℓ+1)(α) (1113957ζ minus α)

(ℓ+2)minus

1113938b

a(ζ minus α)

(ℓ+2)dλ(ζ)

1113938b

adλ(ζ)

⎛⎝ ⎞⎠ minus Ψ(ℓ+1)(β) (1113957ζ minus β)

(ℓ+2)+

1113938b

a(ζ minus β)

(ℓ+2)dλ(ζ)

1113938b

adλ(ζ)

⎛⎝ ⎞⎠⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

(19)

If the inequality (18) holds in reverse direction then (19)also holds reversely

e special case of above corollary can be given in theform of following remark

Remark 3 It is interesting to see that substituting λ(ζ) ζgives 1113938

b

adλ(ζ) b minus a and 1113957ζ a + b2 Using these substi-

tutions in (2) and by following remark (20) we get the LHSinequality of renowned HermitendashHadamard inequality fornminusconvex functions

22 New Generalization of Converse of Jensenrsquos IntegralInequality In this section we give the results for the

converse of Jensenrsquos inequality to hold giving the conditionson the real Stieltjes measure dλ such that λ(a)ne λ(b)allowing that the measure can also be negative butemploying Montgomery identity

To start with we need the following assumption for theresults of this section

A3 Let mM isin [α β](mneM) be such thatmle Z(ζ)leM for all ζ isin [a b] where Z is defined in A1

For a given function Ψ [α β]⟶ R we consider thedifference

CJ Ψ Z mM λ1113872 1113873 1113938

b

aΨ(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

minusM minus 1113957Z

M minus mΨ(m) minus

1113957Z minus mM minus mΨ(M) (20)

where 1113957Z is defined in (3)Using Montgomery identity we obtain the following

representation of the converse of Jensenrsquos inequality

Theorem 5 Let Z λ be as defined in A1 A2 and letΨ [α β]⟶ R be such that for nge 1 Ψ(nminus1) is absolutelycontinuous If Ψ is nminusconvex such that

CJ Rn(x s) Z mM λ1113872 1113873le 0 s isin [α β] (21)

or

1113938b

aRn(Z(ζ) s)dλ(ζ)

1113938b

adλ(ζ)

leM minus 1113957Z

M minus mRn(m s)( 1113857 +

1113957Z minus mM minus m

Rn(M s)( 1113857 s isin [α β] (22)

then we get the following extension of the converse of Jensenrsquosdifference

1113938b

aΨ(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

leM minus 1113957Z

M minus mΨ(m) +

1113957Z minus mM minus mΨ(M) + 1113944

nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889

times Ψ(ℓ+1)(α)CJ (x minus α)

ℓ+2 Z mM λ1113872 1113873 minus Ψ(ℓ+1)

(β)CJ (x minus β)ℓ+2

Z mM λ1113872 11138731113872 1113873

(23)

where Rn(middot s) is defined in (5)

Journal of Mathematics 5

Proof AsΨ(nminus1) is absolutely continuous for (nge 1) we canuse the representation of Ψ using Montgomery identity (4)in the difference CJ(Ψ Z mM λ)

CJ Ψ Z mM λ1113872 1113873 CJ1

β minus α1113946β

αΨ(ζ)dζ Z mM λ1113888 1113889

+ 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889Ψ(ℓ+1)(α)CJ (x minus α)

ℓ+2 Z mM λ1113872 1113873

minus 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889Ψ(ℓ+1)(β)CJ (x minus β)

ℓ+2 Z mM λ1113872 1113873 +

1(n minus 1)

1113946β

αCJ Rn(x s) Z mM λ1113872 1113873Ψ(n)

(s)ds

(24)

After simplification and following the fact thatCJ(Ψ Z mM λ) is zero for Ψ to be constant or linear we getthe following generalized identity

CJ Ψ Z mM λ1113872 1113873j 1113944nminus2

ℓ0

ℓ(ℓ + 2)(β minus α)1113888 1113889

times Ψ(ℓ+1)(α)CJ (x minus α)

ℓ+2 Z mM λ1113872 1113873 minus Ψ(ℓ+1)

(β)CJ (x minus β)ℓ+2

Z mM λ1113872 11138731113872 1113873

+1

(n minus 1)1113946β

αCJ Rn(x s) Z mM λ1113872 1113873Ψ(n)

(s)ds(CGI1)

(25)

Now using characterizations of nminusconvex functions likein the proof of eorem 3 we get (23)

e next result gives converse of Jensenrsquos inequality forhigher-order convex functions

Theorem 6 Let Ψ defined in =eorem 5 be nminusconvex and Z

be as defined in A3 =en the following results hold

(i) If λ is non-negative measure on [a b] then for evennge 3 (22) is valid

(ii) Moreover if (22) is valid and the function H(middot)

defined in (12) is convex then we get the followinginequality for nminusconvex function to be valid

1113938b

aΨ(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

leM minus 1113957Z

M minus mΨ(m) minus

1113957Z minus mM minus mΨ(M) (26)

Proof e idea of the proof is similar to that of (6) but we useconverse of Jensenrsquos inequality (see [32] or [30] p 98)

23 Applications of Jensenrsquos Integral Inequality In this sec-tion we give applications of Jensenrsquos integral inequality

Another important consequence of eorem 3 is bysetting the function Z as Z(ζ) ζ gives generalized version ofL H S inequality of the HermitendashHadamard inequality

Corollary 4 Let λ [a b]⟶ R be a function of boundedvariation such that λ(a)ne λ(b) with [a b] sub [α β] and1113957ζ 1113938

b

aζdλ(ζ) 1113938

b

adλ(ζ) isin [α β] Under the assumptions of

=eorem 5 if Ψ is nminusconvex such that

1113938b

aRn(ζ s)dλ(ζ)

1113938b

adλ(ζ)

leb minus 1113957ζb minus a

Rn(a s)( 1113857 +1113957ζ minus a

b minus aRn(b s)( 1113857 s isin [α β]

(27)

then we have

6 Journal of Mathematics

1113938b

aΨ(ζ)dλ(ζ)

1113938b

adλ(ζ)

leb minus 1113957ζb minus aΨ(a) +

1113957ζ minus a

b minus aΨ(b) + 1113944

nminus2

ℓ0

ℓ(ℓ + 2)(β minus α)1113888 1113889times

Ψ(ℓ+1)(α)CJ (x minus α)

ℓ+2 id ab λ1113872 1113873 minus Ψ(ℓ+1)

(β)CJ (x minus β)ℓ+2

id ab λ1113872 11138731113872 1113873

(28)

If the inequality (27) holds in reverse direction then (28)also holds reversely

e special case of above corollary can be given in theform of following remark

Remark 4 It is interesting to see that substituting λ(ζ) ζand by following eorem 6 we get the RHS inequality ofrenowned HermitendashHadamard inequality for nminusconvexfunctions

3 Generalization of Jensenrsquos DiscreteInequality by Using Montgomery Identity

In this section we give generalizations for Jensenrsquos discreteinequality by using Montgomery identity e proofs aresimilar to those of continuous case as given in previoussection therefore we give results directly

31 Generalization of Jensenrsquos Discrete Inequality for RealWeights In discrete case we have that pJ gt 0 for allJ 1 2 m Here we give generalizations of results

allowing pJ to be negative real numbers Also with usualnotations for pJxJ(J 1 2 n) we notate

x x1 x2 xm( 1113857 and p p1 p2 pm( 1113857 (29)

to be mminustuples

Pv 1113944v

J1pJ Pv Pm minus Pvminus1 (v 1 2 m) (30)

and

x 1

Pm

1113944

m

J1pJxJ (31)

Using Montgomery identity (4) we obtain the followingrepresentations of Jensenrsquos discrete inequality

Theorem 7 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also let xJ isin [a b]sube[α β]pJ isin R(J 1 m) be such that Pm ne 0 and x isin [α β]

(i) =en the following generalized identity holds

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

Ψ(ℓ+1)(α) (x minus α)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus α1113872 1113873

ℓ+2⎛⎝ ⎞⎠ minus Ψ(ℓ+1)(β) (x minus β)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus β1113872 1113873

ℓ+2⎛⎝ ⎞⎠⎧⎨

⎩

⎫⎬

⎭

+1

(n minus 1)1113946β

αRn(x s) minus

1Pm

1113944

m

J1pJRn xJ s1113872 1113873⎡⎢⎢⎣ ⎤⎥⎥⎦Ψ(n)

(s)ds (DGI1)

(32)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

Rn(x s)le1

Pm

1113944

m

J1pJRn xJ s1113872 1113873 (33)

holds then we have the following generalized inequality

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873le 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

Ψ(ℓ+1)(α) (x minus α)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus α1113872 1113873

ℓ+2⎛⎝ ⎞⎠ minus Ψ(ℓ+1)(β) (x minus β)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus β1113872 1113873

ℓ+2⎛⎝ ⎞⎠⎧⎨

⎩

⎫⎬

⎭

(34)

Journal of Mathematics 7

If inequality (33) holds in reverse direction then (34) alsoholds reversely

Proof Similar to that of eorem 3In the later part of this section we will vary our con-

ditions on pJxJ(J 1 2 n) to obtain generalizeddiscrete variants of JensenndashSteffensen Jensenrsquos and Jen-senndashPetrovic type inequalities We start with the followinggeneralization of JensenndashSteffensen discrete inequality fornminusconvex functions

Theorem 8 Let Ψ be as defined in =eorem 7 Also let x bemonotonic nminustuple xJ isin [a b]sube[α β] and p be a realnminustuple such that

0lePv lePm (v 1 2 m minus 1) Pm gt 0 (35)

is satisfied

(i) If Ψ is nminusconvex then for even nge 3 (33) is valid(ii) Moreover if (33) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized JensenndashSteffensen discrete inequality

Ψ(x)le1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 (36)

Proof It is interesting to see that under the assumed con-ditions on tuples x and p we have that x isin [a b] Forx1 gex2 ge gexm

Pm x1 minus x( 1113857 1113944m

J2pJ x1 minus xJ1113872 1113873 1113944

m

v2xvminus1 minus xv( 1113857 Pm minus Pvminus1( 1113857ge 0

(37)

is shows that x1 ge x Also xge xn since we have

Pm x minus xm( 1113857 1113944

mminus1

J1pJ xJ minus xm1113872 1113873 1113944

mminus1

v1xv minus xvminus1( 1113857Pv ge 0

(38)

For further details see the proof of JensenndashSteffensendiscrete inequality ([24] p 57) e idea of the rest of theproof is similar to that of eorem 3 but here we employeorem 7 and JensenndashSteffensen discrete inequality

Corollary 5 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube [α β] with p being a positive nminustuple

(i) If Ψ is nminusconvex then for even nge 3 (34) is valid(ii) Moreover if (33) is valid and the function H(middot) de-

fined in (12) is convex then again we get (36) which iscalled Jensenrsquos inequality for nminusconvex functions

Proof For pJ gt 0 xJ isin [a b] (J 1 2 3 m) ensuresthat x isin [a b] So by applying classical Jensenrsquos discrete

inequality (1) and idea ofeorem 8 we will get the requiredresults

Remark 5 Under the assumptions of Corollary 5 if wechoose Pm 1 then Corollary 5 (ii) gives the followinginequality for nminusconvex functions

Ψ 1113944m

J1pJxJ

⎛⎝ ⎞⎠le 1113944m

J1pJΨ xJ1113872 1113873 (39)

Nowwe give following reverses of JensenndashSteffensen andJensen-type inequalities

Corollary 6 Let Ψ be as defined in =eorem 7 Also let x bemonotonic mminustuple xJ isin [a b]sube[α β] and p be a realmminustuple such that there exist m isin 1 2 m such that

0gePv for vltm and 0gePv for vgtm (40)

where Pm gt 0 and x isin [α β]

(i) If is nminusconvex then for even nge 3 then reverse ofinequality (33) holds

(ii) Moreover if (33) holds reversely and the function H(middot)

defined in (12) is convex then we get reverse ofgeneralized JensenndashSteffensen inequality (36) fornminusconvex functions

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ reverse of JensenndashSteffensen inequality to obtain results

In the next corollary we give explicit conditions on realtuple p such that we get reverse of classical Jensen inequality

Corollary 7 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube[α β] such that x isin [α β] Let p be a real nminustuplesuch that

0ltp1 0gep2 p3 pm 0ltPm (41)

is satisfied

(i) If Ψ is nminusconvex then for even nge 3 the reverse ofinequality (33) is valid

(ii) Also if reverse of (33) is valid and the function H(middot)

defined in (12) is convex then we get reverse of (36)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ reverse of Jensen in-equality to obtain results

In [33] (see also [30]) one can find the result which isequivalent to the JensenndashSteffensen and the reverse Jen-senndashSteffensen inequality together It is the so-calledJensenndashPetrovic inequality Here without the proof we givethe adequate corollary which uses that result e proof goesthe same way as in the previous corollaries

8 Journal of Mathematics

Corollary 8 Let Ψ be as defined in =eorem 7 and letxi isin [a b]sube[α β] be such that xm ge xmminus1 x2 lex1 Let pbe a real mminustuple with Pm 1 such that

0lePv for 1le vltm minus 1 and 0lePv for 2le vltm

(42)

is satisfied =en we get the equivalent results given in=eorem 8 (i) and (ii) respectively

Remark 6 Under the assumptions of Corollary 8 if thereexist m isin 1 2 n such that

0gePv for vltm and 0gePv for vgtm (43)

and x isin [α β] then we get the equivalent results for reverseJensenndashSteffensen inequality given in Corollary 6 (i) and(ii) respectively

Remark 7 It is interesting to see that the conditions onpJJ 1 2 m given in Corollary 8 and Remark 6 arecoming from JensenndashPetrovic inequality which becomeequivalent to conditions for pJJ 1 2 m for Jen-senndashSteffensen results given in eorem 8 and Corollary 6respectively when Pm 1

Now we give results for Jensen and its reverses fornminustuples x and p when n is an odd number

Corollary 9 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube[α β] for J 1 2 m be such that x p berealmminustuplesm 2m + 1 m isin N and 1113954x 111139362k+1

J1 pJ11139362k+1J1

pJxJ isin [α β] for all k 1 2 m If for everyk 1 2 m we have

(ilowast) p1 gt 0 p2k le 0 p2k + p2k+1 le 0 11139362kJ1pJ ge 0

11139362k+1J1 pJ gt 0

(iilowast) x2k lex2k+1 11139362k+1J1 pJ (xJ minus x2k+1)ge 0

then we have the following statements to be valid(i) If Ψ is nminusconvex then for even nge 3 the inequality

Rn(1113954x s)ge1

P2m+11113944

2m+1

J1pJRn xJ s1113872 1113873 (44)

(ii) Also if (44) is valid and the function H(middot) defined in(12) is convex then we get the following generalizedinequality

Ψ(1113954x)ge1

P2m+11113944

2m+1

J1pJΨ xJ1113872 1113873 (45)

Proof We employ the idea of the proofs of eorems 7 and8 for n odd along with inequality of Vasic and Janic[34]

Remark 8 We can also discuss the following importantcases by considering the explicit conditions given in [34]

We conclude this section by giving the following im-portant cases

(Case 1)Let the condition (ilowast) hold and the reverse inequalitiesin condition (iilowast) hold en again we can give in-equalities (44) and (45) respectively given in Corollary9(Case 2)If in case of conditions (ilowast) and (iilowast) the following arevalid(iiilowast) p1 gt 0 p2k+1 ge 0 p2k + p 2k+1 ge 0 1113936

2kJ1pJ ge 0

11139362k+1J1 pJ gt 0

(ivlowast) x2k lex2k+1 11139362kminus1J1 pJ(xJ minus x2k)le 0

then we can give reverses of inequalities (44) and (45)respectively given in Corollary 9(Case 3)Finally we can also give reverses of inequalities (44)and (45) respectively given in Corollary 9 providedthat the condition (iiilowast) holds and the reverse in-equalities in condition (ivlowast) hold

e result given in (Case 3) is type of generalization ofinequality by Szego [35]

32 Generalization of Converse JensenrsquosDiscrete Inequality forRealWeights In this section we give the results for converseof Jensenrsquos inequality in discrete case by using the Mont-gomery identity

Let xJ isin [a b]sube[α β] ane b pJ isin R(J 1 n) besuch that Pm ne 0 en we have the following difference ofconverse of Jensenrsquos inequality for Ψ [α β]⟶ R

CJdis(Ψ) 1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 minus

b minus x

b minus aΨ(a) minus

x minus a

b minus aΨ(b)

(46)

Similarly we assume the Giaccardi difference [36] givenas

Gcardi(Ψ) 1113944m

J1pJΨ xJ1113872 1113873 minus AΨ 1113944

m

J1pJxJ

⎛⎝ ⎞⎠ minus B 1113944m

J1pJ minus 1⎛⎝ ⎞⎠Ψ x0( 1113857 (47)

where

Journal of Mathematics 9

A 1113936

mJ1 pJ xJ minus x01113872 11138731113872 1113873

1113936mJ1 pJxJ minus x01113872 1113873

B 1113936

mJ1 pJxJ

1113936mJ1 pJxJ minus x01113872 1113873

and 1113944m

J1pJxJ ne x0 (48)

Theorem 9 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also letx0 xJ isin [a b]sube[α β] pJ isin R(J 1 m) be such that1113936

mJ1 pJxJ ne x0

(i) =en the following generalized identity holds

CJdis(Ψ) 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)CJdis xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)CJdis xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875

+1

(n minus 1)1113946β

αCJdis Rn xJ s1113872 11138731113872 1113873Ψ(n)

(s)ds (DCGI)

(49)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

CJdis Rn xJ s1113872 11138731113872 1113873le 0 (50)

holds then we have the following generalized inequality

CJdis(Ψ)le 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)CJdis xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)CJdis xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875 (51)

If inequality (50) holds in reverse direction then (51) alsoholds reversely

Theorem 10 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also let xJ isin [a b]sube[α β]pJ isin R(J 1 m) be such that Pm ne 0 and x isin [α β]

(i) =en the following generalized Giaccardi identityholds

Gcardi(Ψ) 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)Gcardi xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)Gcardi xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875

+1

(n minus 1)1113946β

αGcardi Rn xJ s1113872 11138731113872 1113873Ψ(n)

(s)ds (GIAGI)

(52)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

Gcardi Rn xJ s1113872 11138731113872 1113873le 0 (53)

holds then we have the following generalized Giaccardiinequality

Gcardi(Ψ)le 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)Gcardi xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)Gcardi xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875 (54)

If inequality (53) holds in reverse direction then (54) alsoholds reversely

In the later part of this section we will vary our con-ditions on pJxJ (J 1 2 m) to obtain generalized

10 Journal of Mathematics

converse discrete variants of Jensenrsquos inequality and Giac-cardi inequality for nminusconvex functions

Theorem 11 Let Ψ be as defined in =eorem 9 Also letxJ isin [a b]sube [α β] and p be a positive mminustuple

(i) If Ψ is nminusconvex then for even nge 3 (50) is valid(ii) Moreover if (50) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized converse of Jensenrsquos inequality

1Pm

1113944

m

J1pJΨ xJ1113872 1113873le

b minus x

b minus aΨ(a) +

x minus a

b minus aΨ(b) (55)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ converse of Jensenrsquosinequality (see [32] or [30] p 98) to obtain results

Finally in this section we give Giaccardi inequality forhigher-order convex functions

Theorem 12 Let Ψ be as defined in =eorem 9 Also letx0 xJ isin [a b]sube[α β] and p be a positive mminustuple such that

1113944

m

J1pJxJ nex0 and xv minus x0( 1113857 1113944

m

J1pJxJ minus xv

⎛⎝ ⎞⎠ge 0 (v 1 m) (56)

(i) If Ψ is nminusconvex then for even nge 3 (53) is valid(ii) Moreover if (53) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized Giaccardi inequality

1113944

m

J1pJΨ xJ1113872 1113873leAΨ 1113944

m

J1pJxJ

⎛⎝ ⎞⎠ + B 1113944m

J1pJ minus 1⎛⎝ ⎞⎠Ψ x0( 1113857

(57)

where A and B are defined in (47)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ Giaccardi inequality(see [36] or [37] p 11) to obtain results

33 Applications in Information =eory for Jensenrsquos DiscreteInequality Jensenrsquos inequality plays a key role in infor-mation theory to construct lower bounds for some notableinequalities but here we will use it to make connectionsbetween inequalities in information theory

Let Ψ R+⟶ R+ be a convex function and letp ≔ (p1 pm) and q ≔ (q1 qm) be positive proba-bility distributions then Ψ-divergence functional is defined(in [38]) as follows

IΨ(p q) 1113944m

J1qJΨ

pJ

qJ1113888 1113889 (58)

Horvath et al in [39] defined the generalized Csiszardivergence functional as follows

Definition 2 Let I be an interval in R and Ψ I⟶ R be afunction Also let p ≔ (p1 pm) isin Rm andq ≔ (q1 qm) isin (0infin)m such that

pJ

qJisin I J 1 m (59)

en let

1113957IΨ(p q) 1113944m

J1qJΨ

pJ

qJ1113888 1113889 (60)

In this section we write Jensenrsquos difference here that weuse in upcoming results

F p xJΨ1113872 1113873 Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 (61)

Theorem 13 Under the assumptions of =eorem 9 (ii) let(51) hold and Ψ be nminusconvex Also letp ≔ (p1 pm) inRm and q ≔ (q1 qm) isin (0infin)mthen we have the following results

Journal of Mathematics 11

1113957IΨ(p q)gePmΨ(1) minus Pm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F q

pJ

qJ (x minus α)

ℓ+21113888 11138891113890 1113891 minus Ψ(ℓ+1)

(β) F qpJ

qJ (x minus β)

ℓ+21113888 11138891113890 11138911113896 1113897

(62)

Proof From eorem 9 by following Jensenrsquos difference(61) we can rearrange (34) as

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873le 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F p xJ (x minus α)

ℓ+21113872 11138731113872 1113873 minus Ψ(ℓ+1)

(β) F p xJ (x minus β)ℓ+2

1113872 11138731113872 11138731113966 1113967

(63)

Now replace pJ with qJ and xJ with pJqJ and we get(62)

For positive n-tuple q (q1 qm) such that1113936

mJ1 qJ 1 the Shannon entropy is defined by

S(q) minus 1113944m

J1qJ ln qJ (64)

Corollary 10 Under the assumptions of =eorem 9 (ii) let(51) hold and Ψ be nminusconvex

(i) If q ≔ (q1 qm) isin (0infin)m then

1113944

m

J1qJ ln qJ lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (65)

(ii) We can get bounds for the Shannon entropy of q if wechoose q ≔ (q1 qn) to be a positive probabilitydistribution

S(q)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (66)

Proof (i) Substituting Ψ(x) ≔ minus lnx and usingp ≔ (1 1 1) in eorem 13 we get (65)

(ii) Since we have 1113936mJ1 qJ 1 by multiplying minus1 on

both sides of (65) and taking into account (64) weget (66)

e KullbackndashLeibler distance [40] between the positiveprobability distributions p (p1 pm) andq (q1 qm) is defined by

D(q p) 1113944

m

J1qJ ln

qJ

pJ

1113888 1113889 (67)

Corollary 11 Under the assumptions of Corollary 10

(i) If q ≔ (q1 qm) p ≔ (p1 pm) isin (0infin)mthen

1113944

m

J1qJ ln

qJ

pJ

1113888 1113889lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

pJ

qJ minusln(middot)1113888 1113889

⎧⎨

⎩

⎫⎬

⎭ (68)

(ii) If q ≔ (q1 qm) p ≔ (p1 pm) are positiveprobability distributions then we have

12 Journal of Mathematics

D(q p)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

βℓ+1F q

pJ

qJ minusln(middot)1113888 11138891113896 1113897 (69)

Proof

(i) UsingΨ(x) ≔ minus lnx (which isn-convex for evenn)in eorem 13 we get (68) after simplification

(ii) It is a special case of (i)

34 Results for Zipf and Hybrid ZipfndashMandelbrot EntropyOne of the basic laws in information science is Zipfrsquos law[4142] which is highly applied in linguistics Let cge 0 dgt 0and N isin 1 2 ZipfndashMandelbrot entropy can be given as

ZM(H c d) d

HNcd

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (70)

where

HNcd 1113944

N

σ1

1(σ + c)

d (71)

Consider

qJ Ψ(J N c d) 1

(J + c)dH

Ncd1113872 1113873

(72)

where Ψ(J m c d) is discrete probability distributionknown as ZipfndashMandelbrot law ZipfndashMandelbrot law hasmany application in linguistics and information sciencesSome of the recent study about ZipfndashMandelbrot law can beseen in the listed references (see [39 43]) Now we state ourresults involving entropy introduced by Mandelbrot law byestablishing the relationship with Shannon and relativeentropies

Theorem 14 Let q be ZipfndashMandelbrot law as defined in(72) with parameters cge 0 dgt 0 and N isin 1 2 and wehave

ZM(H c d) S(q)leN times 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873

⎧⎨

⎩

⎫⎬

⎭

(73)

Proof It is interesting to see that for qJ defined in (72)1113936

NJ1 qJ 1 erefore using above qJ in Shannon entropy

(64) we get Mandelbrot entropy (70)

S(q) minus 1113944

N

J1qJ ln qJ minus 1113944

N

J1

1(J + c)

dH

Ncd1113872 1113873

ln1

(J + c)dH

Ncd1113872 1113873

d

HNcd1113872 1113873

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (74)

Finally substituting this qJ 1((J + c)dHNcd) in

Corollary 10 (ii) we get the desired result Corollary 12 Let q and p be ZipfndashMandelbrot law withparameters c1 c2 isin [0infin) d1 d2 gt 0 and let

Journal of Mathematics 13

HNc1 d1

1113936Nσ1 1(σ + c1)

d1 and HNc2 d2

1113936Nσ1 1(σ + c2)

d2 Now using qJ 1(J + c1)

d1HNc1 d1

andpJ 1(J + c2)

d2HNc2 d2

in Corollary 11 (ii) the followingholds

D(q p) 1113944N

J1

1J + c1( 1113857

d1HNc1d1

lnJ + c2( 1113857

d2HNc2 d2

J + c1( 1113857d1H

mc1 d1

⎛⎝ ⎞⎠

minusZ H c1 d1( 1113857 +d2

HNc1 d1

1113944

N

J1

ln J + c2( 1113857

J + c1( 1113857d1

+ ln Hmc2d2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qJ + c1( 1113857

d1HNc1 d1

J + c2( 1113857d2H

Nc2 d2

minusln(middot)⎛⎝ ⎞⎠ minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qJ + c1( 1113857

d1HNc1d1

J + c2( 1113857d2H

Nc2d2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(75)

=e Next Result for Hybrid ZipfndashMandelbrot Entropy Fur-ther generalization of ZipfndashMandelbrot entropy is Hybrid

ZipfndashMandelbrot entropy Let N isin 1 2 cge 0ωgt 0then Hybrid ZipfndashMandelbrot entropy can be given as

1113954ZM Hlowast c dω( 1113857

1Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873 (76)

where

Hlowastcdω 1113944

N

J1

ωJ

(J + c)d (77)

Consider

qJ Ψ(J N c dω) ωJ

(J + c)dHlowastcdω

(78)

which is called Hybrid ZipfndashMandelbrot law =ere is aunified approach maximization of Shannon entropy [44]

that naturally follows the path of generalization from Zipfrsquos toHybrid Zipfrsquos law Extending this idea Jaksetic et al in [45]presented a transition from ZipfndashMandelbrot to HybridZipfndashMandelbrot law by employing maximum entropytechnique with one additional constraint It is interesting thatexamination of its densities provides some new insights ofLerchrsquos transcendent

Theorem 15 Let q be Hybrid ZipfndashMandelbrot law asdefined in (78) with parameters cge 0 dωgt 0 andN isin 1 2 and we have

1113954ZM Hlowast c dω( 1113857 S(q)leN 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠ minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠

⎧⎨

⎩

⎫⎬

⎭

(79)

14 Journal of Mathematics

Proof It is interesting to see that for qJ defined in (78)1113936

NJ1 qJ 1 erefore using above qJ in Shannon en-

tropy (64) we get Hybrid ZipfndashMandelbrot entropy (76)

S(q) minus 1113944N

J1qJ ln qJ minus 1113944

N

J1

ωJ

(J + c)dHlowastcdω

lnωJ

(J + c)dHlowastcdω

minus1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

lnωJ

(J + c)d

1113888 1113889 + ln1

Hlowastcdω

1113888 11138891113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

ln(J + c)

d

ωJ1113888 1113889 + ln H

lowastcdω1113872 11138731113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873

(80)

Finally substituting this qJ ωJ(J + c)dHlowastcdω inCorollary 10 (ii) we get the desired result

Corollary 13 Let q and p be Hybrid ZipfndashMandelbrotlaw with parameters c1 c2 isin [0infin)ω1ω2 d1 d2 gt 0 Nowusing qJ ωJ

1 (J + c1)d1Hlowastc1 d1 ω1

and pJ ωJ2

(J + c2)d2Hlowastc2 d2 ω2

in Corollary 11 (ii) the following holds

D(q p) 1113944N

J1

ωJ1

J + c1( 1113857d1Hlowastc1 d1 ω1

lnωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

ωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

⎛⎝ ⎞⎠

minus1113954ZM Hlowast c1 d1ω1( 1113857 +

1Hlowastc1 d1 ω1

1113944

N

J1

ωJ1

J + c1( 1113857d1ln

J + c2( 1113857d2

ωJ2

⎛⎝ ⎞⎠ + ln Hlowastc2 d2 ω2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎫⎪⎬

⎪⎭

(81)

Remark 9 Similarly we can give results for Shannonentropy KullbackndashLeibler distance ZipfndashMandelbrotentropy and Hybrid ZipfndashMandelbrot entropy by usinggeneralized Giaccardi inequality defined in (54) on thesame steps

4 Concluding Remarks

In this paper we gave generalization of Jensenrsquos inequality aswell as converse of Jensenrsquos inequality by using Montgomeryidentity We also formulate results for other inequalities like

JensenndashSteffensen inequality JensenndashBoas inequality andJensenndashBrunk inequality We can obtain JensenndashSteffenseninequality JensenndashBoas inequality and JensenndashBrunk in-equality by changing the assumption of Jensenrsquos inequalityAt the end we gave applications in information theory forour obtained results especially we gave results for HybridZipfndashMandelbrot entropy for our obtained results [46]

Data Availability

No data were used to support this study

Journal of Mathematics 15

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is research is supported by HEC Pakistan under NRPUProject 7906 and RUDN University Strategic AcademicLeadership Program

References

[1] X L Zhu and G H Yang ldquoJensen inequality approach tostability analysis of discrete-time systems with time-varyingdelayrdquo in Proceedings of the American Control Conferencepp 1644ndash1649 Seattle WA USA June 2008

[2] S I Butt M Klaricic Bakula ETH Pecaric and J PecaricldquoJensen-gruss inequality and its applications for the zipf-mandelbrot lawrdquo Mathematical Methods in the Applied Sci-ences vol 44 no 2 pp 1664ndash1673 2021

[3] M A Khan D Pecaric and J Pecaric ldquoNew refinement of theJensen inequality associated to certain functions with appli-cationsrdquo Journal of Inequalities and Applications vol 762020

[4] S Khan M Adil Khan S I Butt and Y-M Chu ldquoA newbound for the Jensen gap pertaining twice differentiablefunctions with applicationsrdquo Advances in Difference Equa-tions vol 1 2020

[5] H Budak andM Z Sarikaya ldquoOn generalized Ostrowski-typeinequalities for functions whose first derivatives absolutevalues are convexrdquo Turkish Journal of Mathematics vol 40pp 1193ndash1210 2016

[6] P Cerone and S S Dragomir ldquoOn some inequalities arisingfrom Montgomeryrsquos identityrdquo Journal of ComputationalAnalysis and Applications vol 5 no 4 pp 341ndash367 2003

[7] S Bernstein ldquoSur les fonctions absolument monotonesrdquo ActaMathematica vol 52 pp 1ndash66 1929

[8] V Culjak and J Pecaric ldquoInterpolation polynomials andinequalities for convex functions of higher orderrdquo ActaMathematica vol 5 pp 369ndash386 2012

[9] A M Fink ldquoBounds of the deviation of a function from itsaveragesrdquo Czechoslovak Mathematical Journal vol 42no 117 pp 289ndash310 1992

[10] J Pecaric and V Culjak ldquoInterpolation polinomials and in-equalities for convex functions of higher orderrdquo Mathe-matical Inequalities amp Applications vol 5 no 3 pp 369ndash3862002

[11] J Pecaric V Culjak and A M Fink ldquoOn some inequalitiesfor convex function of higher orderrdquo Advanced NonlinearStudies vol 6 no 2 pp 131ndash140 1999

[12] J Pecaric V Culjak andM Rogina ldquoOn some inequalities forconvex function of higher order IIrdquo Advanced NonlinearStudies vol 45 pp 281ndash294 2001

[13] J Pecaric I Peric and M Rodic-Lipanovic ldquoUniformtreatment of Jensen type inequalitiesrdquo Mathematical Reportsvol 16 no 66 pp 183ndash205 2014

[14] J Pecaric and J Peric ldquoImprovement of the Giaccardi and thepetrovic inequality and related stolarsky type meansrdquo Annalsof the University of Craiova - Mathematics and ComputerScience vol 39 no 1 pp 65ndash75 2012

[15] J Pecaric M Praljak and A Witkowski ldquoLinear operatorinequality for nminusconvex functions at a pointrdquo MathematicalInequalities amp Applications Appl vol 18 pp 1201ndash1217 2015

[16] Z Pavic ldquoRefinements of Jensenrsquos inequality for infiniteconvex combinationsrdquo Turkish Journal of Inequalities vol 2no 2 pp 44ndash53 2018

[17] R Bibi A Nosheen and J Pecaric ldquoExtended Jensenrsquos typeinequalities for diamond integrals via Taylors formulardquoTurkish Journal of Inequalities vol 3 no 1 pp 7ndash18 2019

[18] T Niaz K A Khan and J Pecaric ldquoOn refinement of Jensenrsquosinequality for 3minusconvex function at a pointrdquo Turkish Journalof Inequalities vol 4 no 1 pp 70ndash80 2020

[19] A Ekinci A O Akdemir and M E Ozdemir ldquoIntegralinequalities for different kinds of convexity via classical in-equalitiesrdquo Turkish Journal of Science vol 5 no 3 pp 305ndash313 2020

[20] Y M Chu S Talib E Set M U Awan and M A Noorldquo(p q)minus Analysis of Montgomery identity and estimates of(p q)minusbounds with applicationsrdquo Journal of Inequalities andApplications vol 2021 9 pages 2021

[21] M Z Sarikaya H Yaldrz and E Set ldquoOn fractional in-equalities via Montgomery identitiesrdquo International Journalof Open Problems in Complex Analysis vol 6 no 2 pp 36ndash432014

[22] A Bnouhachem ldquoA descent SQP alternating directionmethod for minimizing the sum of three convex functionsrdquoJournal of Nonlinear Functional Analysis vol 4 pp 469ndash4822020

[23] A O Akdemir S I Butt M Nadeem and M A RagusaldquoNew general variants of Chebyshev type inequalities viageneralized fractional integral operatorsrdquo Mathematicsvol 92 2021

[24] S I Butt M Umar S Rashid A O Akdemir and Y MingChu ldquoNew Hermite Jensen Mercer type inequalities viakminusfractional integralsrdquo Advances in Difference Equationsvol 635 2020

[25] S I Butt M Nadeem S Qaisar A O Akdemir andT Abdeljawad ldquoHermite Jensen Mercer type inequalities forconformable integrals and related resultsrdquo Advances in Dif-ference Equations vol 501 2020

[26] A Bnouhachem and X Qin ldquoAn inertial proximal Peaceman-Rachford splittingmethod with SQP regularization for convexprogrammingrdquo Journal of Nonlinear Functional Analysisvol 50 2020

[27] S Hamann ldquoMinimality conditions for convex compositefunctions and efficiency conditions in vector optimizationrdquoApplied Set-Valued Analysis and Optimization vol 1pp 221ndash229 2019

[28] A R Khan J Pecaric and M Praljak ldquoPopoviciu type in-equalities for n-convex functions via extension of Mont-gomery identitynminusconvex functions via extension ofMontgomery identityrdquo Analele Universitatii ldquoOvidiusrdquo Con-stanta-Seria Matematica vol 24 no 3 pp 161ndash188 2016

[29] R P Boas ldquoe Jensen-Steffensen inequalityrdquo ElektrotehnFak Ser Mat Fiz vol 302ndash319 pp 1ndash8 1970

[30] J Pecaric F Proschan and Y L Tong Convex FunctionsPartial Orderings and Statistical Applications Academic PressNew York NY USA 1992

[31] H D Brunk ldquoOn an inequality for convex functionsrdquo Pro-ceedings of the American Mathematical Society vol 7 no 5p 817 1956

[32] P R Beesack and J E Pecaric ldquoOn Jessenrsquos inequality forconvex functionsrdquo Journal of Mathematical Analysis andApplications vol 110 no 2 pp 536ndash552 1985

[33] R Barlow A Marshall and F Proschan ldquoSome inequalitiesfor starshaped and convex functionsrdquo Pacific Journal ofMathematics vol 29 no 1 pp 19ndash42 1969

16 Journal of Mathematics

[34] J E Pecaric ldquoOn an inequality of PMV Janicrdquo Publications delrsquoInstitut Mathematique vol 28 no 42 pp 145ndash149 1980

[35] G Szego ldquoUber eine Verallgemeinerung des DirichletschenIntegralsrdquo Mathematische Zeitschrift vol 52 no 1pp 676ndash685 1950

[36] F Giaccardi ldquoSu alcune disuguaglianierdquo Journal of Researchof the National Bureau of Standards vol 1 no 4 pp 139ndash1531955

[37] D S Mitrinovic J E Pecaric and A M Fink Classical andnew inequalities in analysis Mathematics and its Applications(East European Series) Vol 61 Kluwer Academic PublishersGroup Dordrecht Netherlands 1993

[38] I Csiszar ldquoInformation-type measures of diference ofprobability distributions and indirect observationsrdquo StudiaScience Math Hungar vol 2 pp 299ndash318 1967

[39] L Horvath D Pecaric and J Pecaric ldquoEstimations of f - andrenyi divergences by using a cyclic refinement of the Jensenrsquosinequalityrdquo Bulletin of the Malaysian Mathematical SciencesSociety vol 42 2017

[40] S Kullback Information =eory and Statistics Wiley NewYork NY USA 1959

[41] S T Piantadosi ldquoZipfrsquos word frequency law in natural lan-guage a critical review and future directionsrdquo PsychonomicBulletin and Review vol 21 no 5 pp 1112ndash1130 2014

[42] Z K Silagadze ldquoCitations and the zipfndashmandelbrot lawrdquoComplex Systems no 11 pp 487ndash499 1997

[43] J Jaksetic D Pecaric and J Pecaric ldquoSome properties of zipf-mandelbrot law and hurwitz ζ-functionrdquo Mathematical In-equalities amp Applications vol 21 no 2 pp 575ndash584 2018

[44] M Visser ldquoZipfrsquos law power laws and maximum entropyrdquoMathematical Inequalities amp Applications vol 15 Article ID043021 2013

[45] J Jaksetic D Pecaric and J Pecaric ldquoHybrid zipf-mandelbrotlawrdquo Mathematical Inequalities amp Applications vol 13 no 1pp 275ndash286 2019

[46] D V Widder ldquoCompletely convex functions and Lidstoneseriesrdquo Transactions of the American Mathematical Societyvol 51 p 387 1942

Journal of Mathematics 17

Proof AsΨ(nminus1) is absolutely continuous for (nge 1) we canuse the representation of Ψ using Montgomery identity (4)in the difference CJ(Ψ Z mM λ)

CJ Ψ Z mM λ1113872 1113873 CJ1

β minus α1113946β

αΨ(ζ)dζ Z mM λ1113888 1113889

+ 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889Ψ(ℓ+1)(α)CJ (x minus α)

ℓ+2 Z mM λ1113872 1113873

minus 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889Ψ(ℓ+1)(β)CJ (x minus β)

ℓ+2 Z mM λ1113872 1113873 +

1(n minus 1)

1113946β

αCJ Rn(x s) Z mM λ1113872 1113873Ψ(n)

(s)ds

(24)

After simplification and following the fact thatCJ(Ψ Z mM λ) is zero for Ψ to be constant or linear we getthe following generalized identity

CJ Ψ Z mM λ1113872 1113873j 1113944nminus2

ℓ0

ℓ(ℓ + 2)(β minus α)1113888 1113889

times Ψ(ℓ+1)(α)CJ (x minus α)

ℓ+2 Z mM λ1113872 1113873 minus Ψ(ℓ+1)

(β)CJ (x minus β)ℓ+2

Z mM λ1113872 11138731113872 1113873

+1

(n minus 1)1113946β

αCJ Rn(x s) Z mM λ1113872 1113873Ψ(n)

(s)ds(CGI1)

(25)

Now using characterizations of nminusconvex functions likein the proof of eorem 3 we get (23)

e next result gives converse of Jensenrsquos inequality forhigher-order convex functions

Theorem 6 Let Ψ defined in =eorem 5 be nminusconvex and Z

be as defined in A3 =en the following results hold

(i) If λ is non-negative measure on [a b] then for evennge 3 (22) is valid

(ii) Moreover if (22) is valid and the function H(middot)

defined in (12) is convex then we get the followinginequality for nminusconvex function to be valid

1113938b

aΨ(Z(ζ))dλ(ζ)

1113938b

adλ(ζ)

leM minus 1113957Z

M minus mΨ(m) minus

1113957Z minus mM minus mΨ(M) (26)

Proof e idea of the proof is similar to that of (6) but we useconverse of Jensenrsquos inequality (see [32] or [30] p 98)

23 Applications of Jensenrsquos Integral Inequality In this sec-tion we give applications of Jensenrsquos integral inequality

Another important consequence of eorem 3 is bysetting the function Z as Z(ζ) ζ gives generalized version ofL H S inequality of the HermitendashHadamard inequality

Corollary 4 Let λ [a b]⟶ R be a function of boundedvariation such that λ(a)ne λ(b) with [a b] sub [α β] and1113957ζ 1113938

b

aζdλ(ζ) 1113938

b

adλ(ζ) isin [α β] Under the assumptions of

=eorem 5 if Ψ is nminusconvex such that

1113938b

aRn(ζ s)dλ(ζ)

1113938b

adλ(ζ)

leb minus 1113957ζb minus a

Rn(a s)( 1113857 +1113957ζ minus a

b minus aRn(b s)( 1113857 s isin [α β]

(27)

then we have

6 Journal of Mathematics

1113938b

aΨ(ζ)dλ(ζ)

1113938b

adλ(ζ)

leb minus 1113957ζb minus aΨ(a) +

1113957ζ minus a

b minus aΨ(b) + 1113944

nminus2

ℓ0

ℓ(ℓ + 2)(β minus α)1113888 1113889times

Ψ(ℓ+1)(α)CJ (x minus α)

ℓ+2 id ab λ1113872 1113873 minus Ψ(ℓ+1)

(β)CJ (x minus β)ℓ+2

id ab λ1113872 11138731113872 1113873

(28)

If the inequality (27) holds in reverse direction then (28)also holds reversely

e special case of above corollary can be given in theform of following remark

Remark 4 It is interesting to see that substituting λ(ζ) ζand by following eorem 6 we get the RHS inequality ofrenowned HermitendashHadamard inequality for nminusconvexfunctions

3 Generalization of Jensenrsquos DiscreteInequality by Using Montgomery Identity

In this section we give generalizations for Jensenrsquos discreteinequality by using Montgomery identity e proofs aresimilar to those of continuous case as given in previoussection therefore we give results directly

31 Generalization of Jensenrsquos Discrete Inequality for RealWeights In discrete case we have that pJ gt 0 for allJ 1 2 m Here we give generalizations of results

allowing pJ to be negative real numbers Also with usualnotations for pJxJ(J 1 2 n) we notate

x x1 x2 xm( 1113857 and p p1 p2 pm( 1113857 (29)

to be mminustuples

Pv 1113944v

J1pJ Pv Pm minus Pvminus1 (v 1 2 m) (30)

and

x 1

Pm

1113944

m

J1pJxJ (31)

Using Montgomery identity (4) we obtain the followingrepresentations of Jensenrsquos discrete inequality

Theorem 7 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also let xJ isin [a b]sube[α β]pJ isin R(J 1 m) be such that Pm ne 0 and x isin [α β]

(i) =en the following generalized identity holds

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

Ψ(ℓ+1)(α) (x minus α)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus α1113872 1113873

ℓ+2⎛⎝ ⎞⎠ minus Ψ(ℓ+1)(β) (x minus β)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus β1113872 1113873

ℓ+2⎛⎝ ⎞⎠⎧⎨

⎩

⎫⎬

⎭

+1

(n minus 1)1113946β

αRn(x s) minus

1Pm

1113944

m

J1pJRn xJ s1113872 1113873⎡⎢⎢⎣ ⎤⎥⎥⎦Ψ(n)

(s)ds (DGI1)

(32)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

Rn(x s)le1

Pm

1113944

m

J1pJRn xJ s1113872 1113873 (33)

holds then we have the following generalized inequality

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873le 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

Ψ(ℓ+1)(α) (x minus α)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus α1113872 1113873

ℓ+2⎛⎝ ⎞⎠ minus Ψ(ℓ+1)(β) (x minus β)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus β1113872 1113873

ℓ+2⎛⎝ ⎞⎠⎧⎨

⎩

⎫⎬

⎭

(34)

Journal of Mathematics 7

If inequality (33) holds in reverse direction then (34) alsoholds reversely

Proof Similar to that of eorem 3In the later part of this section we will vary our con-

ditions on pJxJ(J 1 2 n) to obtain generalizeddiscrete variants of JensenndashSteffensen Jensenrsquos and Jen-senndashPetrovic type inequalities We start with the followinggeneralization of JensenndashSteffensen discrete inequality fornminusconvex functions

Theorem 8 Let Ψ be as defined in =eorem 7 Also let x bemonotonic nminustuple xJ isin [a b]sube[α β] and p be a realnminustuple such that

0lePv lePm (v 1 2 m minus 1) Pm gt 0 (35)

is satisfied

(i) If Ψ is nminusconvex then for even nge 3 (33) is valid(ii) Moreover if (33) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized JensenndashSteffensen discrete inequality

Ψ(x)le1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 (36)

Proof It is interesting to see that under the assumed con-ditions on tuples x and p we have that x isin [a b] Forx1 gex2 ge gexm

Pm x1 minus x( 1113857 1113944m

J2pJ x1 minus xJ1113872 1113873 1113944

m

v2xvminus1 minus xv( 1113857 Pm minus Pvminus1( 1113857ge 0

(37)

is shows that x1 ge x Also xge xn since we have

Pm x minus xm( 1113857 1113944

mminus1

J1pJ xJ minus xm1113872 1113873 1113944

mminus1

v1xv minus xvminus1( 1113857Pv ge 0

(38)

For further details see the proof of JensenndashSteffensendiscrete inequality ([24] p 57) e idea of the rest of theproof is similar to that of eorem 3 but here we employeorem 7 and JensenndashSteffensen discrete inequality

Corollary 5 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube [α β] with p being a positive nminustuple

(i) If Ψ is nminusconvex then for even nge 3 (34) is valid(ii) Moreover if (33) is valid and the function H(middot) de-

fined in (12) is convex then again we get (36) which iscalled Jensenrsquos inequality for nminusconvex functions

Proof For pJ gt 0 xJ isin [a b] (J 1 2 3 m) ensuresthat x isin [a b] So by applying classical Jensenrsquos discrete

inequality (1) and idea ofeorem 8 we will get the requiredresults

Remark 5 Under the assumptions of Corollary 5 if wechoose Pm 1 then Corollary 5 (ii) gives the followinginequality for nminusconvex functions

Ψ 1113944m

J1pJxJ

⎛⎝ ⎞⎠le 1113944m

J1pJΨ xJ1113872 1113873 (39)

Nowwe give following reverses of JensenndashSteffensen andJensen-type inequalities

Corollary 6 Let Ψ be as defined in =eorem 7 Also let x bemonotonic mminustuple xJ isin [a b]sube[α β] and p be a realmminustuple such that there exist m isin 1 2 m such that

0gePv for vltm and 0gePv for vgtm (40)

where Pm gt 0 and x isin [α β]

(i) If is nminusconvex then for even nge 3 then reverse ofinequality (33) holds

(ii) Moreover if (33) holds reversely and the function H(middot)

defined in (12) is convex then we get reverse ofgeneralized JensenndashSteffensen inequality (36) fornminusconvex functions

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ reverse of JensenndashSteffensen inequality to obtain results

In the next corollary we give explicit conditions on realtuple p such that we get reverse of classical Jensen inequality

Corollary 7 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube[α β] such that x isin [α β] Let p be a real nminustuplesuch that

0ltp1 0gep2 p3 pm 0ltPm (41)

is satisfied

(i) If Ψ is nminusconvex then for even nge 3 the reverse ofinequality (33) is valid

(ii) Also if reverse of (33) is valid and the function H(middot)

defined in (12) is convex then we get reverse of (36)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ reverse of Jensen in-equality to obtain results

In [33] (see also [30]) one can find the result which isequivalent to the JensenndashSteffensen and the reverse Jen-senndashSteffensen inequality together It is the so-calledJensenndashPetrovic inequality Here without the proof we givethe adequate corollary which uses that result e proof goesthe same way as in the previous corollaries

8 Journal of Mathematics

Corollary 8 Let Ψ be as defined in =eorem 7 and letxi isin [a b]sube[α β] be such that xm ge xmminus1 x2 lex1 Let pbe a real mminustuple with Pm 1 such that

0lePv for 1le vltm minus 1 and 0lePv for 2le vltm

(42)

is satisfied =en we get the equivalent results given in=eorem 8 (i) and (ii) respectively

Remark 6 Under the assumptions of Corollary 8 if thereexist m isin 1 2 n such that

0gePv for vltm and 0gePv for vgtm (43)

and x isin [α β] then we get the equivalent results for reverseJensenndashSteffensen inequality given in Corollary 6 (i) and(ii) respectively

Remark 7 It is interesting to see that the conditions onpJJ 1 2 m given in Corollary 8 and Remark 6 arecoming from JensenndashPetrovic inequality which becomeequivalent to conditions for pJJ 1 2 m for Jen-senndashSteffensen results given in eorem 8 and Corollary 6respectively when Pm 1

Now we give results for Jensen and its reverses fornminustuples x and p when n is an odd number

Corollary 9 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube[α β] for J 1 2 m be such that x p berealmminustuplesm 2m + 1 m isin N and 1113954x 111139362k+1

J1 pJ11139362k+1J1

pJxJ isin [α β] for all k 1 2 m If for everyk 1 2 m we have

(ilowast) p1 gt 0 p2k le 0 p2k + p2k+1 le 0 11139362kJ1pJ ge 0

11139362k+1J1 pJ gt 0

(iilowast) x2k lex2k+1 11139362k+1J1 pJ (xJ minus x2k+1)ge 0

then we have the following statements to be valid(i) If Ψ is nminusconvex then for even nge 3 the inequality

Rn(1113954x s)ge1

P2m+11113944

2m+1

J1pJRn xJ s1113872 1113873 (44)

(ii) Also if (44) is valid and the function H(middot) defined in(12) is convex then we get the following generalizedinequality

Ψ(1113954x)ge1

P2m+11113944

2m+1

J1pJΨ xJ1113872 1113873 (45)

Proof We employ the idea of the proofs of eorems 7 and8 for n odd along with inequality of Vasic and Janic[34]

Remark 8 We can also discuss the following importantcases by considering the explicit conditions given in [34]

We conclude this section by giving the following im-portant cases

(Case 1)Let the condition (ilowast) hold and the reverse inequalitiesin condition (iilowast) hold en again we can give in-equalities (44) and (45) respectively given in Corollary9(Case 2)If in case of conditions (ilowast) and (iilowast) the following arevalid(iiilowast) p1 gt 0 p2k+1 ge 0 p2k + p 2k+1 ge 0 1113936

2kJ1pJ ge 0

11139362k+1J1 pJ gt 0

(ivlowast) x2k lex2k+1 11139362kminus1J1 pJ(xJ minus x2k)le 0

then we can give reverses of inequalities (44) and (45)respectively given in Corollary 9(Case 3)Finally we can also give reverses of inequalities (44)and (45) respectively given in Corollary 9 providedthat the condition (iiilowast) holds and the reverse in-equalities in condition (ivlowast) hold

e result given in (Case 3) is type of generalization ofinequality by Szego [35]

32 Generalization of Converse JensenrsquosDiscrete Inequality forRealWeights In this section we give the results for converseof Jensenrsquos inequality in discrete case by using the Mont-gomery identity

Let xJ isin [a b]sube[α β] ane b pJ isin R(J 1 n) besuch that Pm ne 0 en we have the following difference ofconverse of Jensenrsquos inequality for Ψ [α β]⟶ R

CJdis(Ψ) 1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 minus

b minus x

b minus aΨ(a) minus

x minus a

b minus aΨ(b)

(46)

Similarly we assume the Giaccardi difference [36] givenas

Gcardi(Ψ) 1113944m

J1pJΨ xJ1113872 1113873 minus AΨ 1113944

m

J1pJxJ

⎛⎝ ⎞⎠ minus B 1113944m

J1pJ minus 1⎛⎝ ⎞⎠Ψ x0( 1113857 (47)

where

Journal of Mathematics 9

A 1113936

mJ1 pJ xJ minus x01113872 11138731113872 1113873

1113936mJ1 pJxJ minus x01113872 1113873

B 1113936

mJ1 pJxJ

1113936mJ1 pJxJ minus x01113872 1113873

and 1113944m

J1pJxJ ne x0 (48)

Theorem 9 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also letx0 xJ isin [a b]sube[α β] pJ isin R(J 1 m) be such that1113936

mJ1 pJxJ ne x0

(i) =en the following generalized identity holds

CJdis(Ψ) 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)CJdis xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)CJdis xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875

+1

(n minus 1)1113946β

αCJdis Rn xJ s1113872 11138731113872 1113873Ψ(n)

(s)ds (DCGI)

(49)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

CJdis Rn xJ s1113872 11138731113872 1113873le 0 (50)

holds then we have the following generalized inequality

CJdis(Ψ)le 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)CJdis xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)CJdis xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875 (51)

If inequality (50) holds in reverse direction then (51) alsoholds reversely

Theorem 10 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also let xJ isin [a b]sube[α β]pJ isin R(J 1 m) be such that Pm ne 0 and x isin [α β]

(i) =en the following generalized Giaccardi identityholds

Gcardi(Ψ) 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)Gcardi xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)Gcardi xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875

+1

(n minus 1)1113946β

αGcardi Rn xJ s1113872 11138731113872 1113873Ψ(n)

(s)ds (GIAGI)

(52)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

Gcardi Rn xJ s1113872 11138731113872 1113873le 0 (53)

holds then we have the following generalized Giaccardiinequality

Gcardi(Ψ)le 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)Gcardi xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)Gcardi xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875 (54)

If inequality (53) holds in reverse direction then (54) alsoholds reversely

In the later part of this section we will vary our con-ditions on pJxJ (J 1 2 m) to obtain generalized

10 Journal of Mathematics

converse discrete variants of Jensenrsquos inequality and Giac-cardi inequality for nminusconvex functions

Theorem 11 Let Ψ be as defined in =eorem 9 Also letxJ isin [a b]sube [α β] and p be a positive mminustuple

(i) If Ψ is nminusconvex then for even nge 3 (50) is valid(ii) Moreover if (50) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized converse of Jensenrsquos inequality

1Pm

1113944

m

J1pJΨ xJ1113872 1113873le

b minus x

b minus aΨ(a) +

x minus a

b minus aΨ(b) (55)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ converse of Jensenrsquosinequality (see [32] or [30] p 98) to obtain results

Finally in this section we give Giaccardi inequality forhigher-order convex functions

Theorem 12 Let Ψ be as defined in =eorem 9 Also letx0 xJ isin [a b]sube[α β] and p be a positive mminustuple such that

1113944

m

J1pJxJ nex0 and xv minus x0( 1113857 1113944

m

J1pJxJ minus xv

⎛⎝ ⎞⎠ge 0 (v 1 m) (56)

(i) If Ψ is nminusconvex then for even nge 3 (53) is valid(ii) Moreover if (53) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized Giaccardi inequality

1113944

m

J1pJΨ xJ1113872 1113873leAΨ 1113944

m

J1pJxJ

⎛⎝ ⎞⎠ + B 1113944m

J1pJ minus 1⎛⎝ ⎞⎠Ψ x0( 1113857

(57)

where A and B are defined in (47)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ Giaccardi inequality(see [36] or [37] p 11) to obtain results

33 Applications in Information =eory for Jensenrsquos DiscreteInequality Jensenrsquos inequality plays a key role in infor-mation theory to construct lower bounds for some notableinequalities but here we will use it to make connectionsbetween inequalities in information theory

Let Ψ R+⟶ R+ be a convex function and letp ≔ (p1 pm) and q ≔ (q1 qm) be positive proba-bility distributions then Ψ-divergence functional is defined(in [38]) as follows

IΨ(p q) 1113944m

J1qJΨ

pJ

qJ1113888 1113889 (58)

Horvath et al in [39] defined the generalized Csiszardivergence functional as follows

Definition 2 Let I be an interval in R and Ψ I⟶ R be afunction Also let p ≔ (p1 pm) isin Rm andq ≔ (q1 qm) isin (0infin)m such that

pJ

qJisin I J 1 m (59)

en let

1113957IΨ(p q) 1113944m

J1qJΨ

pJ

qJ1113888 1113889 (60)

In this section we write Jensenrsquos difference here that weuse in upcoming results

F p xJΨ1113872 1113873 Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 (61)

Theorem 13 Under the assumptions of =eorem 9 (ii) let(51) hold and Ψ be nminusconvex Also letp ≔ (p1 pm) inRm and q ≔ (q1 qm) isin (0infin)mthen we have the following results

Journal of Mathematics 11

1113957IΨ(p q)gePmΨ(1) minus Pm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F q

pJ

qJ (x minus α)

ℓ+21113888 11138891113890 1113891 minus Ψ(ℓ+1)

(β) F qpJ

qJ (x minus β)

ℓ+21113888 11138891113890 11138911113896 1113897

(62)

Proof From eorem 9 by following Jensenrsquos difference(61) we can rearrange (34) as

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873le 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F p xJ (x minus α)

ℓ+21113872 11138731113872 1113873 minus Ψ(ℓ+1)

(β) F p xJ (x minus β)ℓ+2

1113872 11138731113872 11138731113966 1113967

(63)

Now replace pJ with qJ and xJ with pJqJ and we get(62)

For positive n-tuple q (q1 qm) such that1113936

mJ1 qJ 1 the Shannon entropy is defined by

S(q) minus 1113944m

J1qJ ln qJ (64)

Corollary 10 Under the assumptions of =eorem 9 (ii) let(51) hold and Ψ be nminusconvex

(i) If q ≔ (q1 qm) isin (0infin)m then

1113944

m

J1qJ ln qJ lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (65)

(ii) We can get bounds for the Shannon entropy of q if wechoose q ≔ (q1 qn) to be a positive probabilitydistribution

S(q)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (66)

Proof (i) Substituting Ψ(x) ≔ minus lnx and usingp ≔ (1 1 1) in eorem 13 we get (65)

(ii) Since we have 1113936mJ1 qJ 1 by multiplying minus1 on

both sides of (65) and taking into account (64) weget (66)

e KullbackndashLeibler distance [40] between the positiveprobability distributions p (p1 pm) andq (q1 qm) is defined by

D(q p) 1113944

m

J1qJ ln

qJ

pJ

1113888 1113889 (67)

Corollary 11 Under the assumptions of Corollary 10

(i) If q ≔ (q1 qm) p ≔ (p1 pm) isin (0infin)mthen

1113944

m

J1qJ ln

qJ

pJ

1113888 1113889lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

pJ

qJ minusln(middot)1113888 1113889

⎧⎨

⎩

⎫⎬

⎭ (68)

(ii) If q ≔ (q1 qm) p ≔ (p1 pm) are positiveprobability distributions then we have

12 Journal of Mathematics

D(q p)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

βℓ+1F q

pJ

qJ minusln(middot)1113888 11138891113896 1113897 (69)

Proof

(i) UsingΨ(x) ≔ minus lnx (which isn-convex for evenn)in eorem 13 we get (68) after simplification

(ii) It is a special case of (i)

34 Results for Zipf and Hybrid ZipfndashMandelbrot EntropyOne of the basic laws in information science is Zipfrsquos law[4142] which is highly applied in linguistics Let cge 0 dgt 0and N isin 1 2 ZipfndashMandelbrot entropy can be given as

ZM(H c d) d

HNcd

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (70)

where

HNcd 1113944

N

σ1

1(σ + c)

d (71)

Consider

qJ Ψ(J N c d) 1

(J + c)dH

Ncd1113872 1113873

(72)

where Ψ(J m c d) is discrete probability distributionknown as ZipfndashMandelbrot law ZipfndashMandelbrot law hasmany application in linguistics and information sciencesSome of the recent study about ZipfndashMandelbrot law can beseen in the listed references (see [39 43]) Now we state ourresults involving entropy introduced by Mandelbrot law byestablishing the relationship with Shannon and relativeentropies

Theorem 14 Let q be ZipfndashMandelbrot law as defined in(72) with parameters cge 0 dgt 0 and N isin 1 2 and wehave

ZM(H c d) S(q)leN times 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873

⎧⎨

⎩

⎫⎬

⎭

(73)

Proof It is interesting to see that for qJ defined in (72)1113936

NJ1 qJ 1 erefore using above qJ in Shannon entropy

(64) we get Mandelbrot entropy (70)

S(q) minus 1113944

N

J1qJ ln qJ minus 1113944

N

J1

1(J + c)

dH

Ncd1113872 1113873

ln1

(J + c)dH

Ncd1113872 1113873

d

HNcd1113872 1113873

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (74)

Finally substituting this qJ 1((J + c)dHNcd) in

Corollary 10 (ii) we get the desired result Corollary 12 Let q and p be ZipfndashMandelbrot law withparameters c1 c2 isin [0infin) d1 d2 gt 0 and let

Journal of Mathematics 13

HNc1 d1

1113936Nσ1 1(σ + c1)

d1 and HNc2 d2

1113936Nσ1 1(σ + c2)

d2 Now using qJ 1(J + c1)

d1HNc1 d1

andpJ 1(J + c2)

d2HNc2 d2

in Corollary 11 (ii) the followingholds

D(q p) 1113944N

J1

1J + c1( 1113857

d1HNc1d1

lnJ + c2( 1113857

d2HNc2 d2

J + c1( 1113857d1H

mc1 d1

⎛⎝ ⎞⎠

minusZ H c1 d1( 1113857 +d2

HNc1 d1

1113944

N

J1

ln J + c2( 1113857

J + c1( 1113857d1

+ ln Hmc2d2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qJ + c1( 1113857

d1HNc1 d1

J + c2( 1113857d2H

Nc2 d2

minusln(middot)⎛⎝ ⎞⎠ minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qJ + c1( 1113857

d1HNc1d1

J + c2( 1113857d2H

Nc2d2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(75)

=e Next Result for Hybrid ZipfndashMandelbrot Entropy Fur-ther generalization of ZipfndashMandelbrot entropy is Hybrid

ZipfndashMandelbrot entropy Let N isin 1 2 cge 0ωgt 0then Hybrid ZipfndashMandelbrot entropy can be given as

1113954ZM Hlowast c dω( 1113857

1Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873 (76)

where

Hlowastcdω 1113944

N

J1

ωJ

(J + c)d (77)

Consider

qJ Ψ(J N c dω) ωJ

(J + c)dHlowastcdω

(78)

which is called Hybrid ZipfndashMandelbrot law =ere is aunified approach maximization of Shannon entropy [44]

that naturally follows the path of generalization from Zipfrsquos toHybrid Zipfrsquos law Extending this idea Jaksetic et al in [45]presented a transition from ZipfndashMandelbrot to HybridZipfndashMandelbrot law by employing maximum entropytechnique with one additional constraint It is interesting thatexamination of its densities provides some new insights ofLerchrsquos transcendent

Theorem 15 Let q be Hybrid ZipfndashMandelbrot law asdefined in (78) with parameters cge 0 dωgt 0 andN isin 1 2 and we have

1113954ZM Hlowast c dω( 1113857 S(q)leN 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠ minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠

⎧⎨

⎩

⎫⎬

⎭

(79)

14 Journal of Mathematics

Proof It is interesting to see that for qJ defined in (78)1113936

NJ1 qJ 1 erefore using above qJ in Shannon en-

tropy (64) we get Hybrid ZipfndashMandelbrot entropy (76)

S(q) minus 1113944N

J1qJ ln qJ minus 1113944

N

J1

ωJ

(J + c)dHlowastcdω

lnωJ

(J + c)dHlowastcdω

minus1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

lnωJ

(J + c)d

1113888 1113889 + ln1

Hlowastcdω

1113888 11138891113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

ln(J + c)

d

ωJ1113888 1113889 + ln H

lowastcdω1113872 11138731113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873

(80)

Finally substituting this qJ ωJ(J + c)dHlowastcdω inCorollary 10 (ii) we get the desired result

Corollary 13 Let q and p be Hybrid ZipfndashMandelbrotlaw with parameters c1 c2 isin [0infin)ω1ω2 d1 d2 gt 0 Nowusing qJ ωJ

1 (J + c1)d1Hlowastc1 d1 ω1

and pJ ωJ2

(J + c2)d2Hlowastc2 d2 ω2

in Corollary 11 (ii) the following holds

D(q p) 1113944N

J1

ωJ1

J + c1( 1113857d1Hlowastc1 d1 ω1

lnωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

ωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

⎛⎝ ⎞⎠

minus1113954ZM Hlowast c1 d1ω1( 1113857 +

1Hlowastc1 d1 ω1

1113944

N

J1

ωJ1

J + c1( 1113857d1ln

J + c2( 1113857d2

ωJ2

⎛⎝ ⎞⎠ + ln Hlowastc2 d2 ω2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎫⎪⎬

⎪⎭

(81)

Remark 9 Similarly we can give results for Shannonentropy KullbackndashLeibler distance ZipfndashMandelbrotentropy and Hybrid ZipfndashMandelbrot entropy by usinggeneralized Giaccardi inequality defined in (54) on thesame steps

4 Concluding Remarks

In this paper we gave generalization of Jensenrsquos inequality aswell as converse of Jensenrsquos inequality by using Montgomeryidentity We also formulate results for other inequalities like

JensenndashSteffensen inequality JensenndashBoas inequality andJensenndashBrunk inequality We can obtain JensenndashSteffenseninequality JensenndashBoas inequality and JensenndashBrunk in-equality by changing the assumption of Jensenrsquos inequalityAt the end we gave applications in information theory forour obtained results especially we gave results for HybridZipfndashMandelbrot entropy for our obtained results [46]

Data Availability

No data were used to support this study

Journal of Mathematics 15

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is research is supported by HEC Pakistan under NRPUProject 7906 and RUDN University Strategic AcademicLeadership Program

References

[1] X L Zhu and G H Yang ldquoJensen inequality approach tostability analysis of discrete-time systems with time-varyingdelayrdquo in Proceedings of the American Control Conferencepp 1644ndash1649 Seattle WA USA June 2008

[2] S I Butt M Klaricic Bakula ETH Pecaric and J PecaricldquoJensen-gruss inequality and its applications for the zipf-mandelbrot lawrdquo Mathematical Methods in the Applied Sci-ences vol 44 no 2 pp 1664ndash1673 2021

[3] M A Khan D Pecaric and J Pecaric ldquoNew refinement of theJensen inequality associated to certain functions with appli-cationsrdquo Journal of Inequalities and Applications vol 762020

[4] S Khan M Adil Khan S I Butt and Y-M Chu ldquoA newbound for the Jensen gap pertaining twice differentiablefunctions with applicationsrdquo Advances in Difference Equa-tions vol 1 2020

[5] H Budak andM Z Sarikaya ldquoOn generalized Ostrowski-typeinequalities for functions whose first derivatives absolutevalues are convexrdquo Turkish Journal of Mathematics vol 40pp 1193ndash1210 2016

[6] P Cerone and S S Dragomir ldquoOn some inequalities arisingfrom Montgomeryrsquos identityrdquo Journal of ComputationalAnalysis and Applications vol 5 no 4 pp 341ndash367 2003

[7] S Bernstein ldquoSur les fonctions absolument monotonesrdquo ActaMathematica vol 52 pp 1ndash66 1929

[8] V Culjak and J Pecaric ldquoInterpolation polynomials andinequalities for convex functions of higher orderrdquo ActaMathematica vol 5 pp 369ndash386 2012

[9] A M Fink ldquoBounds of the deviation of a function from itsaveragesrdquo Czechoslovak Mathematical Journal vol 42no 117 pp 289ndash310 1992

[10] J Pecaric and V Culjak ldquoInterpolation polinomials and in-equalities for convex functions of higher orderrdquo Mathe-matical Inequalities amp Applications vol 5 no 3 pp 369ndash3862002

[11] J Pecaric V Culjak and A M Fink ldquoOn some inequalitiesfor convex function of higher orderrdquo Advanced NonlinearStudies vol 6 no 2 pp 131ndash140 1999

[12] J Pecaric V Culjak andM Rogina ldquoOn some inequalities forconvex function of higher order IIrdquo Advanced NonlinearStudies vol 45 pp 281ndash294 2001

[13] J Pecaric I Peric and M Rodic-Lipanovic ldquoUniformtreatment of Jensen type inequalitiesrdquo Mathematical Reportsvol 16 no 66 pp 183ndash205 2014

[14] J Pecaric and J Peric ldquoImprovement of the Giaccardi and thepetrovic inequality and related stolarsky type meansrdquo Annalsof the University of Craiova - Mathematics and ComputerScience vol 39 no 1 pp 65ndash75 2012

[15] J Pecaric M Praljak and A Witkowski ldquoLinear operatorinequality for nminusconvex functions at a pointrdquo MathematicalInequalities amp Applications Appl vol 18 pp 1201ndash1217 2015

[16] Z Pavic ldquoRefinements of Jensenrsquos inequality for infiniteconvex combinationsrdquo Turkish Journal of Inequalities vol 2no 2 pp 44ndash53 2018

[17] R Bibi A Nosheen and J Pecaric ldquoExtended Jensenrsquos typeinequalities for diamond integrals via Taylors formulardquoTurkish Journal of Inequalities vol 3 no 1 pp 7ndash18 2019

[18] T Niaz K A Khan and J Pecaric ldquoOn refinement of Jensenrsquosinequality for 3minusconvex function at a pointrdquo Turkish Journalof Inequalities vol 4 no 1 pp 70ndash80 2020

[19] A Ekinci A O Akdemir and M E Ozdemir ldquoIntegralinequalities for different kinds of convexity via classical in-equalitiesrdquo Turkish Journal of Science vol 5 no 3 pp 305ndash313 2020

[20] Y M Chu S Talib E Set M U Awan and M A Noorldquo(p q)minus Analysis of Montgomery identity and estimates of(p q)minusbounds with applicationsrdquo Journal of Inequalities andApplications vol 2021 9 pages 2021

[21] M Z Sarikaya H Yaldrz and E Set ldquoOn fractional in-equalities via Montgomery identitiesrdquo International Journalof Open Problems in Complex Analysis vol 6 no 2 pp 36ndash432014

[22] A Bnouhachem ldquoA descent SQP alternating directionmethod for minimizing the sum of three convex functionsrdquoJournal of Nonlinear Functional Analysis vol 4 pp 469ndash4822020

[23] A O Akdemir S I Butt M Nadeem and M A RagusaldquoNew general variants of Chebyshev type inequalities viageneralized fractional integral operatorsrdquo Mathematicsvol 92 2021

[24] S I Butt M Umar S Rashid A O Akdemir and Y MingChu ldquoNew Hermite Jensen Mercer type inequalities viakminusfractional integralsrdquo Advances in Difference Equationsvol 635 2020

[25] S I Butt M Nadeem S Qaisar A O Akdemir andT Abdeljawad ldquoHermite Jensen Mercer type inequalities forconformable integrals and related resultsrdquo Advances in Dif-ference Equations vol 501 2020

[26] A Bnouhachem and X Qin ldquoAn inertial proximal Peaceman-Rachford splittingmethod with SQP regularization for convexprogrammingrdquo Journal of Nonlinear Functional Analysisvol 50 2020

[27] S Hamann ldquoMinimality conditions for convex compositefunctions and efficiency conditions in vector optimizationrdquoApplied Set-Valued Analysis and Optimization vol 1pp 221ndash229 2019

[28] A R Khan J Pecaric and M Praljak ldquoPopoviciu type in-equalities for n-convex functions via extension of Mont-gomery identitynminusconvex functions via extension ofMontgomery identityrdquo Analele Universitatii ldquoOvidiusrdquo Con-stanta-Seria Matematica vol 24 no 3 pp 161ndash188 2016

[29] R P Boas ldquoe Jensen-Steffensen inequalityrdquo ElektrotehnFak Ser Mat Fiz vol 302ndash319 pp 1ndash8 1970

[30] J Pecaric F Proschan and Y L Tong Convex FunctionsPartial Orderings and Statistical Applications Academic PressNew York NY USA 1992

[31] H D Brunk ldquoOn an inequality for convex functionsrdquo Pro-ceedings of the American Mathematical Society vol 7 no 5p 817 1956

[32] P R Beesack and J E Pecaric ldquoOn Jessenrsquos inequality forconvex functionsrdquo Journal of Mathematical Analysis andApplications vol 110 no 2 pp 536ndash552 1985

[33] R Barlow A Marshall and F Proschan ldquoSome inequalitiesfor starshaped and convex functionsrdquo Pacific Journal ofMathematics vol 29 no 1 pp 19ndash42 1969

16 Journal of Mathematics

[34] J E Pecaric ldquoOn an inequality of PMV Janicrdquo Publications delrsquoInstitut Mathematique vol 28 no 42 pp 145ndash149 1980

[35] G Szego ldquoUber eine Verallgemeinerung des DirichletschenIntegralsrdquo Mathematische Zeitschrift vol 52 no 1pp 676ndash685 1950

[36] F Giaccardi ldquoSu alcune disuguaglianierdquo Journal of Researchof the National Bureau of Standards vol 1 no 4 pp 139ndash1531955

[37] D S Mitrinovic J E Pecaric and A M Fink Classical andnew inequalities in analysis Mathematics and its Applications(East European Series) Vol 61 Kluwer Academic PublishersGroup Dordrecht Netherlands 1993

[38] I Csiszar ldquoInformation-type measures of diference ofprobability distributions and indirect observationsrdquo StudiaScience Math Hungar vol 2 pp 299ndash318 1967

[39] L Horvath D Pecaric and J Pecaric ldquoEstimations of f - andrenyi divergences by using a cyclic refinement of the Jensenrsquosinequalityrdquo Bulletin of the Malaysian Mathematical SciencesSociety vol 42 2017

[40] S Kullback Information =eory and Statistics Wiley NewYork NY USA 1959

[41] S T Piantadosi ldquoZipfrsquos word frequency law in natural lan-guage a critical review and future directionsrdquo PsychonomicBulletin and Review vol 21 no 5 pp 1112ndash1130 2014

[42] Z K Silagadze ldquoCitations and the zipfndashmandelbrot lawrdquoComplex Systems no 11 pp 487ndash499 1997

[43] J Jaksetic D Pecaric and J Pecaric ldquoSome properties of zipf-mandelbrot law and hurwitz ζ-functionrdquo Mathematical In-equalities amp Applications vol 21 no 2 pp 575ndash584 2018

[44] M Visser ldquoZipfrsquos law power laws and maximum entropyrdquoMathematical Inequalities amp Applications vol 15 Article ID043021 2013

[45] J Jaksetic D Pecaric and J Pecaric ldquoHybrid zipf-mandelbrotlawrdquo Mathematical Inequalities amp Applications vol 13 no 1pp 275ndash286 2019

[46] D V Widder ldquoCompletely convex functions and Lidstoneseriesrdquo Transactions of the American Mathematical Societyvol 51 p 387 1942

Journal of Mathematics 17

1113938b

aΨ(ζ)dλ(ζ)

1113938b

adλ(ζ)

leb minus 1113957ζb minus aΨ(a) +

1113957ζ minus a

b minus aΨ(b) + 1113944

nminus2

ℓ0

ℓ(ℓ + 2)(β minus α)1113888 1113889times

Ψ(ℓ+1)(α)CJ (x minus α)

ℓ+2 id ab λ1113872 1113873 minus Ψ(ℓ+1)

(β)CJ (x minus β)ℓ+2

id ab λ1113872 11138731113872 1113873

(28)

If the inequality (27) holds in reverse direction then (28)also holds reversely

e special case of above corollary can be given in theform of following remark

Remark 4 It is interesting to see that substituting λ(ζ) ζand by following eorem 6 we get the RHS inequality ofrenowned HermitendashHadamard inequality for nminusconvexfunctions

3 Generalization of Jensenrsquos DiscreteInequality by Using Montgomery Identity

In this section we give generalizations for Jensenrsquos discreteinequality by using Montgomery identity e proofs aresimilar to those of continuous case as given in previoussection therefore we give results directly

31 Generalization of Jensenrsquos Discrete Inequality for RealWeights In discrete case we have that pJ gt 0 for allJ 1 2 m Here we give generalizations of results

allowing pJ to be negative real numbers Also with usualnotations for pJxJ(J 1 2 n) we notate

x x1 x2 xm( 1113857 and p p1 p2 pm( 1113857 (29)

to be mminustuples

Pv 1113944v

J1pJ Pv Pm minus Pvminus1 (v 1 2 m) (30)

and

x 1

Pm

1113944

m

J1pJxJ (31)

Using Montgomery identity (4) we obtain the followingrepresentations of Jensenrsquos discrete inequality

Theorem 7 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also let xJ isin [a b]sube[α β]pJ isin R(J 1 m) be such that Pm ne 0 and x isin [α β]

(i) =en the following generalized identity holds

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

Ψ(ℓ+1)(α) (x minus α)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus α1113872 1113873

ℓ+2⎛⎝ ⎞⎠ minus Ψ(ℓ+1)(β) (x minus β)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus β1113872 1113873

ℓ+2⎛⎝ ⎞⎠⎧⎨

⎩

⎫⎬

⎭

+1

(n minus 1)1113946β

αRn(x s) minus

1Pm

1113944

m

J1pJRn xJ s1113872 1113873⎡⎢⎢⎣ ⎤⎥⎥⎦Ψ(n)

(s)ds (DGI1)

(32)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

Rn(x s)le1

Pm

1113944

m

J1pJRn xJ s1113872 1113873 (33)

holds then we have the following generalized inequality

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873le 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

Ψ(ℓ+1)(α) (x minus α)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus α1113872 1113873

ℓ+2⎛⎝ ⎞⎠ minus Ψ(ℓ+1)(β) (x minus β)

ℓ+2minus

1Pm

1113944

m

J1pJ xJ minus β1113872 1113873

ℓ+2⎛⎝ ⎞⎠⎧⎨

⎩

⎫⎬

⎭

(34)

Journal of Mathematics 7

If inequality (33) holds in reverse direction then (34) alsoholds reversely

Proof Similar to that of eorem 3In the later part of this section we will vary our con-

ditions on pJxJ(J 1 2 n) to obtain generalizeddiscrete variants of JensenndashSteffensen Jensenrsquos and Jen-senndashPetrovic type inequalities We start with the followinggeneralization of JensenndashSteffensen discrete inequality fornminusconvex functions

Theorem 8 Let Ψ be as defined in =eorem 7 Also let x bemonotonic nminustuple xJ isin [a b]sube[α β] and p be a realnminustuple such that

0lePv lePm (v 1 2 m minus 1) Pm gt 0 (35)

is satisfied

(i) If Ψ is nminusconvex then for even nge 3 (33) is valid(ii) Moreover if (33) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized JensenndashSteffensen discrete inequality

Ψ(x)le1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 (36)

Proof It is interesting to see that under the assumed con-ditions on tuples x and p we have that x isin [a b] Forx1 gex2 ge gexm

Pm x1 minus x( 1113857 1113944m

J2pJ x1 minus xJ1113872 1113873 1113944

m

v2xvminus1 minus xv( 1113857 Pm minus Pvminus1( 1113857ge 0

(37)

is shows that x1 ge x Also xge xn since we have

Pm x minus xm( 1113857 1113944

mminus1

J1pJ xJ minus xm1113872 1113873 1113944

mminus1

v1xv minus xvminus1( 1113857Pv ge 0

(38)

For further details see the proof of JensenndashSteffensendiscrete inequality ([24] p 57) e idea of the rest of theproof is similar to that of eorem 3 but here we employeorem 7 and JensenndashSteffensen discrete inequality

Corollary 5 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube [α β] with p being a positive nminustuple

(i) If Ψ is nminusconvex then for even nge 3 (34) is valid(ii) Moreover if (33) is valid and the function H(middot) de-

fined in (12) is convex then again we get (36) which iscalled Jensenrsquos inequality for nminusconvex functions

Proof For pJ gt 0 xJ isin [a b] (J 1 2 3 m) ensuresthat x isin [a b] So by applying classical Jensenrsquos discrete

inequality (1) and idea ofeorem 8 we will get the requiredresults

Remark 5 Under the assumptions of Corollary 5 if wechoose Pm 1 then Corollary 5 (ii) gives the followinginequality for nminusconvex functions

Ψ 1113944m

J1pJxJ

⎛⎝ ⎞⎠le 1113944m

J1pJΨ xJ1113872 1113873 (39)

Nowwe give following reverses of JensenndashSteffensen andJensen-type inequalities

Corollary 6 Let Ψ be as defined in =eorem 7 Also let x bemonotonic mminustuple xJ isin [a b]sube[α β] and p be a realmminustuple such that there exist m isin 1 2 m such that

0gePv for vltm and 0gePv for vgtm (40)

where Pm gt 0 and x isin [α β]

(i) If is nminusconvex then for even nge 3 then reverse ofinequality (33) holds

(ii) Moreover if (33) holds reversely and the function H(middot)

defined in (12) is convex then we get reverse ofgeneralized JensenndashSteffensen inequality (36) fornminusconvex functions

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ reverse of JensenndashSteffensen inequality to obtain results

In the next corollary we give explicit conditions on realtuple p such that we get reverse of classical Jensen inequality

Corollary 7 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube[α β] such that x isin [α β] Let p be a real nminustuplesuch that

0ltp1 0gep2 p3 pm 0ltPm (41)

is satisfied

(i) If Ψ is nminusconvex then for even nge 3 the reverse ofinequality (33) is valid

(ii) Also if reverse of (33) is valid and the function H(middot)

defined in (12) is convex then we get reverse of (36)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ reverse of Jensen in-equality to obtain results

In [33] (see also [30]) one can find the result which isequivalent to the JensenndashSteffensen and the reverse Jen-senndashSteffensen inequality together It is the so-calledJensenndashPetrovic inequality Here without the proof we givethe adequate corollary which uses that result e proof goesthe same way as in the previous corollaries

8 Journal of Mathematics

Corollary 8 Let Ψ be as defined in =eorem 7 and letxi isin [a b]sube[α β] be such that xm ge xmminus1 x2 lex1 Let pbe a real mminustuple with Pm 1 such that

0lePv for 1le vltm minus 1 and 0lePv for 2le vltm

(42)

is satisfied =en we get the equivalent results given in=eorem 8 (i) and (ii) respectively

Remark 6 Under the assumptions of Corollary 8 if thereexist m isin 1 2 n such that

0gePv for vltm and 0gePv for vgtm (43)

and x isin [α β] then we get the equivalent results for reverseJensenndashSteffensen inequality given in Corollary 6 (i) and(ii) respectively

Remark 7 It is interesting to see that the conditions onpJJ 1 2 m given in Corollary 8 and Remark 6 arecoming from JensenndashPetrovic inequality which becomeequivalent to conditions for pJJ 1 2 m for Jen-senndashSteffensen results given in eorem 8 and Corollary 6respectively when Pm 1

Now we give results for Jensen and its reverses fornminustuples x and p when n is an odd number

Corollary 9 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube[α β] for J 1 2 m be such that x p berealmminustuplesm 2m + 1 m isin N and 1113954x 111139362k+1

J1 pJ11139362k+1J1

pJxJ isin [α β] for all k 1 2 m If for everyk 1 2 m we have

(ilowast) p1 gt 0 p2k le 0 p2k + p2k+1 le 0 11139362kJ1pJ ge 0

11139362k+1J1 pJ gt 0

(iilowast) x2k lex2k+1 11139362k+1J1 pJ (xJ minus x2k+1)ge 0

then we have the following statements to be valid(i) If Ψ is nminusconvex then for even nge 3 the inequality

Rn(1113954x s)ge1

P2m+11113944

2m+1

J1pJRn xJ s1113872 1113873 (44)

(ii) Also if (44) is valid and the function H(middot) defined in(12) is convex then we get the following generalizedinequality

Ψ(1113954x)ge1

P2m+11113944

2m+1

J1pJΨ xJ1113872 1113873 (45)

Proof We employ the idea of the proofs of eorems 7 and8 for n odd along with inequality of Vasic and Janic[34]

Remark 8 We can also discuss the following importantcases by considering the explicit conditions given in [34]

We conclude this section by giving the following im-portant cases

(Case 1)Let the condition (ilowast) hold and the reverse inequalitiesin condition (iilowast) hold en again we can give in-equalities (44) and (45) respectively given in Corollary9(Case 2)If in case of conditions (ilowast) and (iilowast) the following arevalid(iiilowast) p1 gt 0 p2k+1 ge 0 p2k + p 2k+1 ge 0 1113936

2kJ1pJ ge 0

11139362k+1J1 pJ gt 0

(ivlowast) x2k lex2k+1 11139362kminus1J1 pJ(xJ minus x2k)le 0

then we can give reverses of inequalities (44) and (45)respectively given in Corollary 9(Case 3)Finally we can also give reverses of inequalities (44)and (45) respectively given in Corollary 9 providedthat the condition (iiilowast) holds and the reverse in-equalities in condition (ivlowast) hold

e result given in (Case 3) is type of generalization ofinequality by Szego [35]

32 Generalization of Converse JensenrsquosDiscrete Inequality forRealWeights In this section we give the results for converseof Jensenrsquos inequality in discrete case by using the Mont-gomery identity

Let xJ isin [a b]sube[α β] ane b pJ isin R(J 1 n) besuch that Pm ne 0 en we have the following difference ofconverse of Jensenrsquos inequality for Ψ [α β]⟶ R

CJdis(Ψ) 1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 minus

b minus x

b minus aΨ(a) minus

x minus a

b minus aΨ(b)

(46)

Similarly we assume the Giaccardi difference [36] givenas

Gcardi(Ψ) 1113944m

J1pJΨ xJ1113872 1113873 minus AΨ 1113944

m

J1pJxJ

⎛⎝ ⎞⎠ minus B 1113944m

J1pJ minus 1⎛⎝ ⎞⎠Ψ x0( 1113857 (47)

where

Journal of Mathematics 9

A 1113936

mJ1 pJ xJ minus x01113872 11138731113872 1113873

1113936mJ1 pJxJ minus x01113872 1113873

B 1113936

mJ1 pJxJ

1113936mJ1 pJxJ minus x01113872 1113873

and 1113944m

J1pJxJ ne x0 (48)

Theorem 9 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also letx0 xJ isin [a b]sube[α β] pJ isin R(J 1 m) be such that1113936

mJ1 pJxJ ne x0

(i) =en the following generalized identity holds

CJdis(Ψ) 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)CJdis xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)CJdis xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875

+1

(n minus 1)1113946β

αCJdis Rn xJ s1113872 11138731113872 1113873Ψ(n)

(s)ds (DCGI)

(49)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

CJdis Rn xJ s1113872 11138731113872 1113873le 0 (50)

holds then we have the following generalized inequality

CJdis(Ψ)le 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)CJdis xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)CJdis xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875 (51)

If inequality (50) holds in reverse direction then (51) alsoholds reversely

Theorem 10 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also let xJ isin [a b]sube[α β]pJ isin R(J 1 m) be such that Pm ne 0 and x isin [α β]

(i) =en the following generalized Giaccardi identityholds

Gcardi(Ψ) 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)Gcardi xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)Gcardi xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875

+1

(n minus 1)1113946β

αGcardi Rn xJ s1113872 11138731113872 1113873Ψ(n)

(s)ds (GIAGI)

(52)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

Gcardi Rn xJ s1113872 11138731113872 1113873le 0 (53)

holds then we have the following generalized Giaccardiinequality

Gcardi(Ψ)le 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)Gcardi xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)Gcardi xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875 (54)

If inequality (53) holds in reverse direction then (54) alsoholds reversely

In the later part of this section we will vary our con-ditions on pJxJ (J 1 2 m) to obtain generalized

10 Journal of Mathematics

converse discrete variants of Jensenrsquos inequality and Giac-cardi inequality for nminusconvex functions

Theorem 11 Let Ψ be as defined in =eorem 9 Also letxJ isin [a b]sube [α β] and p be a positive mminustuple

(i) If Ψ is nminusconvex then for even nge 3 (50) is valid(ii) Moreover if (50) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized converse of Jensenrsquos inequality

1Pm

1113944

m

J1pJΨ xJ1113872 1113873le

b minus x

b minus aΨ(a) +

x minus a

b minus aΨ(b) (55)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ converse of Jensenrsquosinequality (see [32] or [30] p 98) to obtain results

Finally in this section we give Giaccardi inequality forhigher-order convex functions

Theorem 12 Let Ψ be as defined in =eorem 9 Also letx0 xJ isin [a b]sube[α β] and p be a positive mminustuple such that

1113944

m

J1pJxJ nex0 and xv minus x0( 1113857 1113944

m

J1pJxJ minus xv

⎛⎝ ⎞⎠ge 0 (v 1 m) (56)

(i) If Ψ is nminusconvex then for even nge 3 (53) is valid(ii) Moreover if (53) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized Giaccardi inequality

1113944

m

J1pJΨ xJ1113872 1113873leAΨ 1113944

m

J1pJxJ

⎛⎝ ⎞⎠ + B 1113944m

J1pJ minus 1⎛⎝ ⎞⎠Ψ x0( 1113857

(57)

where A and B are defined in (47)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ Giaccardi inequality(see [36] or [37] p 11) to obtain results

33 Applications in Information =eory for Jensenrsquos DiscreteInequality Jensenrsquos inequality plays a key role in infor-mation theory to construct lower bounds for some notableinequalities but here we will use it to make connectionsbetween inequalities in information theory

Let Ψ R+⟶ R+ be a convex function and letp ≔ (p1 pm) and q ≔ (q1 qm) be positive proba-bility distributions then Ψ-divergence functional is defined(in [38]) as follows

IΨ(p q) 1113944m

J1qJΨ

pJ

qJ1113888 1113889 (58)

Horvath et al in [39] defined the generalized Csiszardivergence functional as follows

Definition 2 Let I be an interval in R and Ψ I⟶ R be afunction Also let p ≔ (p1 pm) isin Rm andq ≔ (q1 qm) isin (0infin)m such that

pJ

qJisin I J 1 m (59)

en let

1113957IΨ(p q) 1113944m

J1qJΨ

pJ

qJ1113888 1113889 (60)

In this section we write Jensenrsquos difference here that weuse in upcoming results

F p xJΨ1113872 1113873 Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 (61)

Theorem 13 Under the assumptions of =eorem 9 (ii) let(51) hold and Ψ be nminusconvex Also letp ≔ (p1 pm) inRm and q ≔ (q1 qm) isin (0infin)mthen we have the following results

Journal of Mathematics 11

1113957IΨ(p q)gePmΨ(1) minus Pm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F q

pJ

qJ (x minus α)

ℓ+21113888 11138891113890 1113891 minus Ψ(ℓ+1)

(β) F qpJ

qJ (x minus β)

ℓ+21113888 11138891113890 11138911113896 1113897

(62)

Proof From eorem 9 by following Jensenrsquos difference(61) we can rearrange (34) as

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873le 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F p xJ (x minus α)

ℓ+21113872 11138731113872 1113873 minus Ψ(ℓ+1)

(β) F p xJ (x minus β)ℓ+2

1113872 11138731113872 11138731113966 1113967

(63)

Now replace pJ with qJ and xJ with pJqJ and we get(62)

For positive n-tuple q (q1 qm) such that1113936

mJ1 qJ 1 the Shannon entropy is defined by

S(q) minus 1113944m

J1qJ ln qJ (64)

Corollary 10 Under the assumptions of =eorem 9 (ii) let(51) hold and Ψ be nminusconvex

(i) If q ≔ (q1 qm) isin (0infin)m then

1113944

m

J1qJ ln qJ lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (65)

(ii) We can get bounds for the Shannon entropy of q if wechoose q ≔ (q1 qn) to be a positive probabilitydistribution

S(q)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (66)

Proof (i) Substituting Ψ(x) ≔ minus lnx and usingp ≔ (1 1 1) in eorem 13 we get (65)

(ii) Since we have 1113936mJ1 qJ 1 by multiplying minus1 on

both sides of (65) and taking into account (64) weget (66)

e KullbackndashLeibler distance [40] between the positiveprobability distributions p (p1 pm) andq (q1 qm) is defined by

D(q p) 1113944

m

J1qJ ln

qJ

pJ

1113888 1113889 (67)

Corollary 11 Under the assumptions of Corollary 10

(i) If q ≔ (q1 qm) p ≔ (p1 pm) isin (0infin)mthen

1113944

m

J1qJ ln

qJ

pJ

1113888 1113889lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

pJ

qJ minusln(middot)1113888 1113889

⎧⎨

⎩

⎫⎬

⎭ (68)

(ii) If q ≔ (q1 qm) p ≔ (p1 pm) are positiveprobability distributions then we have

12 Journal of Mathematics

D(q p)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

βℓ+1F q

pJ

qJ minusln(middot)1113888 11138891113896 1113897 (69)

Proof

(i) UsingΨ(x) ≔ minus lnx (which isn-convex for evenn)in eorem 13 we get (68) after simplification

(ii) It is a special case of (i)

34 Results for Zipf and Hybrid ZipfndashMandelbrot EntropyOne of the basic laws in information science is Zipfrsquos law[4142] which is highly applied in linguistics Let cge 0 dgt 0and N isin 1 2 ZipfndashMandelbrot entropy can be given as

ZM(H c d) d

HNcd

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (70)

where

HNcd 1113944

N

σ1

1(σ + c)

d (71)

Consider

qJ Ψ(J N c d) 1

(J + c)dH

Ncd1113872 1113873

(72)

where Ψ(J m c d) is discrete probability distributionknown as ZipfndashMandelbrot law ZipfndashMandelbrot law hasmany application in linguistics and information sciencesSome of the recent study about ZipfndashMandelbrot law can beseen in the listed references (see [39 43]) Now we state ourresults involving entropy introduced by Mandelbrot law byestablishing the relationship with Shannon and relativeentropies

Theorem 14 Let q be ZipfndashMandelbrot law as defined in(72) with parameters cge 0 dgt 0 and N isin 1 2 and wehave

ZM(H c d) S(q)leN times 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873

⎧⎨

⎩

⎫⎬

⎭

(73)

Proof It is interesting to see that for qJ defined in (72)1113936

NJ1 qJ 1 erefore using above qJ in Shannon entropy

(64) we get Mandelbrot entropy (70)

S(q) minus 1113944

N

J1qJ ln qJ minus 1113944

N

J1

1(J + c)

dH

Ncd1113872 1113873

ln1

(J + c)dH

Ncd1113872 1113873

d

HNcd1113872 1113873

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (74)

Finally substituting this qJ 1((J + c)dHNcd) in

Corollary 10 (ii) we get the desired result Corollary 12 Let q and p be ZipfndashMandelbrot law withparameters c1 c2 isin [0infin) d1 d2 gt 0 and let

Journal of Mathematics 13

HNc1 d1

1113936Nσ1 1(σ + c1)

d1 and HNc2 d2

1113936Nσ1 1(σ + c2)

d2 Now using qJ 1(J + c1)

d1HNc1 d1

andpJ 1(J + c2)

d2HNc2 d2

in Corollary 11 (ii) the followingholds

D(q p) 1113944N

J1

1J + c1( 1113857

d1HNc1d1

lnJ + c2( 1113857

d2HNc2 d2

J + c1( 1113857d1H

mc1 d1

⎛⎝ ⎞⎠

minusZ H c1 d1( 1113857 +d2

HNc1 d1

1113944

N

J1

ln J + c2( 1113857

J + c1( 1113857d1

+ ln Hmc2d2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qJ + c1( 1113857

d1HNc1 d1

J + c2( 1113857d2H

Nc2 d2

minusln(middot)⎛⎝ ⎞⎠ minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qJ + c1( 1113857

d1HNc1d1

J + c2( 1113857d2H

Nc2d2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(75)

=e Next Result for Hybrid ZipfndashMandelbrot Entropy Fur-ther generalization of ZipfndashMandelbrot entropy is Hybrid

ZipfndashMandelbrot entropy Let N isin 1 2 cge 0ωgt 0then Hybrid ZipfndashMandelbrot entropy can be given as

1113954ZM Hlowast c dω( 1113857

1Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873 (76)

where

Hlowastcdω 1113944

N

J1

ωJ

(J + c)d (77)

Consider

qJ Ψ(J N c dω) ωJ

(J + c)dHlowastcdω

(78)

which is called Hybrid ZipfndashMandelbrot law =ere is aunified approach maximization of Shannon entropy [44]

that naturally follows the path of generalization from Zipfrsquos toHybrid Zipfrsquos law Extending this idea Jaksetic et al in [45]presented a transition from ZipfndashMandelbrot to HybridZipfndashMandelbrot law by employing maximum entropytechnique with one additional constraint It is interesting thatexamination of its densities provides some new insights ofLerchrsquos transcendent

Theorem 15 Let q be Hybrid ZipfndashMandelbrot law asdefined in (78) with parameters cge 0 dωgt 0 andN isin 1 2 and we have

1113954ZM Hlowast c dω( 1113857 S(q)leN 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠ minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠

⎧⎨

⎩

⎫⎬

⎭

(79)

14 Journal of Mathematics

Proof It is interesting to see that for qJ defined in (78)1113936

NJ1 qJ 1 erefore using above qJ in Shannon en-

tropy (64) we get Hybrid ZipfndashMandelbrot entropy (76)

S(q) minus 1113944N

J1qJ ln qJ minus 1113944

N

J1

ωJ

(J + c)dHlowastcdω

lnωJ

(J + c)dHlowastcdω

minus1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

lnωJ

(J + c)d

1113888 1113889 + ln1

Hlowastcdω

1113888 11138891113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

ln(J + c)

d

ωJ1113888 1113889 + ln H

lowastcdω1113872 11138731113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873

(80)

Finally substituting this qJ ωJ(J + c)dHlowastcdω inCorollary 10 (ii) we get the desired result

Corollary 13 Let q and p be Hybrid ZipfndashMandelbrotlaw with parameters c1 c2 isin [0infin)ω1ω2 d1 d2 gt 0 Nowusing qJ ωJ

1 (J + c1)d1Hlowastc1 d1 ω1

and pJ ωJ2

(J + c2)d2Hlowastc2 d2 ω2

in Corollary 11 (ii) the following holds

D(q p) 1113944N

J1

ωJ1

J + c1( 1113857d1Hlowastc1 d1 ω1

lnωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

ωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

⎛⎝ ⎞⎠

minus1113954ZM Hlowast c1 d1ω1( 1113857 +

1Hlowastc1 d1 ω1

1113944

N

J1

ωJ1

J + c1( 1113857d1ln

J + c2( 1113857d2

ωJ2

⎛⎝ ⎞⎠ + ln Hlowastc2 d2 ω2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎫⎪⎬

⎪⎭

(81)

Remark 9 Similarly we can give results for Shannonentropy KullbackndashLeibler distance ZipfndashMandelbrotentropy and Hybrid ZipfndashMandelbrot entropy by usinggeneralized Giaccardi inequality defined in (54) on thesame steps

4 Concluding Remarks

In this paper we gave generalization of Jensenrsquos inequality aswell as converse of Jensenrsquos inequality by using Montgomeryidentity We also formulate results for other inequalities like

JensenndashSteffensen inequality JensenndashBoas inequality andJensenndashBrunk inequality We can obtain JensenndashSteffenseninequality JensenndashBoas inequality and JensenndashBrunk in-equality by changing the assumption of Jensenrsquos inequalityAt the end we gave applications in information theory forour obtained results especially we gave results for HybridZipfndashMandelbrot entropy for our obtained results [46]

Data Availability

No data were used to support this study

Journal of Mathematics 15

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is research is supported by HEC Pakistan under NRPUProject 7906 and RUDN University Strategic AcademicLeadership Program

References

[1] X L Zhu and G H Yang ldquoJensen inequality approach tostability analysis of discrete-time systems with time-varyingdelayrdquo in Proceedings of the American Control Conferencepp 1644ndash1649 Seattle WA USA June 2008

[2] S I Butt M Klaricic Bakula ETH Pecaric and J PecaricldquoJensen-gruss inequality and its applications for the zipf-mandelbrot lawrdquo Mathematical Methods in the Applied Sci-ences vol 44 no 2 pp 1664ndash1673 2021

[3] M A Khan D Pecaric and J Pecaric ldquoNew refinement of theJensen inequality associated to certain functions with appli-cationsrdquo Journal of Inequalities and Applications vol 762020

[4] S Khan M Adil Khan S I Butt and Y-M Chu ldquoA newbound for the Jensen gap pertaining twice differentiablefunctions with applicationsrdquo Advances in Difference Equa-tions vol 1 2020

[5] H Budak andM Z Sarikaya ldquoOn generalized Ostrowski-typeinequalities for functions whose first derivatives absolutevalues are convexrdquo Turkish Journal of Mathematics vol 40pp 1193ndash1210 2016

[6] P Cerone and S S Dragomir ldquoOn some inequalities arisingfrom Montgomeryrsquos identityrdquo Journal of ComputationalAnalysis and Applications vol 5 no 4 pp 341ndash367 2003

[7] S Bernstein ldquoSur les fonctions absolument monotonesrdquo ActaMathematica vol 52 pp 1ndash66 1929

[8] V Culjak and J Pecaric ldquoInterpolation polynomials andinequalities for convex functions of higher orderrdquo ActaMathematica vol 5 pp 369ndash386 2012

[9] A M Fink ldquoBounds of the deviation of a function from itsaveragesrdquo Czechoslovak Mathematical Journal vol 42no 117 pp 289ndash310 1992

[10] J Pecaric and V Culjak ldquoInterpolation polinomials and in-equalities for convex functions of higher orderrdquo Mathe-matical Inequalities amp Applications vol 5 no 3 pp 369ndash3862002

[11] J Pecaric V Culjak and A M Fink ldquoOn some inequalitiesfor convex function of higher orderrdquo Advanced NonlinearStudies vol 6 no 2 pp 131ndash140 1999

[12] J Pecaric V Culjak andM Rogina ldquoOn some inequalities forconvex function of higher order IIrdquo Advanced NonlinearStudies vol 45 pp 281ndash294 2001

[13] J Pecaric I Peric and M Rodic-Lipanovic ldquoUniformtreatment of Jensen type inequalitiesrdquo Mathematical Reportsvol 16 no 66 pp 183ndash205 2014

[14] J Pecaric and J Peric ldquoImprovement of the Giaccardi and thepetrovic inequality and related stolarsky type meansrdquo Annalsof the University of Craiova - Mathematics and ComputerScience vol 39 no 1 pp 65ndash75 2012

[15] J Pecaric M Praljak and A Witkowski ldquoLinear operatorinequality for nminusconvex functions at a pointrdquo MathematicalInequalities amp Applications Appl vol 18 pp 1201ndash1217 2015

[16] Z Pavic ldquoRefinements of Jensenrsquos inequality for infiniteconvex combinationsrdquo Turkish Journal of Inequalities vol 2no 2 pp 44ndash53 2018

[17] R Bibi A Nosheen and J Pecaric ldquoExtended Jensenrsquos typeinequalities for diamond integrals via Taylors formulardquoTurkish Journal of Inequalities vol 3 no 1 pp 7ndash18 2019

[18] T Niaz K A Khan and J Pecaric ldquoOn refinement of Jensenrsquosinequality for 3minusconvex function at a pointrdquo Turkish Journalof Inequalities vol 4 no 1 pp 70ndash80 2020

[19] A Ekinci A O Akdemir and M E Ozdemir ldquoIntegralinequalities for different kinds of convexity via classical in-equalitiesrdquo Turkish Journal of Science vol 5 no 3 pp 305ndash313 2020

[20] Y M Chu S Talib E Set M U Awan and M A Noorldquo(p q)minus Analysis of Montgomery identity and estimates of(p q)minusbounds with applicationsrdquo Journal of Inequalities andApplications vol 2021 9 pages 2021

[21] M Z Sarikaya H Yaldrz and E Set ldquoOn fractional in-equalities via Montgomery identitiesrdquo International Journalof Open Problems in Complex Analysis vol 6 no 2 pp 36ndash432014

[22] A Bnouhachem ldquoA descent SQP alternating directionmethod for minimizing the sum of three convex functionsrdquoJournal of Nonlinear Functional Analysis vol 4 pp 469ndash4822020

[23] A O Akdemir S I Butt M Nadeem and M A RagusaldquoNew general variants of Chebyshev type inequalities viageneralized fractional integral operatorsrdquo Mathematicsvol 92 2021

[24] S I Butt M Umar S Rashid A O Akdemir and Y MingChu ldquoNew Hermite Jensen Mercer type inequalities viakminusfractional integralsrdquo Advances in Difference Equationsvol 635 2020

[25] S I Butt M Nadeem S Qaisar A O Akdemir andT Abdeljawad ldquoHermite Jensen Mercer type inequalities forconformable integrals and related resultsrdquo Advances in Dif-ference Equations vol 501 2020

[26] A Bnouhachem and X Qin ldquoAn inertial proximal Peaceman-Rachford splittingmethod with SQP regularization for convexprogrammingrdquo Journal of Nonlinear Functional Analysisvol 50 2020

[27] S Hamann ldquoMinimality conditions for convex compositefunctions and efficiency conditions in vector optimizationrdquoApplied Set-Valued Analysis and Optimization vol 1pp 221ndash229 2019

[28] A R Khan J Pecaric and M Praljak ldquoPopoviciu type in-equalities for n-convex functions via extension of Mont-gomery identitynminusconvex functions via extension ofMontgomery identityrdquo Analele Universitatii ldquoOvidiusrdquo Con-stanta-Seria Matematica vol 24 no 3 pp 161ndash188 2016

[29] R P Boas ldquoe Jensen-Steffensen inequalityrdquo ElektrotehnFak Ser Mat Fiz vol 302ndash319 pp 1ndash8 1970

[30] J Pecaric F Proschan and Y L Tong Convex FunctionsPartial Orderings and Statistical Applications Academic PressNew York NY USA 1992

[31] H D Brunk ldquoOn an inequality for convex functionsrdquo Pro-ceedings of the American Mathematical Society vol 7 no 5p 817 1956

[32] P R Beesack and J E Pecaric ldquoOn Jessenrsquos inequality forconvex functionsrdquo Journal of Mathematical Analysis andApplications vol 110 no 2 pp 536ndash552 1985

[33] R Barlow A Marshall and F Proschan ldquoSome inequalitiesfor starshaped and convex functionsrdquo Pacific Journal ofMathematics vol 29 no 1 pp 19ndash42 1969

16 Journal of Mathematics

[34] J E Pecaric ldquoOn an inequality of PMV Janicrdquo Publications delrsquoInstitut Mathematique vol 28 no 42 pp 145ndash149 1980

[35] G Szego ldquoUber eine Verallgemeinerung des DirichletschenIntegralsrdquo Mathematische Zeitschrift vol 52 no 1pp 676ndash685 1950

[36] F Giaccardi ldquoSu alcune disuguaglianierdquo Journal of Researchof the National Bureau of Standards vol 1 no 4 pp 139ndash1531955

[37] D S Mitrinovic J E Pecaric and A M Fink Classical andnew inequalities in analysis Mathematics and its Applications(East European Series) Vol 61 Kluwer Academic PublishersGroup Dordrecht Netherlands 1993

[38] I Csiszar ldquoInformation-type measures of diference ofprobability distributions and indirect observationsrdquo StudiaScience Math Hungar vol 2 pp 299ndash318 1967

[39] L Horvath D Pecaric and J Pecaric ldquoEstimations of f - andrenyi divergences by using a cyclic refinement of the Jensenrsquosinequalityrdquo Bulletin of the Malaysian Mathematical SciencesSociety vol 42 2017

[40] S Kullback Information =eory and Statistics Wiley NewYork NY USA 1959

[41] S T Piantadosi ldquoZipfrsquos word frequency law in natural lan-guage a critical review and future directionsrdquo PsychonomicBulletin and Review vol 21 no 5 pp 1112ndash1130 2014

[42] Z K Silagadze ldquoCitations and the zipfndashmandelbrot lawrdquoComplex Systems no 11 pp 487ndash499 1997

[43] J Jaksetic D Pecaric and J Pecaric ldquoSome properties of zipf-mandelbrot law and hurwitz ζ-functionrdquo Mathematical In-equalities amp Applications vol 21 no 2 pp 575ndash584 2018

[44] M Visser ldquoZipfrsquos law power laws and maximum entropyrdquoMathematical Inequalities amp Applications vol 15 Article ID043021 2013

[45] J Jaksetic D Pecaric and J Pecaric ldquoHybrid zipf-mandelbrotlawrdquo Mathematical Inequalities amp Applications vol 13 no 1pp 275ndash286 2019

[46] D V Widder ldquoCompletely convex functions and Lidstoneseriesrdquo Transactions of the American Mathematical Societyvol 51 p 387 1942

Journal of Mathematics 17

If inequality (33) holds in reverse direction then (34) alsoholds reversely

Proof Similar to that of eorem 3In the later part of this section we will vary our con-

ditions on pJxJ(J 1 2 n) to obtain generalizeddiscrete variants of JensenndashSteffensen Jensenrsquos and Jen-senndashPetrovic type inequalities We start with the followinggeneralization of JensenndashSteffensen discrete inequality fornminusconvex functions

Theorem 8 Let Ψ be as defined in =eorem 7 Also let x bemonotonic nminustuple xJ isin [a b]sube[α β] and p be a realnminustuple such that

0lePv lePm (v 1 2 m minus 1) Pm gt 0 (35)

is satisfied

(i) If Ψ is nminusconvex then for even nge 3 (33) is valid(ii) Moreover if (33) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized JensenndashSteffensen discrete inequality

Ψ(x)le1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 (36)

Proof It is interesting to see that under the assumed con-ditions on tuples x and p we have that x isin [a b] Forx1 gex2 ge gexm

Pm x1 minus x( 1113857 1113944m

J2pJ x1 minus xJ1113872 1113873 1113944

m

v2xvminus1 minus xv( 1113857 Pm minus Pvminus1( 1113857ge 0

(37)

is shows that x1 ge x Also xge xn since we have

Pm x minus xm( 1113857 1113944

mminus1

J1pJ xJ minus xm1113872 1113873 1113944

mminus1

v1xv minus xvminus1( 1113857Pv ge 0

(38)

For further details see the proof of JensenndashSteffensendiscrete inequality ([24] p 57) e idea of the rest of theproof is similar to that of eorem 3 but here we employeorem 7 and JensenndashSteffensen discrete inequality

Corollary 5 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube [α β] with p being a positive nminustuple

(i) If Ψ is nminusconvex then for even nge 3 (34) is valid(ii) Moreover if (33) is valid and the function H(middot) de-

fined in (12) is convex then again we get (36) which iscalled Jensenrsquos inequality for nminusconvex functions

Proof For pJ gt 0 xJ isin [a b] (J 1 2 3 m) ensuresthat x isin [a b] So by applying classical Jensenrsquos discrete

inequality (1) and idea ofeorem 8 we will get the requiredresults

Remark 5 Under the assumptions of Corollary 5 if wechoose Pm 1 then Corollary 5 (ii) gives the followinginequality for nminusconvex functions

Ψ 1113944m

J1pJxJ

⎛⎝ ⎞⎠le 1113944m

J1pJΨ xJ1113872 1113873 (39)

Nowwe give following reverses of JensenndashSteffensen andJensen-type inequalities

Corollary 6 Let Ψ be as defined in =eorem 7 Also let x bemonotonic mminustuple xJ isin [a b]sube[α β] and p be a realmminustuple such that there exist m isin 1 2 m such that

0gePv for vltm and 0gePv for vgtm (40)

where Pm gt 0 and x isin [α β]

(i) If is nminusconvex then for even nge 3 then reverse ofinequality (33) holds

(ii) Moreover if (33) holds reversely and the function H(middot)

defined in (12) is convex then we get reverse ofgeneralized JensenndashSteffensen inequality (36) fornminusconvex functions

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ reverse of JensenndashSteffensen inequality to obtain results

In the next corollary we give explicit conditions on realtuple p such that we get reverse of classical Jensen inequality

Corollary 7 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube[α β] such that x isin [α β] Let p be a real nminustuplesuch that

0ltp1 0gep2 p3 pm 0ltPm (41)

is satisfied

(i) If Ψ is nminusconvex then for even nge 3 the reverse ofinequality (33) is valid

(ii) Also if reverse of (33) is valid and the function H(middot)

defined in (12) is convex then we get reverse of (36)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ reverse of Jensen in-equality to obtain results

In [33] (see also [30]) one can find the result which isequivalent to the JensenndashSteffensen and the reverse Jen-senndashSteffensen inequality together It is the so-calledJensenndashPetrovic inequality Here without the proof we givethe adequate corollary which uses that result e proof goesthe same way as in the previous corollaries

8 Journal of Mathematics

Corollary 8 Let Ψ be as defined in =eorem 7 and letxi isin [a b]sube[α β] be such that xm ge xmminus1 x2 lex1 Let pbe a real mminustuple with Pm 1 such that

0lePv for 1le vltm minus 1 and 0lePv for 2le vltm

(42)

is satisfied =en we get the equivalent results given in=eorem 8 (i) and (ii) respectively

Remark 6 Under the assumptions of Corollary 8 if thereexist m isin 1 2 n such that

0gePv for vltm and 0gePv for vgtm (43)

and x isin [α β] then we get the equivalent results for reverseJensenndashSteffensen inequality given in Corollary 6 (i) and(ii) respectively

Remark 7 It is interesting to see that the conditions onpJJ 1 2 m given in Corollary 8 and Remark 6 arecoming from JensenndashPetrovic inequality which becomeequivalent to conditions for pJJ 1 2 m for Jen-senndashSteffensen results given in eorem 8 and Corollary 6respectively when Pm 1

Now we give results for Jensen and its reverses fornminustuples x and p when n is an odd number

Corollary 9 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube[α β] for J 1 2 m be such that x p berealmminustuplesm 2m + 1 m isin N and 1113954x 111139362k+1

J1 pJ11139362k+1J1

pJxJ isin [α β] for all k 1 2 m If for everyk 1 2 m we have

(ilowast) p1 gt 0 p2k le 0 p2k + p2k+1 le 0 11139362kJ1pJ ge 0

11139362k+1J1 pJ gt 0

(iilowast) x2k lex2k+1 11139362k+1J1 pJ (xJ minus x2k+1)ge 0

then we have the following statements to be valid(i) If Ψ is nminusconvex then for even nge 3 the inequality

Rn(1113954x s)ge1

P2m+11113944

2m+1

J1pJRn xJ s1113872 1113873 (44)

(ii) Also if (44) is valid and the function H(middot) defined in(12) is convex then we get the following generalizedinequality

Ψ(1113954x)ge1

P2m+11113944

2m+1

J1pJΨ xJ1113872 1113873 (45)

Proof We employ the idea of the proofs of eorems 7 and8 for n odd along with inequality of Vasic and Janic[34]

Remark 8 We can also discuss the following importantcases by considering the explicit conditions given in [34]

We conclude this section by giving the following im-portant cases

(Case 1)Let the condition (ilowast) hold and the reverse inequalitiesin condition (iilowast) hold en again we can give in-equalities (44) and (45) respectively given in Corollary9(Case 2)If in case of conditions (ilowast) and (iilowast) the following arevalid(iiilowast) p1 gt 0 p2k+1 ge 0 p2k + p 2k+1 ge 0 1113936

2kJ1pJ ge 0

11139362k+1J1 pJ gt 0

(ivlowast) x2k lex2k+1 11139362kminus1J1 pJ(xJ minus x2k)le 0

then we can give reverses of inequalities (44) and (45)respectively given in Corollary 9(Case 3)Finally we can also give reverses of inequalities (44)and (45) respectively given in Corollary 9 providedthat the condition (iiilowast) holds and the reverse in-equalities in condition (ivlowast) hold

e result given in (Case 3) is type of generalization ofinequality by Szego [35]

32 Generalization of Converse JensenrsquosDiscrete Inequality forRealWeights In this section we give the results for converseof Jensenrsquos inequality in discrete case by using the Mont-gomery identity

Let xJ isin [a b]sube[α β] ane b pJ isin R(J 1 n) besuch that Pm ne 0 en we have the following difference ofconverse of Jensenrsquos inequality for Ψ [α β]⟶ R

CJdis(Ψ) 1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 minus

b minus x

b minus aΨ(a) minus

x minus a

b minus aΨ(b)

(46)

Similarly we assume the Giaccardi difference [36] givenas

Gcardi(Ψ) 1113944m

J1pJΨ xJ1113872 1113873 minus AΨ 1113944

m

J1pJxJ

⎛⎝ ⎞⎠ minus B 1113944m

J1pJ minus 1⎛⎝ ⎞⎠Ψ x0( 1113857 (47)

where

Journal of Mathematics 9

A 1113936

mJ1 pJ xJ minus x01113872 11138731113872 1113873

1113936mJ1 pJxJ minus x01113872 1113873

B 1113936

mJ1 pJxJ

1113936mJ1 pJxJ minus x01113872 1113873

and 1113944m

J1pJxJ ne x0 (48)

Theorem 9 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also letx0 xJ isin [a b]sube[α β] pJ isin R(J 1 m) be such that1113936

mJ1 pJxJ ne x0

(i) =en the following generalized identity holds

CJdis(Ψ) 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)CJdis xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)CJdis xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875

+1

(n minus 1)1113946β

αCJdis Rn xJ s1113872 11138731113872 1113873Ψ(n)

(s)ds (DCGI)

(49)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

CJdis Rn xJ s1113872 11138731113872 1113873le 0 (50)

holds then we have the following generalized inequality

CJdis(Ψ)le 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)CJdis xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)CJdis xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875 (51)

If inequality (50) holds in reverse direction then (51) alsoholds reversely

Theorem 10 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also let xJ isin [a b]sube[α β]pJ isin R(J 1 m) be such that Pm ne 0 and x isin [α β]

(i) =en the following generalized Giaccardi identityholds

Gcardi(Ψ) 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)Gcardi xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)Gcardi xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875

+1

(n minus 1)1113946β

αGcardi Rn xJ s1113872 11138731113872 1113873Ψ(n)

(s)ds (GIAGI)

(52)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

Gcardi Rn xJ s1113872 11138731113872 1113873le 0 (53)

holds then we have the following generalized Giaccardiinequality

Gcardi(Ψ)le 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)Gcardi xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)Gcardi xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875 (54)

If inequality (53) holds in reverse direction then (54) alsoholds reversely

In the later part of this section we will vary our con-ditions on pJxJ (J 1 2 m) to obtain generalized

10 Journal of Mathematics

converse discrete variants of Jensenrsquos inequality and Giac-cardi inequality for nminusconvex functions

Theorem 11 Let Ψ be as defined in =eorem 9 Also letxJ isin [a b]sube [α β] and p be a positive mminustuple

(i) If Ψ is nminusconvex then for even nge 3 (50) is valid(ii) Moreover if (50) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized converse of Jensenrsquos inequality

1Pm

1113944

m

J1pJΨ xJ1113872 1113873le

b minus x

b minus aΨ(a) +

x minus a

b minus aΨ(b) (55)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ converse of Jensenrsquosinequality (see [32] or [30] p 98) to obtain results

Finally in this section we give Giaccardi inequality forhigher-order convex functions

Theorem 12 Let Ψ be as defined in =eorem 9 Also letx0 xJ isin [a b]sube[α β] and p be a positive mminustuple such that

1113944

m

J1pJxJ nex0 and xv minus x0( 1113857 1113944

m

J1pJxJ minus xv

⎛⎝ ⎞⎠ge 0 (v 1 m) (56)

(i) If Ψ is nminusconvex then for even nge 3 (53) is valid(ii) Moreover if (53) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized Giaccardi inequality

1113944

m

J1pJΨ xJ1113872 1113873leAΨ 1113944

m

J1pJxJ

⎛⎝ ⎞⎠ + B 1113944m

J1pJ minus 1⎛⎝ ⎞⎠Ψ x0( 1113857

(57)

where A and B are defined in (47)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ Giaccardi inequality(see [36] or [37] p 11) to obtain results

33 Applications in Information =eory for Jensenrsquos DiscreteInequality Jensenrsquos inequality plays a key role in infor-mation theory to construct lower bounds for some notableinequalities but here we will use it to make connectionsbetween inequalities in information theory

Let Ψ R+⟶ R+ be a convex function and letp ≔ (p1 pm) and q ≔ (q1 qm) be positive proba-bility distributions then Ψ-divergence functional is defined(in [38]) as follows

IΨ(p q) 1113944m

J1qJΨ

pJ

qJ1113888 1113889 (58)

Horvath et al in [39] defined the generalized Csiszardivergence functional as follows

Definition 2 Let I be an interval in R and Ψ I⟶ R be afunction Also let p ≔ (p1 pm) isin Rm andq ≔ (q1 qm) isin (0infin)m such that

pJ

qJisin I J 1 m (59)

en let

1113957IΨ(p q) 1113944m

J1qJΨ

pJ

qJ1113888 1113889 (60)

In this section we write Jensenrsquos difference here that weuse in upcoming results

F p xJΨ1113872 1113873 Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 (61)

Theorem 13 Under the assumptions of =eorem 9 (ii) let(51) hold and Ψ be nminusconvex Also letp ≔ (p1 pm) inRm and q ≔ (q1 qm) isin (0infin)mthen we have the following results

Journal of Mathematics 11

1113957IΨ(p q)gePmΨ(1) minus Pm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F q

pJ

qJ (x minus α)

ℓ+21113888 11138891113890 1113891 minus Ψ(ℓ+1)

(β) F qpJ

qJ (x minus β)

ℓ+21113888 11138891113890 11138911113896 1113897

(62)

Proof From eorem 9 by following Jensenrsquos difference(61) we can rearrange (34) as

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873le 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F p xJ (x minus α)

ℓ+21113872 11138731113872 1113873 minus Ψ(ℓ+1)

(β) F p xJ (x minus β)ℓ+2

1113872 11138731113872 11138731113966 1113967

(63)

Now replace pJ with qJ and xJ with pJqJ and we get(62)

For positive n-tuple q (q1 qm) such that1113936

mJ1 qJ 1 the Shannon entropy is defined by

S(q) minus 1113944m

J1qJ ln qJ (64)

Corollary 10 Under the assumptions of =eorem 9 (ii) let(51) hold and Ψ be nminusconvex

(i) If q ≔ (q1 qm) isin (0infin)m then

1113944

m

J1qJ ln qJ lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (65)

(ii) We can get bounds for the Shannon entropy of q if wechoose q ≔ (q1 qn) to be a positive probabilitydistribution

S(q)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (66)

Proof (i) Substituting Ψ(x) ≔ minus lnx and usingp ≔ (1 1 1) in eorem 13 we get (65)

(ii) Since we have 1113936mJ1 qJ 1 by multiplying minus1 on

both sides of (65) and taking into account (64) weget (66)

e KullbackndashLeibler distance [40] between the positiveprobability distributions p (p1 pm) andq (q1 qm) is defined by

D(q p) 1113944

m

J1qJ ln

qJ

pJ

1113888 1113889 (67)

Corollary 11 Under the assumptions of Corollary 10

(i) If q ≔ (q1 qm) p ≔ (p1 pm) isin (0infin)mthen

1113944

m

J1qJ ln

qJ

pJ

1113888 1113889lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

pJ

qJ minusln(middot)1113888 1113889

⎧⎨

⎩

⎫⎬

⎭ (68)

(ii) If q ≔ (q1 qm) p ≔ (p1 pm) are positiveprobability distributions then we have

12 Journal of Mathematics

D(q p)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

βℓ+1F q

pJ

qJ minusln(middot)1113888 11138891113896 1113897 (69)

Proof

(i) UsingΨ(x) ≔ minus lnx (which isn-convex for evenn)in eorem 13 we get (68) after simplification

(ii) It is a special case of (i)

34 Results for Zipf and Hybrid ZipfndashMandelbrot EntropyOne of the basic laws in information science is Zipfrsquos law[4142] which is highly applied in linguistics Let cge 0 dgt 0and N isin 1 2 ZipfndashMandelbrot entropy can be given as

ZM(H c d) d

HNcd

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (70)

where

HNcd 1113944

N

σ1

1(σ + c)

d (71)

Consider

qJ Ψ(J N c d) 1

(J + c)dH

Ncd1113872 1113873

(72)

where Ψ(J m c d) is discrete probability distributionknown as ZipfndashMandelbrot law ZipfndashMandelbrot law hasmany application in linguistics and information sciencesSome of the recent study about ZipfndashMandelbrot law can beseen in the listed references (see [39 43]) Now we state ourresults involving entropy introduced by Mandelbrot law byestablishing the relationship with Shannon and relativeentropies

Theorem 14 Let q be ZipfndashMandelbrot law as defined in(72) with parameters cge 0 dgt 0 and N isin 1 2 and wehave

ZM(H c d) S(q)leN times 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873

⎧⎨

⎩

⎫⎬

⎭

(73)

Proof It is interesting to see that for qJ defined in (72)1113936

NJ1 qJ 1 erefore using above qJ in Shannon entropy

(64) we get Mandelbrot entropy (70)

S(q) minus 1113944

N

J1qJ ln qJ minus 1113944

N

J1

1(J + c)

dH

Ncd1113872 1113873

ln1

(J + c)dH

Ncd1113872 1113873

d

HNcd1113872 1113873

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (74)

Finally substituting this qJ 1((J + c)dHNcd) in

Corollary 10 (ii) we get the desired result Corollary 12 Let q and p be ZipfndashMandelbrot law withparameters c1 c2 isin [0infin) d1 d2 gt 0 and let

Journal of Mathematics 13

HNc1 d1

1113936Nσ1 1(σ + c1)

d1 and HNc2 d2

1113936Nσ1 1(σ + c2)

d2 Now using qJ 1(J + c1)

d1HNc1 d1

andpJ 1(J + c2)

d2HNc2 d2

in Corollary 11 (ii) the followingholds

D(q p) 1113944N

J1

1J + c1( 1113857

d1HNc1d1

lnJ + c2( 1113857

d2HNc2 d2

J + c1( 1113857d1H

mc1 d1

⎛⎝ ⎞⎠

minusZ H c1 d1( 1113857 +d2

HNc1 d1

1113944

N

J1

ln J + c2( 1113857

J + c1( 1113857d1

+ ln Hmc2d2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qJ + c1( 1113857

d1HNc1 d1

J + c2( 1113857d2H

Nc2 d2

minusln(middot)⎛⎝ ⎞⎠ minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qJ + c1( 1113857

d1HNc1d1

J + c2( 1113857d2H

Nc2d2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(75)

=e Next Result for Hybrid ZipfndashMandelbrot Entropy Fur-ther generalization of ZipfndashMandelbrot entropy is Hybrid

ZipfndashMandelbrot entropy Let N isin 1 2 cge 0ωgt 0then Hybrid ZipfndashMandelbrot entropy can be given as

1113954ZM Hlowast c dω( 1113857

1Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873 (76)

where

Hlowastcdω 1113944

N

J1

ωJ

(J + c)d (77)

Consider

qJ Ψ(J N c dω) ωJ

(J + c)dHlowastcdω

(78)

which is called Hybrid ZipfndashMandelbrot law =ere is aunified approach maximization of Shannon entropy [44]

that naturally follows the path of generalization from Zipfrsquos toHybrid Zipfrsquos law Extending this idea Jaksetic et al in [45]presented a transition from ZipfndashMandelbrot to HybridZipfndashMandelbrot law by employing maximum entropytechnique with one additional constraint It is interesting thatexamination of its densities provides some new insights ofLerchrsquos transcendent

Theorem 15 Let q be Hybrid ZipfndashMandelbrot law asdefined in (78) with parameters cge 0 dωgt 0 andN isin 1 2 and we have

1113954ZM Hlowast c dω( 1113857 S(q)leN 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠ minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠

⎧⎨

⎩

⎫⎬

⎭

(79)

14 Journal of Mathematics

Proof It is interesting to see that for qJ defined in (78)1113936

NJ1 qJ 1 erefore using above qJ in Shannon en-

tropy (64) we get Hybrid ZipfndashMandelbrot entropy (76)

S(q) minus 1113944N

J1qJ ln qJ minus 1113944

N

J1

ωJ

(J + c)dHlowastcdω

lnωJ

(J + c)dHlowastcdω

minus1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

lnωJ

(J + c)d

1113888 1113889 + ln1

Hlowastcdω

1113888 11138891113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

ln(J + c)

d

ωJ1113888 1113889 + ln H

lowastcdω1113872 11138731113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873

(80)

Finally substituting this qJ ωJ(J + c)dHlowastcdω inCorollary 10 (ii) we get the desired result

Corollary 13 Let q and p be Hybrid ZipfndashMandelbrotlaw with parameters c1 c2 isin [0infin)ω1ω2 d1 d2 gt 0 Nowusing qJ ωJ

1 (J + c1)d1Hlowastc1 d1 ω1

and pJ ωJ2

(J + c2)d2Hlowastc2 d2 ω2

in Corollary 11 (ii) the following holds

D(q p) 1113944N

J1

ωJ1

J + c1( 1113857d1Hlowastc1 d1 ω1

lnωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

ωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

⎛⎝ ⎞⎠

minus1113954ZM Hlowast c1 d1ω1( 1113857 +

1Hlowastc1 d1 ω1

1113944

N

J1

ωJ1

J + c1( 1113857d1ln

J + c2( 1113857d2

ωJ2

⎛⎝ ⎞⎠ + ln Hlowastc2 d2 ω2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎫⎪⎬

⎪⎭

(81)

Remark 9 Similarly we can give results for Shannonentropy KullbackndashLeibler distance ZipfndashMandelbrotentropy and Hybrid ZipfndashMandelbrot entropy by usinggeneralized Giaccardi inequality defined in (54) on thesame steps

4 Concluding Remarks

In this paper we gave generalization of Jensenrsquos inequality aswell as converse of Jensenrsquos inequality by using Montgomeryidentity We also formulate results for other inequalities like

JensenndashSteffensen inequality JensenndashBoas inequality andJensenndashBrunk inequality We can obtain JensenndashSteffenseninequality JensenndashBoas inequality and JensenndashBrunk in-equality by changing the assumption of Jensenrsquos inequalityAt the end we gave applications in information theory forour obtained results especially we gave results for HybridZipfndashMandelbrot entropy for our obtained results [46]

Data Availability

No data were used to support this study

Journal of Mathematics 15

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is research is supported by HEC Pakistan under NRPUProject 7906 and RUDN University Strategic AcademicLeadership Program

References

[1] X L Zhu and G H Yang ldquoJensen inequality approach tostability analysis of discrete-time systems with time-varyingdelayrdquo in Proceedings of the American Control Conferencepp 1644ndash1649 Seattle WA USA June 2008

[2] S I Butt M Klaricic Bakula ETH Pecaric and J PecaricldquoJensen-gruss inequality and its applications for the zipf-mandelbrot lawrdquo Mathematical Methods in the Applied Sci-ences vol 44 no 2 pp 1664ndash1673 2021

[3] M A Khan D Pecaric and J Pecaric ldquoNew refinement of theJensen inequality associated to certain functions with appli-cationsrdquo Journal of Inequalities and Applications vol 762020

[4] S Khan M Adil Khan S I Butt and Y-M Chu ldquoA newbound for the Jensen gap pertaining twice differentiablefunctions with applicationsrdquo Advances in Difference Equa-tions vol 1 2020

[5] H Budak andM Z Sarikaya ldquoOn generalized Ostrowski-typeinequalities for functions whose first derivatives absolutevalues are convexrdquo Turkish Journal of Mathematics vol 40pp 1193ndash1210 2016

[6] P Cerone and S S Dragomir ldquoOn some inequalities arisingfrom Montgomeryrsquos identityrdquo Journal of ComputationalAnalysis and Applications vol 5 no 4 pp 341ndash367 2003

[7] S Bernstein ldquoSur les fonctions absolument monotonesrdquo ActaMathematica vol 52 pp 1ndash66 1929

[8] V Culjak and J Pecaric ldquoInterpolation polynomials andinequalities for convex functions of higher orderrdquo ActaMathematica vol 5 pp 369ndash386 2012

[9] A M Fink ldquoBounds of the deviation of a function from itsaveragesrdquo Czechoslovak Mathematical Journal vol 42no 117 pp 289ndash310 1992

[10] J Pecaric and V Culjak ldquoInterpolation polinomials and in-equalities for convex functions of higher orderrdquo Mathe-matical Inequalities amp Applications vol 5 no 3 pp 369ndash3862002

[11] J Pecaric V Culjak and A M Fink ldquoOn some inequalitiesfor convex function of higher orderrdquo Advanced NonlinearStudies vol 6 no 2 pp 131ndash140 1999

[12] J Pecaric V Culjak andM Rogina ldquoOn some inequalities forconvex function of higher order IIrdquo Advanced NonlinearStudies vol 45 pp 281ndash294 2001

[13] J Pecaric I Peric and M Rodic-Lipanovic ldquoUniformtreatment of Jensen type inequalitiesrdquo Mathematical Reportsvol 16 no 66 pp 183ndash205 2014

[14] J Pecaric and J Peric ldquoImprovement of the Giaccardi and thepetrovic inequality and related stolarsky type meansrdquo Annalsof the University of Craiova - Mathematics and ComputerScience vol 39 no 1 pp 65ndash75 2012

[15] J Pecaric M Praljak and A Witkowski ldquoLinear operatorinequality for nminusconvex functions at a pointrdquo MathematicalInequalities amp Applications Appl vol 18 pp 1201ndash1217 2015

[16] Z Pavic ldquoRefinements of Jensenrsquos inequality for infiniteconvex combinationsrdquo Turkish Journal of Inequalities vol 2no 2 pp 44ndash53 2018

[17] R Bibi A Nosheen and J Pecaric ldquoExtended Jensenrsquos typeinequalities for diamond integrals via Taylors formulardquoTurkish Journal of Inequalities vol 3 no 1 pp 7ndash18 2019

[18] T Niaz K A Khan and J Pecaric ldquoOn refinement of Jensenrsquosinequality for 3minusconvex function at a pointrdquo Turkish Journalof Inequalities vol 4 no 1 pp 70ndash80 2020

[19] A Ekinci A O Akdemir and M E Ozdemir ldquoIntegralinequalities for different kinds of convexity via classical in-equalitiesrdquo Turkish Journal of Science vol 5 no 3 pp 305ndash313 2020

[20] Y M Chu S Talib E Set M U Awan and M A Noorldquo(p q)minus Analysis of Montgomery identity and estimates of(p q)minusbounds with applicationsrdquo Journal of Inequalities andApplications vol 2021 9 pages 2021

[21] M Z Sarikaya H Yaldrz and E Set ldquoOn fractional in-equalities via Montgomery identitiesrdquo International Journalof Open Problems in Complex Analysis vol 6 no 2 pp 36ndash432014

[22] A Bnouhachem ldquoA descent SQP alternating directionmethod for minimizing the sum of three convex functionsrdquoJournal of Nonlinear Functional Analysis vol 4 pp 469ndash4822020

[23] A O Akdemir S I Butt M Nadeem and M A RagusaldquoNew general variants of Chebyshev type inequalities viageneralized fractional integral operatorsrdquo Mathematicsvol 92 2021

[24] S I Butt M Umar S Rashid A O Akdemir and Y MingChu ldquoNew Hermite Jensen Mercer type inequalities viakminusfractional integralsrdquo Advances in Difference Equationsvol 635 2020

[25] S I Butt M Nadeem S Qaisar A O Akdemir andT Abdeljawad ldquoHermite Jensen Mercer type inequalities forconformable integrals and related resultsrdquo Advances in Dif-ference Equations vol 501 2020

[26] A Bnouhachem and X Qin ldquoAn inertial proximal Peaceman-Rachford splittingmethod with SQP regularization for convexprogrammingrdquo Journal of Nonlinear Functional Analysisvol 50 2020

[27] S Hamann ldquoMinimality conditions for convex compositefunctions and efficiency conditions in vector optimizationrdquoApplied Set-Valued Analysis and Optimization vol 1pp 221ndash229 2019

[28] A R Khan J Pecaric and M Praljak ldquoPopoviciu type in-equalities for n-convex functions via extension of Mont-gomery identitynminusconvex functions via extension ofMontgomery identityrdquo Analele Universitatii ldquoOvidiusrdquo Con-stanta-Seria Matematica vol 24 no 3 pp 161ndash188 2016

[29] R P Boas ldquoe Jensen-Steffensen inequalityrdquo ElektrotehnFak Ser Mat Fiz vol 302ndash319 pp 1ndash8 1970

[30] J Pecaric F Proschan and Y L Tong Convex FunctionsPartial Orderings and Statistical Applications Academic PressNew York NY USA 1992

[31] H D Brunk ldquoOn an inequality for convex functionsrdquo Pro-ceedings of the American Mathematical Society vol 7 no 5p 817 1956

[32] P R Beesack and J E Pecaric ldquoOn Jessenrsquos inequality forconvex functionsrdquo Journal of Mathematical Analysis andApplications vol 110 no 2 pp 536ndash552 1985

[33] R Barlow A Marshall and F Proschan ldquoSome inequalitiesfor starshaped and convex functionsrdquo Pacific Journal ofMathematics vol 29 no 1 pp 19ndash42 1969

16 Journal of Mathematics

[34] J E Pecaric ldquoOn an inequality of PMV Janicrdquo Publications delrsquoInstitut Mathematique vol 28 no 42 pp 145ndash149 1980

[35] G Szego ldquoUber eine Verallgemeinerung des DirichletschenIntegralsrdquo Mathematische Zeitschrift vol 52 no 1pp 676ndash685 1950

[36] F Giaccardi ldquoSu alcune disuguaglianierdquo Journal of Researchof the National Bureau of Standards vol 1 no 4 pp 139ndash1531955

[37] D S Mitrinovic J E Pecaric and A M Fink Classical andnew inequalities in analysis Mathematics and its Applications(East European Series) Vol 61 Kluwer Academic PublishersGroup Dordrecht Netherlands 1993

[38] I Csiszar ldquoInformation-type measures of diference ofprobability distributions and indirect observationsrdquo StudiaScience Math Hungar vol 2 pp 299ndash318 1967

[39] L Horvath D Pecaric and J Pecaric ldquoEstimations of f - andrenyi divergences by using a cyclic refinement of the Jensenrsquosinequalityrdquo Bulletin of the Malaysian Mathematical SciencesSociety vol 42 2017

[40] S Kullback Information =eory and Statistics Wiley NewYork NY USA 1959

[41] S T Piantadosi ldquoZipfrsquos word frequency law in natural lan-guage a critical review and future directionsrdquo PsychonomicBulletin and Review vol 21 no 5 pp 1112ndash1130 2014

[42] Z K Silagadze ldquoCitations and the zipfndashmandelbrot lawrdquoComplex Systems no 11 pp 487ndash499 1997

[43] J Jaksetic D Pecaric and J Pecaric ldquoSome properties of zipf-mandelbrot law and hurwitz ζ-functionrdquo Mathematical In-equalities amp Applications vol 21 no 2 pp 575ndash584 2018

[44] M Visser ldquoZipfrsquos law power laws and maximum entropyrdquoMathematical Inequalities amp Applications vol 15 Article ID043021 2013

[45] J Jaksetic D Pecaric and J Pecaric ldquoHybrid zipf-mandelbrotlawrdquo Mathematical Inequalities amp Applications vol 13 no 1pp 275ndash286 2019

[46] D V Widder ldquoCompletely convex functions and Lidstoneseriesrdquo Transactions of the American Mathematical Societyvol 51 p 387 1942

Journal of Mathematics 17

Corollary 8 Let Ψ be as defined in =eorem 7 and letxi isin [a b]sube[α β] be such that xm ge xmminus1 x2 lex1 Let pbe a real mminustuple with Pm 1 such that

0lePv for 1le vltm minus 1 and 0lePv for 2le vltm

(42)

is satisfied =en we get the equivalent results given in=eorem 8 (i) and (ii) respectively

Remark 6 Under the assumptions of Corollary 8 if thereexist m isin 1 2 n such that

0gePv for vltm and 0gePv for vgtm (43)

and x isin [α β] then we get the equivalent results for reverseJensenndashSteffensen inequality given in Corollary 6 (i) and(ii) respectively

Remark 7 It is interesting to see that the conditions onpJJ 1 2 m given in Corollary 8 and Remark 6 arecoming from JensenndashPetrovic inequality which becomeequivalent to conditions for pJJ 1 2 m for Jen-senndashSteffensen results given in eorem 8 and Corollary 6respectively when Pm 1

Now we give results for Jensen and its reverses fornminustuples x and p when n is an odd number

Corollary 9 Let Ψ be as defined in =eorem 7 and letxJ isin [a b]sube[α β] for J 1 2 m be such that x p berealmminustuplesm 2m + 1 m isin N and 1113954x 111139362k+1

J1 pJ11139362k+1J1

pJxJ isin [α β] for all k 1 2 m If for everyk 1 2 m we have

(ilowast) p1 gt 0 p2k le 0 p2k + p2k+1 le 0 11139362kJ1pJ ge 0

11139362k+1J1 pJ gt 0

(iilowast) x2k lex2k+1 11139362k+1J1 pJ (xJ minus x2k+1)ge 0

then we have the following statements to be valid(i) If Ψ is nminusconvex then for even nge 3 the inequality

Rn(1113954x s)ge1

P2m+11113944

2m+1

J1pJRn xJ s1113872 1113873 (44)

(ii) Also if (44) is valid and the function H(middot) defined in(12) is convex then we get the following generalizedinequality

Ψ(1113954x)ge1

P2m+11113944

2m+1

J1pJΨ xJ1113872 1113873 (45)

Proof We employ the idea of the proofs of eorems 7 and8 for n odd along with inequality of Vasic and Janic[34]

Remark 8 We can also discuss the following importantcases by considering the explicit conditions given in [34]

We conclude this section by giving the following im-portant cases

(Case 1)Let the condition (ilowast) hold and the reverse inequalitiesin condition (iilowast) hold en again we can give in-equalities (44) and (45) respectively given in Corollary9(Case 2)If in case of conditions (ilowast) and (iilowast) the following arevalid(iiilowast) p1 gt 0 p2k+1 ge 0 p2k + p 2k+1 ge 0 1113936

2kJ1pJ ge 0

11139362k+1J1 pJ gt 0

(ivlowast) x2k lex2k+1 11139362kminus1J1 pJ(xJ minus x2k)le 0

then we can give reverses of inequalities (44) and (45)respectively given in Corollary 9(Case 3)Finally we can also give reverses of inequalities (44)and (45) respectively given in Corollary 9 providedthat the condition (iiilowast) holds and the reverse in-equalities in condition (ivlowast) hold

e result given in (Case 3) is type of generalization ofinequality by Szego [35]

32 Generalization of Converse JensenrsquosDiscrete Inequality forRealWeights In this section we give the results for converseof Jensenrsquos inequality in discrete case by using the Mont-gomery identity

Let xJ isin [a b]sube[α β] ane b pJ isin R(J 1 n) besuch that Pm ne 0 en we have the following difference ofconverse of Jensenrsquos inequality for Ψ [α β]⟶ R

CJdis(Ψ) 1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 minus

b minus x

b minus aΨ(a) minus

x minus a

b minus aΨ(b)

(46)

Similarly we assume the Giaccardi difference [36] givenas

Gcardi(Ψ) 1113944m

J1pJΨ xJ1113872 1113873 minus AΨ 1113944

m

J1pJxJ

⎛⎝ ⎞⎠ minus B 1113944m

J1pJ minus 1⎛⎝ ⎞⎠Ψ x0( 1113857 (47)

where

Journal of Mathematics 9

A 1113936

mJ1 pJ xJ minus x01113872 11138731113872 1113873

1113936mJ1 pJxJ minus x01113872 1113873

B 1113936

mJ1 pJxJ

1113936mJ1 pJxJ minus x01113872 1113873

and 1113944m

J1pJxJ ne x0 (48)

Theorem 9 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also letx0 xJ isin [a b]sube[α β] pJ isin R(J 1 m) be such that1113936

mJ1 pJxJ ne x0

(i) =en the following generalized identity holds

CJdis(Ψ) 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)CJdis xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)CJdis xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875

+1

(n minus 1)1113946β

αCJdis Rn xJ s1113872 11138731113872 1113873Ψ(n)

(s)ds (DCGI)

(49)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

CJdis Rn xJ s1113872 11138731113872 1113873le 0 (50)

holds then we have the following generalized inequality

CJdis(Ψ)le 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)CJdis xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)CJdis xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875 (51)

If inequality (50) holds in reverse direction then (51) alsoholds reversely

Theorem 10 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also let xJ isin [a b]sube[α β]pJ isin R(J 1 m) be such that Pm ne 0 and x isin [α β]

(i) =en the following generalized Giaccardi identityholds

Gcardi(Ψ) 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)Gcardi xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)Gcardi xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875

+1

(n minus 1)1113946β

αGcardi Rn xJ s1113872 11138731113872 1113873Ψ(n)

(s)ds (GIAGI)

(52)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

Gcardi Rn xJ s1113872 11138731113872 1113873le 0 (53)

holds then we have the following generalized Giaccardiinequality

Gcardi(Ψ)le 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)Gcardi xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)Gcardi xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875 (54)

If inequality (53) holds in reverse direction then (54) alsoholds reversely

In the later part of this section we will vary our con-ditions on pJxJ (J 1 2 m) to obtain generalized

10 Journal of Mathematics

converse discrete variants of Jensenrsquos inequality and Giac-cardi inequality for nminusconvex functions

Theorem 11 Let Ψ be as defined in =eorem 9 Also letxJ isin [a b]sube [α β] and p be a positive mminustuple

(i) If Ψ is nminusconvex then for even nge 3 (50) is valid(ii) Moreover if (50) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized converse of Jensenrsquos inequality

1Pm

1113944

m

J1pJΨ xJ1113872 1113873le

b minus x

b minus aΨ(a) +

x minus a

b minus aΨ(b) (55)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ converse of Jensenrsquosinequality (see [32] or [30] p 98) to obtain results

Finally in this section we give Giaccardi inequality forhigher-order convex functions

Theorem 12 Let Ψ be as defined in =eorem 9 Also letx0 xJ isin [a b]sube[α β] and p be a positive mminustuple such that

1113944

m

J1pJxJ nex0 and xv minus x0( 1113857 1113944

m

J1pJxJ minus xv

⎛⎝ ⎞⎠ge 0 (v 1 m) (56)

(i) If Ψ is nminusconvex then for even nge 3 (53) is valid(ii) Moreover if (53) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized Giaccardi inequality

1113944

m

J1pJΨ xJ1113872 1113873leAΨ 1113944

m

J1pJxJ

⎛⎝ ⎞⎠ + B 1113944m

J1pJ minus 1⎛⎝ ⎞⎠Ψ x0( 1113857

(57)

where A and B are defined in (47)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ Giaccardi inequality(see [36] or [37] p 11) to obtain results

33 Applications in Information =eory for Jensenrsquos DiscreteInequality Jensenrsquos inequality plays a key role in infor-mation theory to construct lower bounds for some notableinequalities but here we will use it to make connectionsbetween inequalities in information theory

Let Ψ R+⟶ R+ be a convex function and letp ≔ (p1 pm) and q ≔ (q1 qm) be positive proba-bility distributions then Ψ-divergence functional is defined(in [38]) as follows

IΨ(p q) 1113944m

J1qJΨ

pJ

qJ1113888 1113889 (58)

Horvath et al in [39] defined the generalized Csiszardivergence functional as follows

Definition 2 Let I be an interval in R and Ψ I⟶ R be afunction Also let p ≔ (p1 pm) isin Rm andq ≔ (q1 qm) isin (0infin)m such that

pJ

qJisin I J 1 m (59)

en let

1113957IΨ(p q) 1113944m

J1qJΨ

pJ

qJ1113888 1113889 (60)

In this section we write Jensenrsquos difference here that weuse in upcoming results

F p xJΨ1113872 1113873 Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 (61)

Theorem 13 Under the assumptions of =eorem 9 (ii) let(51) hold and Ψ be nminusconvex Also letp ≔ (p1 pm) inRm and q ≔ (q1 qm) isin (0infin)mthen we have the following results

Journal of Mathematics 11

1113957IΨ(p q)gePmΨ(1) minus Pm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F q

pJ

qJ (x minus α)

ℓ+21113888 11138891113890 1113891 minus Ψ(ℓ+1)

(β) F qpJ

qJ (x minus β)

ℓ+21113888 11138891113890 11138911113896 1113897

(62)

Proof From eorem 9 by following Jensenrsquos difference(61) we can rearrange (34) as

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873le 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F p xJ (x minus α)

ℓ+21113872 11138731113872 1113873 minus Ψ(ℓ+1)

(β) F p xJ (x minus β)ℓ+2

1113872 11138731113872 11138731113966 1113967

(63)

Now replace pJ with qJ and xJ with pJqJ and we get(62)

For positive n-tuple q (q1 qm) such that1113936

mJ1 qJ 1 the Shannon entropy is defined by

S(q) minus 1113944m

J1qJ ln qJ (64)

Corollary 10 Under the assumptions of =eorem 9 (ii) let(51) hold and Ψ be nminusconvex

(i) If q ≔ (q1 qm) isin (0infin)m then

1113944

m

J1qJ ln qJ lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (65)

(ii) We can get bounds for the Shannon entropy of q if wechoose q ≔ (q1 qn) to be a positive probabilitydistribution

S(q)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (66)

Proof (i) Substituting Ψ(x) ≔ minus lnx and usingp ≔ (1 1 1) in eorem 13 we get (65)

(ii) Since we have 1113936mJ1 qJ 1 by multiplying minus1 on

both sides of (65) and taking into account (64) weget (66)

e KullbackndashLeibler distance [40] between the positiveprobability distributions p (p1 pm) andq (q1 qm) is defined by

D(q p) 1113944

m

J1qJ ln

qJ

pJ

1113888 1113889 (67)

Corollary 11 Under the assumptions of Corollary 10

(i) If q ≔ (q1 qm) p ≔ (p1 pm) isin (0infin)mthen

1113944

m

J1qJ ln

qJ

pJ

1113888 1113889lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

pJ

qJ minusln(middot)1113888 1113889

⎧⎨

⎩

⎫⎬

⎭ (68)

(ii) If q ≔ (q1 qm) p ≔ (p1 pm) are positiveprobability distributions then we have

12 Journal of Mathematics

D(q p)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

βℓ+1F q

pJ

qJ minusln(middot)1113888 11138891113896 1113897 (69)

Proof

(i) UsingΨ(x) ≔ minus lnx (which isn-convex for evenn)in eorem 13 we get (68) after simplification

(ii) It is a special case of (i)

34 Results for Zipf and Hybrid ZipfndashMandelbrot EntropyOne of the basic laws in information science is Zipfrsquos law[4142] which is highly applied in linguistics Let cge 0 dgt 0and N isin 1 2 ZipfndashMandelbrot entropy can be given as

ZM(H c d) d

HNcd

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (70)

where

HNcd 1113944

N

σ1

1(σ + c)

d (71)

Consider

qJ Ψ(J N c d) 1

(J + c)dH

Ncd1113872 1113873

(72)

where Ψ(J m c d) is discrete probability distributionknown as ZipfndashMandelbrot law ZipfndashMandelbrot law hasmany application in linguistics and information sciencesSome of the recent study about ZipfndashMandelbrot law can beseen in the listed references (see [39 43]) Now we state ourresults involving entropy introduced by Mandelbrot law byestablishing the relationship with Shannon and relativeentropies

Theorem 14 Let q be ZipfndashMandelbrot law as defined in(72) with parameters cge 0 dgt 0 and N isin 1 2 and wehave

ZM(H c d) S(q)leN times 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873

⎧⎨

⎩

⎫⎬

⎭

(73)

Proof It is interesting to see that for qJ defined in (72)1113936

NJ1 qJ 1 erefore using above qJ in Shannon entropy

(64) we get Mandelbrot entropy (70)

S(q) minus 1113944

N

J1qJ ln qJ minus 1113944

N

J1

1(J + c)

dH

Ncd1113872 1113873

ln1

(J + c)dH

Ncd1113872 1113873

d

HNcd1113872 1113873

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (74)

Finally substituting this qJ 1((J + c)dHNcd) in

Corollary 10 (ii) we get the desired result Corollary 12 Let q and p be ZipfndashMandelbrot law withparameters c1 c2 isin [0infin) d1 d2 gt 0 and let

Journal of Mathematics 13

HNc1 d1

1113936Nσ1 1(σ + c1)

d1 and HNc2 d2

1113936Nσ1 1(σ + c2)

d2 Now using qJ 1(J + c1)

d1HNc1 d1

andpJ 1(J + c2)

d2HNc2 d2

in Corollary 11 (ii) the followingholds

D(q p) 1113944N

J1

1J + c1( 1113857

d1HNc1d1

lnJ + c2( 1113857

d2HNc2 d2

J + c1( 1113857d1H

mc1 d1

⎛⎝ ⎞⎠

minusZ H c1 d1( 1113857 +d2

HNc1 d1

1113944

N

J1

ln J + c2( 1113857

J + c1( 1113857d1

+ ln Hmc2d2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qJ + c1( 1113857

d1HNc1 d1

J + c2( 1113857d2H

Nc2 d2

minusln(middot)⎛⎝ ⎞⎠ minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qJ + c1( 1113857

d1HNc1d1

J + c2( 1113857d2H

Nc2d2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(75)

=e Next Result for Hybrid ZipfndashMandelbrot Entropy Fur-ther generalization of ZipfndashMandelbrot entropy is Hybrid

ZipfndashMandelbrot entropy Let N isin 1 2 cge 0ωgt 0then Hybrid ZipfndashMandelbrot entropy can be given as

1113954ZM Hlowast c dω( 1113857

1Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873 (76)

where

Hlowastcdω 1113944

N

J1

ωJ

(J + c)d (77)

Consider

qJ Ψ(J N c dω) ωJ

(J + c)dHlowastcdω

(78)

which is called Hybrid ZipfndashMandelbrot law =ere is aunified approach maximization of Shannon entropy [44]

that naturally follows the path of generalization from Zipfrsquos toHybrid Zipfrsquos law Extending this idea Jaksetic et al in [45]presented a transition from ZipfndashMandelbrot to HybridZipfndashMandelbrot law by employing maximum entropytechnique with one additional constraint It is interesting thatexamination of its densities provides some new insights ofLerchrsquos transcendent

Theorem 15 Let q be Hybrid ZipfndashMandelbrot law asdefined in (78) with parameters cge 0 dωgt 0 andN isin 1 2 and we have

1113954ZM Hlowast c dω( 1113857 S(q)leN 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠ minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠

⎧⎨

⎩

⎫⎬

⎭

(79)

14 Journal of Mathematics

Proof It is interesting to see that for qJ defined in (78)1113936

NJ1 qJ 1 erefore using above qJ in Shannon en-

tropy (64) we get Hybrid ZipfndashMandelbrot entropy (76)

S(q) minus 1113944N

J1qJ ln qJ minus 1113944

N

J1

ωJ

(J + c)dHlowastcdω

lnωJ

(J + c)dHlowastcdω

minus1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

lnωJ

(J + c)d

1113888 1113889 + ln1

Hlowastcdω

1113888 11138891113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

ln(J + c)

d

ωJ1113888 1113889 + ln H

lowastcdω1113872 11138731113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873

(80)

Finally substituting this qJ ωJ(J + c)dHlowastcdω inCorollary 10 (ii) we get the desired result

Corollary 13 Let q and p be Hybrid ZipfndashMandelbrotlaw with parameters c1 c2 isin [0infin)ω1ω2 d1 d2 gt 0 Nowusing qJ ωJ

1 (J + c1)d1Hlowastc1 d1 ω1

and pJ ωJ2

(J + c2)d2Hlowastc2 d2 ω2

in Corollary 11 (ii) the following holds

D(q p) 1113944N

J1

ωJ1

J + c1( 1113857d1Hlowastc1 d1 ω1

lnωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

ωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

⎛⎝ ⎞⎠

minus1113954ZM Hlowast c1 d1ω1( 1113857 +

1Hlowastc1 d1 ω1

1113944

N

J1

ωJ1

J + c1( 1113857d1ln

J + c2( 1113857d2

ωJ2

⎛⎝ ⎞⎠ + ln Hlowastc2 d2 ω2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎫⎪⎬

⎪⎭

(81)

Remark 9 Similarly we can give results for Shannonentropy KullbackndashLeibler distance ZipfndashMandelbrotentropy and Hybrid ZipfndashMandelbrot entropy by usinggeneralized Giaccardi inequality defined in (54) on thesame steps

4 Concluding Remarks

In this paper we gave generalization of Jensenrsquos inequality aswell as converse of Jensenrsquos inequality by using Montgomeryidentity We also formulate results for other inequalities like

JensenndashSteffensen inequality JensenndashBoas inequality andJensenndashBrunk inequality We can obtain JensenndashSteffenseninequality JensenndashBoas inequality and JensenndashBrunk in-equality by changing the assumption of Jensenrsquos inequalityAt the end we gave applications in information theory forour obtained results especially we gave results for HybridZipfndashMandelbrot entropy for our obtained results [46]

Data Availability

No data were used to support this study

Journal of Mathematics 15

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is research is supported by HEC Pakistan under NRPUProject 7906 and RUDN University Strategic AcademicLeadership Program

References

[1] X L Zhu and G H Yang ldquoJensen inequality approach tostability analysis of discrete-time systems with time-varyingdelayrdquo in Proceedings of the American Control Conferencepp 1644ndash1649 Seattle WA USA June 2008

[2] S I Butt M Klaricic Bakula ETH Pecaric and J PecaricldquoJensen-gruss inequality and its applications for the zipf-mandelbrot lawrdquo Mathematical Methods in the Applied Sci-ences vol 44 no 2 pp 1664ndash1673 2021

[3] M A Khan D Pecaric and J Pecaric ldquoNew refinement of theJensen inequality associated to certain functions with appli-cationsrdquo Journal of Inequalities and Applications vol 762020

[4] S Khan M Adil Khan S I Butt and Y-M Chu ldquoA newbound for the Jensen gap pertaining twice differentiablefunctions with applicationsrdquo Advances in Difference Equa-tions vol 1 2020

[5] H Budak andM Z Sarikaya ldquoOn generalized Ostrowski-typeinequalities for functions whose first derivatives absolutevalues are convexrdquo Turkish Journal of Mathematics vol 40pp 1193ndash1210 2016

[6] P Cerone and S S Dragomir ldquoOn some inequalities arisingfrom Montgomeryrsquos identityrdquo Journal of ComputationalAnalysis and Applications vol 5 no 4 pp 341ndash367 2003

[7] S Bernstein ldquoSur les fonctions absolument monotonesrdquo ActaMathematica vol 52 pp 1ndash66 1929

[8] V Culjak and J Pecaric ldquoInterpolation polynomials andinequalities for convex functions of higher orderrdquo ActaMathematica vol 5 pp 369ndash386 2012

[9] A M Fink ldquoBounds of the deviation of a function from itsaveragesrdquo Czechoslovak Mathematical Journal vol 42no 117 pp 289ndash310 1992

[10] J Pecaric and V Culjak ldquoInterpolation polinomials and in-equalities for convex functions of higher orderrdquo Mathe-matical Inequalities amp Applications vol 5 no 3 pp 369ndash3862002

[11] J Pecaric V Culjak and A M Fink ldquoOn some inequalitiesfor convex function of higher orderrdquo Advanced NonlinearStudies vol 6 no 2 pp 131ndash140 1999

[12] J Pecaric V Culjak andM Rogina ldquoOn some inequalities forconvex function of higher order IIrdquo Advanced NonlinearStudies vol 45 pp 281ndash294 2001

[13] J Pecaric I Peric and M Rodic-Lipanovic ldquoUniformtreatment of Jensen type inequalitiesrdquo Mathematical Reportsvol 16 no 66 pp 183ndash205 2014

[14] J Pecaric and J Peric ldquoImprovement of the Giaccardi and thepetrovic inequality and related stolarsky type meansrdquo Annalsof the University of Craiova - Mathematics and ComputerScience vol 39 no 1 pp 65ndash75 2012

[15] J Pecaric M Praljak and A Witkowski ldquoLinear operatorinequality for nminusconvex functions at a pointrdquo MathematicalInequalities amp Applications Appl vol 18 pp 1201ndash1217 2015

[16] Z Pavic ldquoRefinements of Jensenrsquos inequality for infiniteconvex combinationsrdquo Turkish Journal of Inequalities vol 2no 2 pp 44ndash53 2018

[17] R Bibi A Nosheen and J Pecaric ldquoExtended Jensenrsquos typeinequalities for diamond integrals via Taylors formulardquoTurkish Journal of Inequalities vol 3 no 1 pp 7ndash18 2019

[18] T Niaz K A Khan and J Pecaric ldquoOn refinement of Jensenrsquosinequality for 3minusconvex function at a pointrdquo Turkish Journalof Inequalities vol 4 no 1 pp 70ndash80 2020

[19] A Ekinci A O Akdemir and M E Ozdemir ldquoIntegralinequalities for different kinds of convexity via classical in-equalitiesrdquo Turkish Journal of Science vol 5 no 3 pp 305ndash313 2020

[20] Y M Chu S Talib E Set M U Awan and M A Noorldquo(p q)minus Analysis of Montgomery identity and estimates of(p q)minusbounds with applicationsrdquo Journal of Inequalities andApplications vol 2021 9 pages 2021

[21] M Z Sarikaya H Yaldrz and E Set ldquoOn fractional in-equalities via Montgomery identitiesrdquo International Journalof Open Problems in Complex Analysis vol 6 no 2 pp 36ndash432014

[22] A Bnouhachem ldquoA descent SQP alternating directionmethod for minimizing the sum of three convex functionsrdquoJournal of Nonlinear Functional Analysis vol 4 pp 469ndash4822020

[23] A O Akdemir S I Butt M Nadeem and M A RagusaldquoNew general variants of Chebyshev type inequalities viageneralized fractional integral operatorsrdquo Mathematicsvol 92 2021

[24] S I Butt M Umar S Rashid A O Akdemir and Y MingChu ldquoNew Hermite Jensen Mercer type inequalities viakminusfractional integralsrdquo Advances in Difference Equationsvol 635 2020

[25] S I Butt M Nadeem S Qaisar A O Akdemir andT Abdeljawad ldquoHermite Jensen Mercer type inequalities forconformable integrals and related resultsrdquo Advances in Dif-ference Equations vol 501 2020

[26] A Bnouhachem and X Qin ldquoAn inertial proximal Peaceman-Rachford splittingmethod with SQP regularization for convexprogrammingrdquo Journal of Nonlinear Functional Analysisvol 50 2020

[27] S Hamann ldquoMinimality conditions for convex compositefunctions and efficiency conditions in vector optimizationrdquoApplied Set-Valued Analysis and Optimization vol 1pp 221ndash229 2019

[28] A R Khan J Pecaric and M Praljak ldquoPopoviciu type in-equalities for n-convex functions via extension of Mont-gomery identitynminusconvex functions via extension ofMontgomery identityrdquo Analele Universitatii ldquoOvidiusrdquo Con-stanta-Seria Matematica vol 24 no 3 pp 161ndash188 2016

[29] R P Boas ldquoe Jensen-Steffensen inequalityrdquo ElektrotehnFak Ser Mat Fiz vol 302ndash319 pp 1ndash8 1970

[30] J Pecaric F Proschan and Y L Tong Convex FunctionsPartial Orderings and Statistical Applications Academic PressNew York NY USA 1992

[31] H D Brunk ldquoOn an inequality for convex functionsrdquo Pro-ceedings of the American Mathematical Society vol 7 no 5p 817 1956

[32] P R Beesack and J E Pecaric ldquoOn Jessenrsquos inequality forconvex functionsrdquo Journal of Mathematical Analysis andApplications vol 110 no 2 pp 536ndash552 1985

[33] R Barlow A Marshall and F Proschan ldquoSome inequalitiesfor starshaped and convex functionsrdquo Pacific Journal ofMathematics vol 29 no 1 pp 19ndash42 1969

16 Journal of Mathematics

[34] J E Pecaric ldquoOn an inequality of PMV Janicrdquo Publications delrsquoInstitut Mathematique vol 28 no 42 pp 145ndash149 1980

[35] G Szego ldquoUber eine Verallgemeinerung des DirichletschenIntegralsrdquo Mathematische Zeitschrift vol 52 no 1pp 676ndash685 1950

[36] F Giaccardi ldquoSu alcune disuguaglianierdquo Journal of Researchof the National Bureau of Standards vol 1 no 4 pp 139ndash1531955

[37] D S Mitrinovic J E Pecaric and A M Fink Classical andnew inequalities in analysis Mathematics and its Applications(East European Series) Vol 61 Kluwer Academic PublishersGroup Dordrecht Netherlands 1993

[38] I Csiszar ldquoInformation-type measures of diference ofprobability distributions and indirect observationsrdquo StudiaScience Math Hungar vol 2 pp 299ndash318 1967

[39] L Horvath D Pecaric and J Pecaric ldquoEstimations of f - andrenyi divergences by using a cyclic refinement of the Jensenrsquosinequalityrdquo Bulletin of the Malaysian Mathematical SciencesSociety vol 42 2017

[40] S Kullback Information =eory and Statistics Wiley NewYork NY USA 1959

[41] S T Piantadosi ldquoZipfrsquos word frequency law in natural lan-guage a critical review and future directionsrdquo PsychonomicBulletin and Review vol 21 no 5 pp 1112ndash1130 2014

[42] Z K Silagadze ldquoCitations and the zipfndashmandelbrot lawrdquoComplex Systems no 11 pp 487ndash499 1997

[43] J Jaksetic D Pecaric and J Pecaric ldquoSome properties of zipf-mandelbrot law and hurwitz ζ-functionrdquo Mathematical In-equalities amp Applications vol 21 no 2 pp 575ndash584 2018

[44] M Visser ldquoZipfrsquos law power laws and maximum entropyrdquoMathematical Inequalities amp Applications vol 15 Article ID043021 2013

[45] J Jaksetic D Pecaric and J Pecaric ldquoHybrid zipf-mandelbrotlawrdquo Mathematical Inequalities amp Applications vol 13 no 1pp 275ndash286 2019

[46] D V Widder ldquoCompletely convex functions and Lidstoneseriesrdquo Transactions of the American Mathematical Societyvol 51 p 387 1942

Journal of Mathematics 17

A 1113936

mJ1 pJ xJ minus x01113872 11138731113872 1113873

1113936mJ1 pJxJ minus x01113872 1113873

B 1113936

mJ1 pJxJ

1113936mJ1 pJxJ minus x01113872 1113873

and 1113944m

J1pJxJ ne x0 (48)

Theorem 9 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also letx0 xJ isin [a b]sube[α β] pJ isin R(J 1 m) be such that1113936

mJ1 pJxJ ne x0

(i) =en the following generalized identity holds

CJdis(Ψ) 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)CJdis xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)CJdis xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875

+1

(n minus 1)1113946β

αCJdis Rn xJ s1113872 11138731113872 1113873Ψ(n)

(s)ds (DCGI)

(49)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

CJdis Rn xJ s1113872 11138731113872 1113873le 0 (50)

holds then we have the following generalized inequality

CJdis(Ψ)le 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)CJdis xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)CJdis xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875 (51)

If inequality (50) holds in reverse direction then (51) alsoholds reversely

Theorem 10 Let Ψ [α β]⟶ R be such that for nge 1Ψ(nminus1) is absolutely continuous Also let xJ isin [a b]sube[α β]pJ isin R(J 1 m) be such that Pm ne 0 and x isin [α β]

(i) =en the following generalized Giaccardi identityholds

Gcardi(Ψ) 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)Gcardi xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)Gcardi xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875

+1

(n minus 1)1113946β

αGcardi Rn xJ s1113872 11138731113872 1113873Ψ(n)

(s)ds (GIAGI)

(52)

where Rn(middot s) is defined in (5)(ii) Moreover if Ψ is nminusconvex and the inequality

Gcardi Rn xJ s1113872 11138731113872 1113873le 0 (53)

holds then we have the following generalized Giaccardiinequality

Gcardi(Ψ)le 1113944nminus2

ℓ0

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 Ψ(ℓ+1)(α)Gcardi xJ minus α1113872 1113873

ℓ+21113874 1113875 minus Ψ(ℓ+1)

(β)Gcardi xJ minus β1113872 1113873ℓ+2

1113874 11138751113874 1113875 (54)

If inequality (53) holds in reverse direction then (54) alsoholds reversely

In the later part of this section we will vary our con-ditions on pJxJ (J 1 2 m) to obtain generalized

10 Journal of Mathematics

converse discrete variants of Jensenrsquos inequality and Giac-cardi inequality for nminusconvex functions

Theorem 11 Let Ψ be as defined in =eorem 9 Also letxJ isin [a b]sube [α β] and p be a positive mminustuple

(i) If Ψ is nminusconvex then for even nge 3 (50) is valid(ii) Moreover if (50) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized converse of Jensenrsquos inequality

1Pm

1113944

m

J1pJΨ xJ1113872 1113873le

b minus x

b minus aΨ(a) +

x minus a

b minus aΨ(b) (55)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ converse of Jensenrsquosinequality (see [32] or [30] p 98) to obtain results

Finally in this section we give Giaccardi inequality forhigher-order convex functions

Theorem 12 Let Ψ be as defined in =eorem 9 Also letx0 xJ isin [a b]sube[α β] and p be a positive mminustuple such that

1113944

m

J1pJxJ nex0 and xv minus x0( 1113857 1113944

m

J1pJxJ minus xv

⎛⎝ ⎞⎠ge 0 (v 1 m) (56)

(i) If Ψ is nminusconvex then for even nge 3 (53) is valid(ii) Moreover if (53) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized Giaccardi inequality

1113944

m

J1pJΨ xJ1113872 1113873leAΨ 1113944

m

J1pJxJ

⎛⎝ ⎞⎠ + B 1113944m

J1pJ minus 1⎛⎝ ⎞⎠Ψ x0( 1113857

(57)

where A and B are defined in (47)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ Giaccardi inequality(see [36] or [37] p 11) to obtain results

33 Applications in Information =eory for Jensenrsquos DiscreteInequality Jensenrsquos inequality plays a key role in infor-mation theory to construct lower bounds for some notableinequalities but here we will use it to make connectionsbetween inequalities in information theory

Let Ψ R+⟶ R+ be a convex function and letp ≔ (p1 pm) and q ≔ (q1 qm) be positive proba-bility distributions then Ψ-divergence functional is defined(in [38]) as follows

IΨ(p q) 1113944m

J1qJΨ

pJ

qJ1113888 1113889 (58)

Horvath et al in [39] defined the generalized Csiszardivergence functional as follows

Definition 2 Let I be an interval in R and Ψ I⟶ R be afunction Also let p ≔ (p1 pm) isin Rm andq ≔ (q1 qm) isin (0infin)m such that

pJ

qJisin I J 1 m (59)

en let

1113957IΨ(p q) 1113944m

J1qJΨ

pJ

qJ1113888 1113889 (60)

In this section we write Jensenrsquos difference here that weuse in upcoming results

F p xJΨ1113872 1113873 Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 (61)

Theorem 13 Under the assumptions of =eorem 9 (ii) let(51) hold and Ψ be nminusconvex Also letp ≔ (p1 pm) inRm and q ≔ (q1 qm) isin (0infin)mthen we have the following results

Journal of Mathematics 11

1113957IΨ(p q)gePmΨ(1) minus Pm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F q

pJ

qJ (x minus α)

ℓ+21113888 11138891113890 1113891 minus Ψ(ℓ+1)

(β) F qpJ

qJ (x minus β)

ℓ+21113888 11138891113890 11138911113896 1113897

(62)

Proof From eorem 9 by following Jensenrsquos difference(61) we can rearrange (34) as

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873le 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F p xJ (x minus α)

ℓ+21113872 11138731113872 1113873 minus Ψ(ℓ+1)

(β) F p xJ (x minus β)ℓ+2

1113872 11138731113872 11138731113966 1113967

(63)

Now replace pJ with qJ and xJ with pJqJ and we get(62)

For positive n-tuple q (q1 qm) such that1113936

mJ1 qJ 1 the Shannon entropy is defined by

S(q) minus 1113944m

J1qJ ln qJ (64)

Corollary 10 Under the assumptions of =eorem 9 (ii) let(51) hold and Ψ be nminusconvex

(i) If q ≔ (q1 qm) isin (0infin)m then

1113944

m

J1qJ ln qJ lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (65)

(ii) We can get bounds for the Shannon entropy of q if wechoose q ≔ (q1 qn) to be a positive probabilitydistribution

S(q)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (66)

Proof (i) Substituting Ψ(x) ≔ minus lnx and usingp ≔ (1 1 1) in eorem 13 we get (65)

(ii) Since we have 1113936mJ1 qJ 1 by multiplying minus1 on

both sides of (65) and taking into account (64) weget (66)

e KullbackndashLeibler distance [40] between the positiveprobability distributions p (p1 pm) andq (q1 qm) is defined by

D(q p) 1113944

m

J1qJ ln

qJ

pJ

1113888 1113889 (67)

Corollary 11 Under the assumptions of Corollary 10

(i) If q ≔ (q1 qm) p ≔ (p1 pm) isin (0infin)mthen

1113944

m

J1qJ ln

qJ

pJ

1113888 1113889lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

pJ

qJ minusln(middot)1113888 1113889

⎧⎨

⎩

⎫⎬

⎭ (68)

(ii) If q ≔ (q1 qm) p ≔ (p1 pm) are positiveprobability distributions then we have

12 Journal of Mathematics

D(q p)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

βℓ+1F q

pJ

qJ minusln(middot)1113888 11138891113896 1113897 (69)

Proof

(i) UsingΨ(x) ≔ minus lnx (which isn-convex for evenn)in eorem 13 we get (68) after simplification

(ii) It is a special case of (i)

34 Results for Zipf and Hybrid ZipfndashMandelbrot EntropyOne of the basic laws in information science is Zipfrsquos law[4142] which is highly applied in linguistics Let cge 0 dgt 0and N isin 1 2 ZipfndashMandelbrot entropy can be given as

ZM(H c d) d

HNcd

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (70)

where

HNcd 1113944

N

σ1

1(σ + c)

d (71)

Consider

qJ Ψ(J N c d) 1

(J + c)dH

Ncd1113872 1113873

(72)

where Ψ(J m c d) is discrete probability distributionknown as ZipfndashMandelbrot law ZipfndashMandelbrot law hasmany application in linguistics and information sciencesSome of the recent study about ZipfndashMandelbrot law can beseen in the listed references (see [39 43]) Now we state ourresults involving entropy introduced by Mandelbrot law byestablishing the relationship with Shannon and relativeentropies

Theorem 14 Let q be ZipfndashMandelbrot law as defined in(72) with parameters cge 0 dgt 0 and N isin 1 2 and wehave

ZM(H c d) S(q)leN times 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873

⎧⎨

⎩

⎫⎬

⎭

(73)

Proof It is interesting to see that for qJ defined in (72)1113936

NJ1 qJ 1 erefore using above qJ in Shannon entropy

(64) we get Mandelbrot entropy (70)

S(q) minus 1113944

N

J1qJ ln qJ minus 1113944

N

J1

1(J + c)

dH

Ncd1113872 1113873

ln1

(J + c)dH

Ncd1113872 1113873

d

HNcd1113872 1113873

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (74)

Finally substituting this qJ 1((J + c)dHNcd) in

Corollary 10 (ii) we get the desired result Corollary 12 Let q and p be ZipfndashMandelbrot law withparameters c1 c2 isin [0infin) d1 d2 gt 0 and let

Journal of Mathematics 13

HNc1 d1

1113936Nσ1 1(σ + c1)

d1 and HNc2 d2

1113936Nσ1 1(σ + c2)

d2 Now using qJ 1(J + c1)

d1HNc1 d1

andpJ 1(J + c2)

d2HNc2 d2

in Corollary 11 (ii) the followingholds

D(q p) 1113944N

J1

1J + c1( 1113857

d1HNc1d1

lnJ + c2( 1113857

d2HNc2 d2

J + c1( 1113857d1H

mc1 d1

⎛⎝ ⎞⎠

minusZ H c1 d1( 1113857 +d2

HNc1 d1

1113944

N

J1

ln J + c2( 1113857

J + c1( 1113857d1

+ ln Hmc2d2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qJ + c1( 1113857

d1HNc1 d1

J + c2( 1113857d2H

Nc2 d2

minusln(middot)⎛⎝ ⎞⎠ minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qJ + c1( 1113857

d1HNc1d1

J + c2( 1113857d2H

Nc2d2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(75)

=e Next Result for Hybrid ZipfndashMandelbrot Entropy Fur-ther generalization of ZipfndashMandelbrot entropy is Hybrid

ZipfndashMandelbrot entropy Let N isin 1 2 cge 0ωgt 0then Hybrid ZipfndashMandelbrot entropy can be given as

1113954ZM Hlowast c dω( 1113857

1Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873 (76)

where

Hlowastcdω 1113944

N

J1

ωJ

(J + c)d (77)

Consider

qJ Ψ(J N c dω) ωJ

(J + c)dHlowastcdω

(78)

which is called Hybrid ZipfndashMandelbrot law =ere is aunified approach maximization of Shannon entropy [44]

that naturally follows the path of generalization from Zipfrsquos toHybrid Zipfrsquos law Extending this idea Jaksetic et al in [45]presented a transition from ZipfndashMandelbrot to HybridZipfndashMandelbrot law by employing maximum entropytechnique with one additional constraint It is interesting thatexamination of its densities provides some new insights ofLerchrsquos transcendent

Theorem 15 Let q be Hybrid ZipfndashMandelbrot law asdefined in (78) with parameters cge 0 dωgt 0 andN isin 1 2 and we have

1113954ZM Hlowast c dω( 1113857 S(q)leN 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠ minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠

⎧⎨

⎩

⎫⎬

⎭

(79)

14 Journal of Mathematics

Proof It is interesting to see that for qJ defined in (78)1113936

NJ1 qJ 1 erefore using above qJ in Shannon en-

tropy (64) we get Hybrid ZipfndashMandelbrot entropy (76)

S(q) minus 1113944N

J1qJ ln qJ minus 1113944

N

J1

ωJ

(J + c)dHlowastcdω

lnωJ

(J + c)dHlowastcdω

minus1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

lnωJ

(J + c)d

1113888 1113889 + ln1

Hlowastcdω

1113888 11138891113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

ln(J + c)

d

ωJ1113888 1113889 + ln H

lowastcdω1113872 11138731113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873

(80)

Finally substituting this qJ ωJ(J + c)dHlowastcdω inCorollary 10 (ii) we get the desired result

Corollary 13 Let q and p be Hybrid ZipfndashMandelbrotlaw with parameters c1 c2 isin [0infin)ω1ω2 d1 d2 gt 0 Nowusing qJ ωJ

1 (J + c1)d1Hlowastc1 d1 ω1

and pJ ωJ2

(J + c2)d2Hlowastc2 d2 ω2

in Corollary 11 (ii) the following holds

D(q p) 1113944N

J1

ωJ1

J + c1( 1113857d1Hlowastc1 d1 ω1

lnωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

ωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

⎛⎝ ⎞⎠

minus1113954ZM Hlowast c1 d1ω1( 1113857 +

1Hlowastc1 d1 ω1

1113944

N

J1

ωJ1

J + c1( 1113857d1ln

J + c2( 1113857d2

ωJ2

⎛⎝ ⎞⎠ + ln Hlowastc2 d2 ω2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎫⎪⎬

⎪⎭

(81)

Remark 9 Similarly we can give results for Shannonentropy KullbackndashLeibler distance ZipfndashMandelbrotentropy and Hybrid ZipfndashMandelbrot entropy by usinggeneralized Giaccardi inequality defined in (54) on thesame steps

4 Concluding Remarks

In this paper we gave generalization of Jensenrsquos inequality aswell as converse of Jensenrsquos inequality by using Montgomeryidentity We also formulate results for other inequalities like

JensenndashSteffensen inequality JensenndashBoas inequality andJensenndashBrunk inequality We can obtain JensenndashSteffenseninequality JensenndashBoas inequality and JensenndashBrunk in-equality by changing the assumption of Jensenrsquos inequalityAt the end we gave applications in information theory forour obtained results especially we gave results for HybridZipfndashMandelbrot entropy for our obtained results [46]

Data Availability

No data were used to support this study

Journal of Mathematics 15

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is research is supported by HEC Pakistan under NRPUProject 7906 and RUDN University Strategic AcademicLeadership Program

References

[1] X L Zhu and G H Yang ldquoJensen inequality approach tostability analysis of discrete-time systems with time-varyingdelayrdquo in Proceedings of the American Control Conferencepp 1644ndash1649 Seattle WA USA June 2008

[2] S I Butt M Klaricic Bakula ETH Pecaric and J PecaricldquoJensen-gruss inequality and its applications for the zipf-mandelbrot lawrdquo Mathematical Methods in the Applied Sci-ences vol 44 no 2 pp 1664ndash1673 2021

[3] M A Khan D Pecaric and J Pecaric ldquoNew refinement of theJensen inequality associated to certain functions with appli-cationsrdquo Journal of Inequalities and Applications vol 762020

[4] S Khan M Adil Khan S I Butt and Y-M Chu ldquoA newbound for the Jensen gap pertaining twice differentiablefunctions with applicationsrdquo Advances in Difference Equa-tions vol 1 2020

[5] H Budak andM Z Sarikaya ldquoOn generalized Ostrowski-typeinequalities for functions whose first derivatives absolutevalues are convexrdquo Turkish Journal of Mathematics vol 40pp 1193ndash1210 2016

[6] P Cerone and S S Dragomir ldquoOn some inequalities arisingfrom Montgomeryrsquos identityrdquo Journal of ComputationalAnalysis and Applications vol 5 no 4 pp 341ndash367 2003

[7] S Bernstein ldquoSur les fonctions absolument monotonesrdquo ActaMathematica vol 52 pp 1ndash66 1929

[8] V Culjak and J Pecaric ldquoInterpolation polynomials andinequalities for convex functions of higher orderrdquo ActaMathematica vol 5 pp 369ndash386 2012

[9] A M Fink ldquoBounds of the deviation of a function from itsaveragesrdquo Czechoslovak Mathematical Journal vol 42no 117 pp 289ndash310 1992

[10] J Pecaric and V Culjak ldquoInterpolation polinomials and in-equalities for convex functions of higher orderrdquo Mathe-matical Inequalities amp Applications vol 5 no 3 pp 369ndash3862002

[11] J Pecaric V Culjak and A M Fink ldquoOn some inequalitiesfor convex function of higher orderrdquo Advanced NonlinearStudies vol 6 no 2 pp 131ndash140 1999

[12] J Pecaric V Culjak andM Rogina ldquoOn some inequalities forconvex function of higher order IIrdquo Advanced NonlinearStudies vol 45 pp 281ndash294 2001

[13] J Pecaric I Peric and M Rodic-Lipanovic ldquoUniformtreatment of Jensen type inequalitiesrdquo Mathematical Reportsvol 16 no 66 pp 183ndash205 2014

[14] J Pecaric and J Peric ldquoImprovement of the Giaccardi and thepetrovic inequality and related stolarsky type meansrdquo Annalsof the University of Craiova - Mathematics and ComputerScience vol 39 no 1 pp 65ndash75 2012

[15] J Pecaric M Praljak and A Witkowski ldquoLinear operatorinequality for nminusconvex functions at a pointrdquo MathematicalInequalities amp Applications Appl vol 18 pp 1201ndash1217 2015

[16] Z Pavic ldquoRefinements of Jensenrsquos inequality for infiniteconvex combinationsrdquo Turkish Journal of Inequalities vol 2no 2 pp 44ndash53 2018

[17] R Bibi A Nosheen and J Pecaric ldquoExtended Jensenrsquos typeinequalities for diamond integrals via Taylors formulardquoTurkish Journal of Inequalities vol 3 no 1 pp 7ndash18 2019

[18] T Niaz K A Khan and J Pecaric ldquoOn refinement of Jensenrsquosinequality for 3minusconvex function at a pointrdquo Turkish Journalof Inequalities vol 4 no 1 pp 70ndash80 2020

[19] A Ekinci A O Akdemir and M E Ozdemir ldquoIntegralinequalities for different kinds of convexity via classical in-equalitiesrdquo Turkish Journal of Science vol 5 no 3 pp 305ndash313 2020

[20] Y M Chu S Talib E Set M U Awan and M A Noorldquo(p q)minus Analysis of Montgomery identity and estimates of(p q)minusbounds with applicationsrdquo Journal of Inequalities andApplications vol 2021 9 pages 2021

[21] M Z Sarikaya H Yaldrz and E Set ldquoOn fractional in-equalities via Montgomery identitiesrdquo International Journalof Open Problems in Complex Analysis vol 6 no 2 pp 36ndash432014

[22] A Bnouhachem ldquoA descent SQP alternating directionmethod for minimizing the sum of three convex functionsrdquoJournal of Nonlinear Functional Analysis vol 4 pp 469ndash4822020

[23] A O Akdemir S I Butt M Nadeem and M A RagusaldquoNew general variants of Chebyshev type inequalities viageneralized fractional integral operatorsrdquo Mathematicsvol 92 2021

[24] S I Butt M Umar S Rashid A O Akdemir and Y MingChu ldquoNew Hermite Jensen Mercer type inequalities viakminusfractional integralsrdquo Advances in Difference Equationsvol 635 2020

[25] S I Butt M Nadeem S Qaisar A O Akdemir andT Abdeljawad ldquoHermite Jensen Mercer type inequalities forconformable integrals and related resultsrdquo Advances in Dif-ference Equations vol 501 2020

[26] A Bnouhachem and X Qin ldquoAn inertial proximal Peaceman-Rachford splittingmethod with SQP regularization for convexprogrammingrdquo Journal of Nonlinear Functional Analysisvol 50 2020

[27] S Hamann ldquoMinimality conditions for convex compositefunctions and efficiency conditions in vector optimizationrdquoApplied Set-Valued Analysis and Optimization vol 1pp 221ndash229 2019

[28] A R Khan J Pecaric and M Praljak ldquoPopoviciu type in-equalities for n-convex functions via extension of Mont-gomery identitynminusconvex functions via extension ofMontgomery identityrdquo Analele Universitatii ldquoOvidiusrdquo Con-stanta-Seria Matematica vol 24 no 3 pp 161ndash188 2016

[29] R P Boas ldquoe Jensen-Steffensen inequalityrdquo ElektrotehnFak Ser Mat Fiz vol 302ndash319 pp 1ndash8 1970

[30] J Pecaric F Proschan and Y L Tong Convex FunctionsPartial Orderings and Statistical Applications Academic PressNew York NY USA 1992

[31] H D Brunk ldquoOn an inequality for convex functionsrdquo Pro-ceedings of the American Mathematical Society vol 7 no 5p 817 1956

[32] P R Beesack and J E Pecaric ldquoOn Jessenrsquos inequality forconvex functionsrdquo Journal of Mathematical Analysis andApplications vol 110 no 2 pp 536ndash552 1985

[33] R Barlow A Marshall and F Proschan ldquoSome inequalitiesfor starshaped and convex functionsrdquo Pacific Journal ofMathematics vol 29 no 1 pp 19ndash42 1969

16 Journal of Mathematics

[34] J E Pecaric ldquoOn an inequality of PMV Janicrdquo Publications delrsquoInstitut Mathematique vol 28 no 42 pp 145ndash149 1980

[35] G Szego ldquoUber eine Verallgemeinerung des DirichletschenIntegralsrdquo Mathematische Zeitschrift vol 52 no 1pp 676ndash685 1950

[36] F Giaccardi ldquoSu alcune disuguaglianierdquo Journal of Researchof the National Bureau of Standards vol 1 no 4 pp 139ndash1531955

[37] D S Mitrinovic J E Pecaric and A M Fink Classical andnew inequalities in analysis Mathematics and its Applications(East European Series) Vol 61 Kluwer Academic PublishersGroup Dordrecht Netherlands 1993

[38] I Csiszar ldquoInformation-type measures of diference ofprobability distributions and indirect observationsrdquo StudiaScience Math Hungar vol 2 pp 299ndash318 1967

[39] L Horvath D Pecaric and J Pecaric ldquoEstimations of f - andrenyi divergences by using a cyclic refinement of the Jensenrsquosinequalityrdquo Bulletin of the Malaysian Mathematical SciencesSociety vol 42 2017

[40] S Kullback Information =eory and Statistics Wiley NewYork NY USA 1959

[41] S T Piantadosi ldquoZipfrsquos word frequency law in natural lan-guage a critical review and future directionsrdquo PsychonomicBulletin and Review vol 21 no 5 pp 1112ndash1130 2014

[42] Z K Silagadze ldquoCitations and the zipfndashmandelbrot lawrdquoComplex Systems no 11 pp 487ndash499 1997

[43] J Jaksetic D Pecaric and J Pecaric ldquoSome properties of zipf-mandelbrot law and hurwitz ζ-functionrdquo Mathematical In-equalities amp Applications vol 21 no 2 pp 575ndash584 2018

[44] M Visser ldquoZipfrsquos law power laws and maximum entropyrdquoMathematical Inequalities amp Applications vol 15 Article ID043021 2013

[45] J Jaksetic D Pecaric and J Pecaric ldquoHybrid zipf-mandelbrotlawrdquo Mathematical Inequalities amp Applications vol 13 no 1pp 275ndash286 2019

[46] D V Widder ldquoCompletely convex functions and Lidstoneseriesrdquo Transactions of the American Mathematical Societyvol 51 p 387 1942

Journal of Mathematics 17

converse discrete variants of Jensenrsquos inequality and Giac-cardi inequality for nminusconvex functions

Theorem 11 Let Ψ be as defined in =eorem 9 Also letxJ isin [a b]sube [α β] and p be a positive mminustuple

(i) If Ψ is nminusconvex then for even nge 3 (50) is valid(ii) Moreover if (50) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized converse of Jensenrsquos inequality

1Pm

1113944

m

J1pJΨ xJ1113872 1113873le

b minus x

b minus aΨ(a) +

x minus a

b minus aΨ(b) (55)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ converse of Jensenrsquosinequality (see [32] or [30] p 98) to obtain results

Finally in this section we give Giaccardi inequality forhigher-order convex functions

Theorem 12 Let Ψ be as defined in =eorem 9 Also letx0 xJ isin [a b]sube[α β] and p be a positive mminustuple such that

1113944

m

J1pJxJ nex0 and xv minus x0( 1113857 1113944

m

J1pJxJ minus xv

⎛⎝ ⎞⎠ge 0 (v 1 m) (56)

(i) If Ψ is nminusconvex then for even nge 3 (53) is valid(ii) Moreover if (53) is valid and the function H(middot) de-

fined in (12) is convex then we get the followinggeneralized Giaccardi inequality

1113944

m

J1pJΨ xJ1113872 1113873leAΨ 1113944

m

J1pJxJ

⎛⎝ ⎞⎠ + B 1113944m

J1pJ minus 1⎛⎝ ⎞⎠Ψ x0( 1113857

(57)

where A and B are defined in (47)

Proof We follow the idea of eorem 8 but according toour assumed conditions we employ Giaccardi inequality(see [36] or [37] p 11) to obtain results

33 Applications in Information =eory for Jensenrsquos DiscreteInequality Jensenrsquos inequality plays a key role in infor-mation theory to construct lower bounds for some notableinequalities but here we will use it to make connectionsbetween inequalities in information theory

Let Ψ R+⟶ R+ be a convex function and letp ≔ (p1 pm) and q ≔ (q1 qm) be positive proba-bility distributions then Ψ-divergence functional is defined(in [38]) as follows

IΨ(p q) 1113944m

J1qJΨ

pJ

qJ1113888 1113889 (58)

Horvath et al in [39] defined the generalized Csiszardivergence functional as follows

Definition 2 Let I be an interval in R and Ψ I⟶ R be afunction Also let p ≔ (p1 pm) isin Rm andq ≔ (q1 qm) isin (0infin)m such that

pJ

qJisin I J 1 m (59)

en let

1113957IΨ(p q) 1113944m

J1qJΨ

pJ

qJ1113888 1113889 (60)

In this section we write Jensenrsquos difference here that weuse in upcoming results

F p xJΨ1113872 1113873 Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873 (61)

Theorem 13 Under the assumptions of =eorem 9 (ii) let(51) hold and Ψ be nminusconvex Also letp ≔ (p1 pm) inRm and q ≔ (q1 qm) isin (0infin)mthen we have the following results

Journal of Mathematics 11

1113957IΨ(p q)gePmΨ(1) minus Pm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F q

pJ

qJ (x minus α)

ℓ+21113888 11138891113890 1113891 minus Ψ(ℓ+1)

(β) F qpJ

qJ (x minus β)

ℓ+21113888 11138891113890 11138911113896 1113897

(62)

Proof From eorem 9 by following Jensenrsquos difference(61) we can rearrange (34) as

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873le 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F p xJ (x minus α)

ℓ+21113872 11138731113872 1113873 minus Ψ(ℓ+1)

(β) F p xJ (x minus β)ℓ+2

1113872 11138731113872 11138731113966 1113967

(63)

Now replace pJ with qJ and xJ with pJqJ and we get(62)

For positive n-tuple q (q1 qm) such that1113936

mJ1 qJ 1 the Shannon entropy is defined by

S(q) minus 1113944m

J1qJ ln qJ (64)

Corollary 10 Under the assumptions of =eorem 9 (ii) let(51) hold and Ψ be nminusconvex

(i) If q ≔ (q1 qm) isin (0infin)m then

1113944

m

J1qJ ln qJ lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (65)

(ii) We can get bounds for the Shannon entropy of q if wechoose q ≔ (q1 qn) to be a positive probabilitydistribution

S(q)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (66)

Proof (i) Substituting Ψ(x) ≔ minus lnx and usingp ≔ (1 1 1) in eorem 13 we get (65)

(ii) Since we have 1113936mJ1 qJ 1 by multiplying minus1 on

both sides of (65) and taking into account (64) weget (66)

e KullbackndashLeibler distance [40] between the positiveprobability distributions p (p1 pm) andq (q1 qm) is defined by

D(q p) 1113944

m

J1qJ ln

qJ

pJ

1113888 1113889 (67)

Corollary 11 Under the assumptions of Corollary 10

(i) If q ≔ (q1 qm) p ≔ (p1 pm) isin (0infin)mthen

1113944

m

J1qJ ln

qJ

pJ

1113888 1113889lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

pJ

qJ minusln(middot)1113888 1113889

⎧⎨

⎩

⎫⎬

⎭ (68)

(ii) If q ≔ (q1 qm) p ≔ (p1 pm) are positiveprobability distributions then we have

12 Journal of Mathematics

D(q p)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

βℓ+1F q

pJ

qJ minusln(middot)1113888 11138891113896 1113897 (69)

Proof

(i) UsingΨ(x) ≔ minus lnx (which isn-convex for evenn)in eorem 13 we get (68) after simplification

(ii) It is a special case of (i)

34 Results for Zipf and Hybrid ZipfndashMandelbrot EntropyOne of the basic laws in information science is Zipfrsquos law[4142] which is highly applied in linguistics Let cge 0 dgt 0and N isin 1 2 ZipfndashMandelbrot entropy can be given as

ZM(H c d) d

HNcd

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (70)

where

HNcd 1113944

N

σ1

1(σ + c)

d (71)

Consider

qJ Ψ(J N c d) 1

(J + c)dH

Ncd1113872 1113873

(72)

where Ψ(J m c d) is discrete probability distributionknown as ZipfndashMandelbrot law ZipfndashMandelbrot law hasmany application in linguistics and information sciencesSome of the recent study about ZipfndashMandelbrot law can beseen in the listed references (see [39 43]) Now we state ourresults involving entropy introduced by Mandelbrot law byestablishing the relationship with Shannon and relativeentropies

Theorem 14 Let q be ZipfndashMandelbrot law as defined in(72) with parameters cge 0 dgt 0 and N isin 1 2 and wehave

ZM(H c d) S(q)leN times 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873

⎧⎨

⎩

⎫⎬

⎭

(73)

Proof It is interesting to see that for qJ defined in (72)1113936

NJ1 qJ 1 erefore using above qJ in Shannon entropy

(64) we get Mandelbrot entropy (70)

S(q) minus 1113944

N

J1qJ ln qJ minus 1113944

N

J1

1(J + c)

dH

Ncd1113872 1113873

ln1

(J + c)dH

Ncd1113872 1113873

d

HNcd1113872 1113873

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (74)

Finally substituting this qJ 1((J + c)dHNcd) in

Corollary 10 (ii) we get the desired result Corollary 12 Let q and p be ZipfndashMandelbrot law withparameters c1 c2 isin [0infin) d1 d2 gt 0 and let

Journal of Mathematics 13

HNc1 d1

1113936Nσ1 1(σ + c1)

d1 and HNc2 d2

1113936Nσ1 1(σ + c2)

d2 Now using qJ 1(J + c1)

d1HNc1 d1

andpJ 1(J + c2)

d2HNc2 d2

in Corollary 11 (ii) the followingholds

D(q p) 1113944N

J1

1J + c1( 1113857

d1HNc1d1

lnJ + c2( 1113857

d2HNc2 d2

J + c1( 1113857d1H

mc1 d1

⎛⎝ ⎞⎠

minusZ H c1 d1( 1113857 +d2

HNc1 d1

1113944

N

J1

ln J + c2( 1113857

J + c1( 1113857d1

+ ln Hmc2d2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qJ + c1( 1113857

d1HNc1 d1

J + c2( 1113857d2H

Nc2 d2

minusln(middot)⎛⎝ ⎞⎠ minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qJ + c1( 1113857

d1HNc1d1

J + c2( 1113857d2H

Nc2d2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(75)

=e Next Result for Hybrid ZipfndashMandelbrot Entropy Fur-ther generalization of ZipfndashMandelbrot entropy is Hybrid

ZipfndashMandelbrot entropy Let N isin 1 2 cge 0ωgt 0then Hybrid ZipfndashMandelbrot entropy can be given as

1113954ZM Hlowast c dω( 1113857

1Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873 (76)

where

Hlowastcdω 1113944

N

J1

ωJ

(J + c)d (77)

Consider

qJ Ψ(J N c dω) ωJ

(J + c)dHlowastcdω

(78)

which is called Hybrid ZipfndashMandelbrot law =ere is aunified approach maximization of Shannon entropy [44]

that naturally follows the path of generalization from Zipfrsquos toHybrid Zipfrsquos law Extending this idea Jaksetic et al in [45]presented a transition from ZipfndashMandelbrot to HybridZipfndashMandelbrot law by employing maximum entropytechnique with one additional constraint It is interesting thatexamination of its densities provides some new insights ofLerchrsquos transcendent

Theorem 15 Let q be Hybrid ZipfndashMandelbrot law asdefined in (78) with parameters cge 0 dωgt 0 andN isin 1 2 and we have

1113954ZM Hlowast c dω( 1113857 S(q)leN 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠ minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠

⎧⎨

⎩

⎫⎬

⎭

(79)

14 Journal of Mathematics

Proof It is interesting to see that for qJ defined in (78)1113936

NJ1 qJ 1 erefore using above qJ in Shannon en-

tropy (64) we get Hybrid ZipfndashMandelbrot entropy (76)

S(q) minus 1113944N

J1qJ ln qJ minus 1113944

N

J1

ωJ

(J + c)dHlowastcdω

lnωJ

(J + c)dHlowastcdω

minus1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

lnωJ

(J + c)d

1113888 1113889 + ln1

Hlowastcdω

1113888 11138891113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

ln(J + c)

d

ωJ1113888 1113889 + ln H

lowastcdω1113872 11138731113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873

(80)

Finally substituting this qJ ωJ(J + c)dHlowastcdω inCorollary 10 (ii) we get the desired result

Corollary 13 Let q and p be Hybrid ZipfndashMandelbrotlaw with parameters c1 c2 isin [0infin)ω1ω2 d1 d2 gt 0 Nowusing qJ ωJ

1 (J + c1)d1Hlowastc1 d1 ω1

and pJ ωJ2

(J + c2)d2Hlowastc2 d2 ω2

in Corollary 11 (ii) the following holds

D(q p) 1113944N

J1

ωJ1

J + c1( 1113857d1Hlowastc1 d1 ω1

lnωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

ωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

⎛⎝ ⎞⎠

minus1113954ZM Hlowast c1 d1ω1( 1113857 +

1Hlowastc1 d1 ω1

1113944

N

J1

ωJ1

J + c1( 1113857d1ln

J + c2( 1113857d2

ωJ2

⎛⎝ ⎞⎠ + ln Hlowastc2 d2 ω2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎫⎪⎬

⎪⎭

(81)

Remark 9 Similarly we can give results for Shannonentropy KullbackndashLeibler distance ZipfndashMandelbrotentropy and Hybrid ZipfndashMandelbrot entropy by usinggeneralized Giaccardi inequality defined in (54) on thesame steps

4 Concluding Remarks

In this paper we gave generalization of Jensenrsquos inequality aswell as converse of Jensenrsquos inequality by using Montgomeryidentity We also formulate results for other inequalities like

JensenndashSteffensen inequality JensenndashBoas inequality andJensenndashBrunk inequality We can obtain JensenndashSteffenseninequality JensenndashBoas inequality and JensenndashBrunk in-equality by changing the assumption of Jensenrsquos inequalityAt the end we gave applications in information theory forour obtained results especially we gave results for HybridZipfndashMandelbrot entropy for our obtained results [46]

Data Availability

No data were used to support this study

Journal of Mathematics 15

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is research is supported by HEC Pakistan under NRPUProject 7906 and RUDN University Strategic AcademicLeadership Program

References

[1] X L Zhu and G H Yang ldquoJensen inequality approach tostability analysis of discrete-time systems with time-varyingdelayrdquo in Proceedings of the American Control Conferencepp 1644ndash1649 Seattle WA USA June 2008

[2] S I Butt M Klaricic Bakula ETH Pecaric and J PecaricldquoJensen-gruss inequality and its applications for the zipf-mandelbrot lawrdquo Mathematical Methods in the Applied Sci-ences vol 44 no 2 pp 1664ndash1673 2021

[3] M A Khan D Pecaric and J Pecaric ldquoNew refinement of theJensen inequality associated to certain functions with appli-cationsrdquo Journal of Inequalities and Applications vol 762020

[4] S Khan M Adil Khan S I Butt and Y-M Chu ldquoA newbound for the Jensen gap pertaining twice differentiablefunctions with applicationsrdquo Advances in Difference Equa-tions vol 1 2020

[5] H Budak andM Z Sarikaya ldquoOn generalized Ostrowski-typeinequalities for functions whose first derivatives absolutevalues are convexrdquo Turkish Journal of Mathematics vol 40pp 1193ndash1210 2016

[6] P Cerone and S S Dragomir ldquoOn some inequalities arisingfrom Montgomeryrsquos identityrdquo Journal of ComputationalAnalysis and Applications vol 5 no 4 pp 341ndash367 2003

[7] S Bernstein ldquoSur les fonctions absolument monotonesrdquo ActaMathematica vol 52 pp 1ndash66 1929

[8] V Culjak and J Pecaric ldquoInterpolation polynomials andinequalities for convex functions of higher orderrdquo ActaMathematica vol 5 pp 369ndash386 2012

[9] A M Fink ldquoBounds of the deviation of a function from itsaveragesrdquo Czechoslovak Mathematical Journal vol 42no 117 pp 289ndash310 1992

[10] J Pecaric and V Culjak ldquoInterpolation polinomials and in-equalities for convex functions of higher orderrdquo Mathe-matical Inequalities amp Applications vol 5 no 3 pp 369ndash3862002

[11] J Pecaric V Culjak and A M Fink ldquoOn some inequalitiesfor convex function of higher orderrdquo Advanced NonlinearStudies vol 6 no 2 pp 131ndash140 1999

[12] J Pecaric V Culjak andM Rogina ldquoOn some inequalities forconvex function of higher order IIrdquo Advanced NonlinearStudies vol 45 pp 281ndash294 2001

[13] J Pecaric I Peric and M Rodic-Lipanovic ldquoUniformtreatment of Jensen type inequalitiesrdquo Mathematical Reportsvol 16 no 66 pp 183ndash205 2014

[14] J Pecaric and J Peric ldquoImprovement of the Giaccardi and thepetrovic inequality and related stolarsky type meansrdquo Annalsof the University of Craiova - Mathematics and ComputerScience vol 39 no 1 pp 65ndash75 2012

[15] J Pecaric M Praljak and A Witkowski ldquoLinear operatorinequality for nminusconvex functions at a pointrdquo MathematicalInequalities amp Applications Appl vol 18 pp 1201ndash1217 2015

[16] Z Pavic ldquoRefinements of Jensenrsquos inequality for infiniteconvex combinationsrdquo Turkish Journal of Inequalities vol 2no 2 pp 44ndash53 2018

[17] R Bibi A Nosheen and J Pecaric ldquoExtended Jensenrsquos typeinequalities for diamond integrals via Taylors formulardquoTurkish Journal of Inequalities vol 3 no 1 pp 7ndash18 2019

[18] T Niaz K A Khan and J Pecaric ldquoOn refinement of Jensenrsquosinequality for 3minusconvex function at a pointrdquo Turkish Journalof Inequalities vol 4 no 1 pp 70ndash80 2020

[19] A Ekinci A O Akdemir and M E Ozdemir ldquoIntegralinequalities for different kinds of convexity via classical in-equalitiesrdquo Turkish Journal of Science vol 5 no 3 pp 305ndash313 2020

[20] Y M Chu S Talib E Set M U Awan and M A Noorldquo(p q)minus Analysis of Montgomery identity and estimates of(p q)minusbounds with applicationsrdquo Journal of Inequalities andApplications vol 2021 9 pages 2021

[21] M Z Sarikaya H Yaldrz and E Set ldquoOn fractional in-equalities via Montgomery identitiesrdquo International Journalof Open Problems in Complex Analysis vol 6 no 2 pp 36ndash432014

[22] A Bnouhachem ldquoA descent SQP alternating directionmethod for minimizing the sum of three convex functionsrdquoJournal of Nonlinear Functional Analysis vol 4 pp 469ndash4822020

[23] A O Akdemir S I Butt M Nadeem and M A RagusaldquoNew general variants of Chebyshev type inequalities viageneralized fractional integral operatorsrdquo Mathematicsvol 92 2021

[24] S I Butt M Umar S Rashid A O Akdemir and Y MingChu ldquoNew Hermite Jensen Mercer type inequalities viakminusfractional integralsrdquo Advances in Difference Equationsvol 635 2020

[25] S I Butt M Nadeem S Qaisar A O Akdemir andT Abdeljawad ldquoHermite Jensen Mercer type inequalities forconformable integrals and related resultsrdquo Advances in Dif-ference Equations vol 501 2020

[26] A Bnouhachem and X Qin ldquoAn inertial proximal Peaceman-Rachford splittingmethod with SQP regularization for convexprogrammingrdquo Journal of Nonlinear Functional Analysisvol 50 2020

[27] S Hamann ldquoMinimality conditions for convex compositefunctions and efficiency conditions in vector optimizationrdquoApplied Set-Valued Analysis and Optimization vol 1pp 221ndash229 2019

[28] A R Khan J Pecaric and M Praljak ldquoPopoviciu type in-equalities for n-convex functions via extension of Mont-gomery identitynminusconvex functions via extension ofMontgomery identityrdquo Analele Universitatii ldquoOvidiusrdquo Con-stanta-Seria Matematica vol 24 no 3 pp 161ndash188 2016

[29] R P Boas ldquoe Jensen-Steffensen inequalityrdquo ElektrotehnFak Ser Mat Fiz vol 302ndash319 pp 1ndash8 1970

[30] J Pecaric F Proschan and Y L Tong Convex FunctionsPartial Orderings and Statistical Applications Academic PressNew York NY USA 1992

[31] H D Brunk ldquoOn an inequality for convex functionsrdquo Pro-ceedings of the American Mathematical Society vol 7 no 5p 817 1956

[32] P R Beesack and J E Pecaric ldquoOn Jessenrsquos inequality forconvex functionsrdquo Journal of Mathematical Analysis andApplications vol 110 no 2 pp 536ndash552 1985

[33] R Barlow A Marshall and F Proschan ldquoSome inequalitiesfor starshaped and convex functionsrdquo Pacific Journal ofMathematics vol 29 no 1 pp 19ndash42 1969

16 Journal of Mathematics

[34] J E Pecaric ldquoOn an inequality of PMV Janicrdquo Publications delrsquoInstitut Mathematique vol 28 no 42 pp 145ndash149 1980

[35] G Szego ldquoUber eine Verallgemeinerung des DirichletschenIntegralsrdquo Mathematische Zeitschrift vol 52 no 1pp 676ndash685 1950

[36] F Giaccardi ldquoSu alcune disuguaglianierdquo Journal of Researchof the National Bureau of Standards vol 1 no 4 pp 139ndash1531955

[37] D S Mitrinovic J E Pecaric and A M Fink Classical andnew inequalities in analysis Mathematics and its Applications(East European Series) Vol 61 Kluwer Academic PublishersGroup Dordrecht Netherlands 1993

[38] I Csiszar ldquoInformation-type measures of diference ofprobability distributions and indirect observationsrdquo StudiaScience Math Hungar vol 2 pp 299ndash318 1967

[39] L Horvath D Pecaric and J Pecaric ldquoEstimations of f - andrenyi divergences by using a cyclic refinement of the Jensenrsquosinequalityrdquo Bulletin of the Malaysian Mathematical SciencesSociety vol 42 2017

[40] S Kullback Information =eory and Statistics Wiley NewYork NY USA 1959

[41] S T Piantadosi ldquoZipfrsquos word frequency law in natural lan-guage a critical review and future directionsrdquo PsychonomicBulletin and Review vol 21 no 5 pp 1112ndash1130 2014

[42] Z K Silagadze ldquoCitations and the zipfndashmandelbrot lawrdquoComplex Systems no 11 pp 487ndash499 1997

[43] J Jaksetic D Pecaric and J Pecaric ldquoSome properties of zipf-mandelbrot law and hurwitz ζ-functionrdquo Mathematical In-equalities amp Applications vol 21 no 2 pp 575ndash584 2018

[44] M Visser ldquoZipfrsquos law power laws and maximum entropyrdquoMathematical Inequalities amp Applications vol 15 Article ID043021 2013

[45] J Jaksetic D Pecaric and J Pecaric ldquoHybrid zipf-mandelbrotlawrdquo Mathematical Inequalities amp Applications vol 13 no 1pp 275ndash286 2019

[46] D V Widder ldquoCompletely convex functions and Lidstoneseriesrdquo Transactions of the American Mathematical Societyvol 51 p 387 1942

Journal of Mathematics 17

1113957IΨ(p q)gePmΨ(1) minus Pm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F q

pJ

qJ (x minus α)

ℓ+21113888 11138891113890 1113891 minus Ψ(ℓ+1)

(β) F qpJ

qJ (x minus β)

ℓ+21113888 11138891113890 11138911113896 1113897

(62)

Proof From eorem 9 by following Jensenrsquos difference(61) we can rearrange (34) as

Ψ(x) minus1

Pm

1113944

m

J1pJΨ xJ1113872 1113873le 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times Ψ(ℓ+1)(α) F p xJ (x minus α)

ℓ+21113872 11138731113872 1113873 minus Ψ(ℓ+1)

(β) F p xJ (x minus β)ℓ+2

1113872 11138731113872 11138731113966 1113967

(63)

Now replace pJ with qJ and xJ with pJqJ and we get(62)

For positive n-tuple q (q1 qm) such that1113936

mJ1 qJ 1 the Shannon entropy is defined by

S(q) minus 1113944m

J1qJ ln qJ (64)

Corollary 10 Under the assumptions of =eorem 9 (ii) let(51) hold and Ψ be nminusconvex

(i) If q ≔ (q1 qm) isin (0infin)m then

1113944

m

J1qJ ln qJ lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (65)

(ii) We can get bounds for the Shannon entropy of q if wechoose q ≔ (q1 qn) to be a positive probabilitydistribution

S(q)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F q1

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

1qJ

minusln(middot)1113888 1113889⎧⎨

⎩

⎫⎬

⎭ (66)

Proof (i) Substituting Ψ(x) ≔ minus lnx and usingp ≔ (1 1 1) in eorem 13 we get (65)

(ii) Since we have 1113936mJ1 qJ 1 by multiplying minus1 on

both sides of (65) and taking into account (64) weget (66)

e KullbackndashLeibler distance [40] between the positiveprobability distributions p (p1 pm) andq (q1 qm) is defined by

D(q p) 1113944

m

J1qJ ln

qJ

pJ

1113888 1113889 (67)

Corollary 11 Under the assumptions of Corollary 10

(i) If q ≔ (q1 qm) p ≔ (p1 pm) isin (0infin)mthen

1113944

m

J1qJ ln

qJ

pJ

1113888 1113889lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

pJ

qJ minusln(middot)1113888 1113889

⎧⎨

⎩

⎫⎬

⎭ (68)

(ii) If q ≔ (q1 qm) p ≔ (p1 pm) are positiveprobability distributions then we have

12 Journal of Mathematics

D(q p)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

βℓ+1F q

pJ

qJ minusln(middot)1113888 11138891113896 1113897 (69)

Proof

(i) UsingΨ(x) ≔ minus lnx (which isn-convex for evenn)in eorem 13 we get (68) after simplification

(ii) It is a special case of (i)

34 Results for Zipf and Hybrid ZipfndashMandelbrot EntropyOne of the basic laws in information science is Zipfrsquos law[4142] which is highly applied in linguistics Let cge 0 dgt 0and N isin 1 2 ZipfndashMandelbrot entropy can be given as

ZM(H c d) d

HNcd

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (70)

where

HNcd 1113944

N

σ1

1(σ + c)

d (71)

Consider

qJ Ψ(J N c d) 1

(J + c)dH

Ncd1113872 1113873

(72)

where Ψ(J m c d) is discrete probability distributionknown as ZipfndashMandelbrot law ZipfndashMandelbrot law hasmany application in linguistics and information sciencesSome of the recent study about ZipfndashMandelbrot law can beseen in the listed references (see [39 43]) Now we state ourresults involving entropy introduced by Mandelbrot law byestablishing the relationship with Shannon and relativeentropies

Theorem 14 Let q be ZipfndashMandelbrot law as defined in(72) with parameters cge 0 dgt 0 and N isin 1 2 and wehave

ZM(H c d) S(q)leN times 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873

⎧⎨

⎩

⎫⎬

⎭

(73)

Proof It is interesting to see that for qJ defined in (72)1113936

NJ1 qJ 1 erefore using above qJ in Shannon entropy

(64) we get Mandelbrot entropy (70)

S(q) minus 1113944

N

J1qJ ln qJ minus 1113944

N

J1

1(J + c)

dH

Ncd1113872 1113873

ln1

(J + c)dH

Ncd1113872 1113873

d

HNcd1113872 1113873

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (74)

Finally substituting this qJ 1((J + c)dHNcd) in

Corollary 10 (ii) we get the desired result Corollary 12 Let q and p be ZipfndashMandelbrot law withparameters c1 c2 isin [0infin) d1 d2 gt 0 and let

Journal of Mathematics 13

HNc1 d1

1113936Nσ1 1(σ + c1)

d1 and HNc2 d2

1113936Nσ1 1(σ + c2)

d2 Now using qJ 1(J + c1)

d1HNc1 d1

andpJ 1(J + c2)

d2HNc2 d2

in Corollary 11 (ii) the followingholds

D(q p) 1113944N

J1

1J + c1( 1113857

d1HNc1d1

lnJ + c2( 1113857

d2HNc2 d2

J + c1( 1113857d1H

mc1 d1

⎛⎝ ⎞⎠

minusZ H c1 d1( 1113857 +d2

HNc1 d1

1113944

N

J1

ln J + c2( 1113857

J + c1( 1113857d1

+ ln Hmc2d2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qJ + c1( 1113857

d1HNc1 d1

J + c2( 1113857d2H

Nc2 d2

minusln(middot)⎛⎝ ⎞⎠ minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qJ + c1( 1113857

d1HNc1d1

J + c2( 1113857d2H

Nc2d2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(75)

=e Next Result for Hybrid ZipfndashMandelbrot Entropy Fur-ther generalization of ZipfndashMandelbrot entropy is Hybrid

ZipfndashMandelbrot entropy Let N isin 1 2 cge 0ωgt 0then Hybrid ZipfndashMandelbrot entropy can be given as

1113954ZM Hlowast c dω( 1113857

1Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873 (76)

where

Hlowastcdω 1113944

N

J1

ωJ

(J + c)d (77)

Consider

qJ Ψ(J N c dω) ωJ

(J + c)dHlowastcdω

(78)

which is called Hybrid ZipfndashMandelbrot law =ere is aunified approach maximization of Shannon entropy [44]

that naturally follows the path of generalization from Zipfrsquos toHybrid Zipfrsquos law Extending this idea Jaksetic et al in [45]presented a transition from ZipfndashMandelbrot to HybridZipfndashMandelbrot law by employing maximum entropytechnique with one additional constraint It is interesting thatexamination of its densities provides some new insights ofLerchrsquos transcendent

Theorem 15 Let q be Hybrid ZipfndashMandelbrot law asdefined in (78) with parameters cge 0 dωgt 0 andN isin 1 2 and we have

1113954ZM Hlowast c dω( 1113857 S(q)leN 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠ minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠

⎧⎨

⎩

⎫⎬

⎭

(79)

14 Journal of Mathematics

Proof It is interesting to see that for qJ defined in (78)1113936

NJ1 qJ 1 erefore using above qJ in Shannon en-

tropy (64) we get Hybrid ZipfndashMandelbrot entropy (76)

S(q) minus 1113944N

J1qJ ln qJ minus 1113944

N

J1

ωJ

(J + c)dHlowastcdω

lnωJ

(J + c)dHlowastcdω

minus1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

lnωJ

(J + c)d

1113888 1113889 + ln1

Hlowastcdω

1113888 11138891113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

ln(J + c)

d

ωJ1113888 1113889 + ln H

lowastcdω1113872 11138731113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873

(80)

Finally substituting this qJ ωJ(J + c)dHlowastcdω inCorollary 10 (ii) we get the desired result

Corollary 13 Let q and p be Hybrid ZipfndashMandelbrotlaw with parameters c1 c2 isin [0infin)ω1ω2 d1 d2 gt 0 Nowusing qJ ωJ

1 (J + c1)d1Hlowastc1 d1 ω1

and pJ ωJ2

(J + c2)d2Hlowastc2 d2 ω2

in Corollary 11 (ii) the following holds

D(q p) 1113944N

J1

ωJ1

J + c1( 1113857d1Hlowastc1 d1 ω1

lnωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

ωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

⎛⎝ ⎞⎠

minus1113954ZM Hlowast c1 d1ω1( 1113857 +

1Hlowastc1 d1 ω1

1113944

N

J1

ωJ1

J + c1( 1113857d1ln

J + c2( 1113857d2

ωJ2

⎛⎝ ⎞⎠ + ln Hlowastc2 d2 ω2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎫⎪⎬

⎪⎭

(81)

Remark 9 Similarly we can give results for Shannonentropy KullbackndashLeibler distance ZipfndashMandelbrotentropy and Hybrid ZipfndashMandelbrot entropy by usinggeneralized Giaccardi inequality defined in (54) on thesame steps

4 Concluding Remarks

In this paper we gave generalization of Jensenrsquos inequality aswell as converse of Jensenrsquos inequality by using Montgomeryidentity We also formulate results for other inequalities like

JensenndashSteffensen inequality JensenndashBoas inequality andJensenndashBrunk inequality We can obtain JensenndashSteffenseninequality JensenndashBoas inequality and JensenndashBrunk in-equality by changing the assumption of Jensenrsquos inequalityAt the end we gave applications in information theory forour obtained results especially we gave results for HybridZipfndashMandelbrot entropy for our obtained results [46]

Data Availability

No data were used to support this study

Journal of Mathematics 15

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is research is supported by HEC Pakistan under NRPUProject 7906 and RUDN University Strategic AcademicLeadership Program

References

[1] X L Zhu and G H Yang ldquoJensen inequality approach tostability analysis of discrete-time systems with time-varyingdelayrdquo in Proceedings of the American Control Conferencepp 1644ndash1649 Seattle WA USA June 2008

[2] S I Butt M Klaricic Bakula ETH Pecaric and J PecaricldquoJensen-gruss inequality and its applications for the zipf-mandelbrot lawrdquo Mathematical Methods in the Applied Sci-ences vol 44 no 2 pp 1664ndash1673 2021

[3] M A Khan D Pecaric and J Pecaric ldquoNew refinement of theJensen inequality associated to certain functions with appli-cationsrdquo Journal of Inequalities and Applications vol 762020

[4] S Khan M Adil Khan S I Butt and Y-M Chu ldquoA newbound for the Jensen gap pertaining twice differentiablefunctions with applicationsrdquo Advances in Difference Equa-tions vol 1 2020

[5] H Budak andM Z Sarikaya ldquoOn generalized Ostrowski-typeinequalities for functions whose first derivatives absolutevalues are convexrdquo Turkish Journal of Mathematics vol 40pp 1193ndash1210 2016

[6] P Cerone and S S Dragomir ldquoOn some inequalities arisingfrom Montgomeryrsquos identityrdquo Journal of ComputationalAnalysis and Applications vol 5 no 4 pp 341ndash367 2003

[7] S Bernstein ldquoSur les fonctions absolument monotonesrdquo ActaMathematica vol 52 pp 1ndash66 1929

[8] V Culjak and J Pecaric ldquoInterpolation polynomials andinequalities for convex functions of higher orderrdquo ActaMathematica vol 5 pp 369ndash386 2012

[9] A M Fink ldquoBounds of the deviation of a function from itsaveragesrdquo Czechoslovak Mathematical Journal vol 42no 117 pp 289ndash310 1992

[10] J Pecaric and V Culjak ldquoInterpolation polinomials and in-equalities for convex functions of higher orderrdquo Mathe-matical Inequalities amp Applications vol 5 no 3 pp 369ndash3862002

[11] J Pecaric V Culjak and A M Fink ldquoOn some inequalitiesfor convex function of higher orderrdquo Advanced NonlinearStudies vol 6 no 2 pp 131ndash140 1999

[12] J Pecaric V Culjak andM Rogina ldquoOn some inequalities forconvex function of higher order IIrdquo Advanced NonlinearStudies vol 45 pp 281ndash294 2001

[13] J Pecaric I Peric and M Rodic-Lipanovic ldquoUniformtreatment of Jensen type inequalitiesrdquo Mathematical Reportsvol 16 no 66 pp 183ndash205 2014

[14] J Pecaric and J Peric ldquoImprovement of the Giaccardi and thepetrovic inequality and related stolarsky type meansrdquo Annalsof the University of Craiova - Mathematics and ComputerScience vol 39 no 1 pp 65ndash75 2012

[15] J Pecaric M Praljak and A Witkowski ldquoLinear operatorinequality for nminusconvex functions at a pointrdquo MathematicalInequalities amp Applications Appl vol 18 pp 1201ndash1217 2015

[16] Z Pavic ldquoRefinements of Jensenrsquos inequality for infiniteconvex combinationsrdquo Turkish Journal of Inequalities vol 2no 2 pp 44ndash53 2018

[17] R Bibi A Nosheen and J Pecaric ldquoExtended Jensenrsquos typeinequalities for diamond integrals via Taylors formulardquoTurkish Journal of Inequalities vol 3 no 1 pp 7ndash18 2019

[18] T Niaz K A Khan and J Pecaric ldquoOn refinement of Jensenrsquosinequality for 3minusconvex function at a pointrdquo Turkish Journalof Inequalities vol 4 no 1 pp 70ndash80 2020

[19] A Ekinci A O Akdemir and M E Ozdemir ldquoIntegralinequalities for different kinds of convexity via classical in-equalitiesrdquo Turkish Journal of Science vol 5 no 3 pp 305ndash313 2020

[20] Y M Chu S Talib E Set M U Awan and M A Noorldquo(p q)minus Analysis of Montgomery identity and estimates of(p q)minusbounds with applicationsrdquo Journal of Inequalities andApplications vol 2021 9 pages 2021

[21] M Z Sarikaya H Yaldrz and E Set ldquoOn fractional in-equalities via Montgomery identitiesrdquo International Journalof Open Problems in Complex Analysis vol 6 no 2 pp 36ndash432014

[22] A Bnouhachem ldquoA descent SQP alternating directionmethod for minimizing the sum of three convex functionsrdquoJournal of Nonlinear Functional Analysis vol 4 pp 469ndash4822020

[23] A O Akdemir S I Butt M Nadeem and M A RagusaldquoNew general variants of Chebyshev type inequalities viageneralized fractional integral operatorsrdquo Mathematicsvol 92 2021

[24] S I Butt M Umar S Rashid A O Akdemir and Y MingChu ldquoNew Hermite Jensen Mercer type inequalities viakminusfractional integralsrdquo Advances in Difference Equationsvol 635 2020

[25] S I Butt M Nadeem S Qaisar A O Akdemir andT Abdeljawad ldquoHermite Jensen Mercer type inequalities forconformable integrals and related resultsrdquo Advances in Dif-ference Equations vol 501 2020

[26] A Bnouhachem and X Qin ldquoAn inertial proximal Peaceman-Rachford splittingmethod with SQP regularization for convexprogrammingrdquo Journal of Nonlinear Functional Analysisvol 50 2020

[27] S Hamann ldquoMinimality conditions for convex compositefunctions and efficiency conditions in vector optimizationrdquoApplied Set-Valued Analysis and Optimization vol 1pp 221ndash229 2019

[28] A R Khan J Pecaric and M Praljak ldquoPopoviciu type in-equalities for n-convex functions via extension of Mont-gomery identitynminusconvex functions via extension ofMontgomery identityrdquo Analele Universitatii ldquoOvidiusrdquo Con-stanta-Seria Matematica vol 24 no 3 pp 161ndash188 2016

[29] R P Boas ldquoe Jensen-Steffensen inequalityrdquo ElektrotehnFak Ser Mat Fiz vol 302ndash319 pp 1ndash8 1970

[30] J Pecaric F Proschan and Y L Tong Convex FunctionsPartial Orderings and Statistical Applications Academic PressNew York NY USA 1992

[31] H D Brunk ldquoOn an inequality for convex functionsrdquo Pro-ceedings of the American Mathematical Society vol 7 no 5p 817 1956

[32] P R Beesack and J E Pecaric ldquoOn Jessenrsquos inequality forconvex functionsrdquo Journal of Mathematical Analysis andApplications vol 110 no 2 pp 536ndash552 1985

[33] R Barlow A Marshall and F Proschan ldquoSome inequalitiesfor starshaped and convex functionsrdquo Pacific Journal ofMathematics vol 29 no 1 pp 19ndash42 1969

16 Journal of Mathematics

[34] J E Pecaric ldquoOn an inequality of PMV Janicrdquo Publications delrsquoInstitut Mathematique vol 28 no 42 pp 145ndash149 1980

[35] G Szego ldquoUber eine Verallgemeinerung des DirichletschenIntegralsrdquo Mathematische Zeitschrift vol 52 no 1pp 676ndash685 1950

[36] F Giaccardi ldquoSu alcune disuguaglianierdquo Journal of Researchof the National Bureau of Standards vol 1 no 4 pp 139ndash1531955

[37] D S Mitrinovic J E Pecaric and A M Fink Classical andnew inequalities in analysis Mathematics and its Applications(East European Series) Vol 61 Kluwer Academic PublishersGroup Dordrecht Netherlands 1993

[38] I Csiszar ldquoInformation-type measures of diference ofprobability distributions and indirect observationsrdquo StudiaScience Math Hungar vol 2 pp 299ndash318 1967

[39] L Horvath D Pecaric and J Pecaric ldquoEstimations of f - andrenyi divergences by using a cyclic refinement of the Jensenrsquosinequalityrdquo Bulletin of the Malaysian Mathematical SciencesSociety vol 42 2017

[40] S Kullback Information =eory and Statistics Wiley NewYork NY USA 1959

[41] S T Piantadosi ldquoZipfrsquos word frequency law in natural lan-guage a critical review and future directionsrdquo PsychonomicBulletin and Review vol 21 no 5 pp 1112ndash1130 2014

[42] Z K Silagadze ldquoCitations and the zipfndashmandelbrot lawrdquoComplex Systems no 11 pp 487ndash499 1997

[43] J Jaksetic D Pecaric and J Pecaric ldquoSome properties of zipf-mandelbrot law and hurwitz ζ-functionrdquo Mathematical In-equalities amp Applications vol 21 no 2 pp 575ndash584 2018

[44] M Visser ldquoZipfrsquos law power laws and maximum entropyrdquoMathematical Inequalities amp Applications vol 15 Article ID043021 2013

[45] J Jaksetic D Pecaric and J Pecaric ldquoHybrid zipf-mandelbrotlawrdquo Mathematical Inequalities amp Applications vol 13 no 1pp 275ndash286 2019

[46] D V Widder ldquoCompletely convex functions and Lidstoneseriesrdquo Transactions of the American Mathematical Societyvol 51 p 387 1942

Journal of Mathematics 17

D(q p)lePm 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qpJ

qJ minusln(middot)1113888 1113889 minus

(minus1)ℓ+1ℓ

βℓ+1F q

pJ

qJ minusln(middot)1113888 11138891113896 1113897 (69)

Proof

(i) UsingΨ(x) ≔ minus lnx (which isn-convex for evenn)in eorem 13 we get (68) after simplification

(ii) It is a special case of (i)

34 Results for Zipf and Hybrid ZipfndashMandelbrot EntropyOne of the basic laws in information science is Zipfrsquos law[4142] which is highly applied in linguistics Let cge 0 dgt 0and N isin 1 2 ZipfndashMandelbrot entropy can be given as

ZM(H c d) d

HNcd

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (70)

where

HNcd 1113944

N

σ1

1(σ + c)

d (71)

Consider

qJ Ψ(J N c d) 1

(J + c)dH

Ncd1113872 1113873

(72)

where Ψ(J m c d) is discrete probability distributionknown as ZipfndashMandelbrot law ZipfndashMandelbrot law hasmany application in linguistics and information sciencesSome of the recent study about ZipfndashMandelbrot law can beseen in the listed references (see [39 43]) Now we state ourresults involving entropy introduced by Mandelbrot law byestablishing the relationship with Shannon and relativeentropies

Theorem 14 Let q be ZipfndashMandelbrot law as defined in(72) with parameters cge 0 dgt 0 and N isin 1 2 and wehave

ZM(H c d) S(q)leN times 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873 minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q (J + c)

dH

Ncd1113872 1113873 minusln(middot)1113872 1113873

⎧⎨

⎩

⎫⎬

⎭

(73)

Proof It is interesting to see that for qJ defined in (72)1113936

NJ1 qJ 1 erefore using above qJ in Shannon entropy

(64) we get Mandelbrot entropy (70)

S(q) minus 1113944

N

J1qJ ln qJ minus 1113944

N

J1

1(J + c)

dH

Ncd1113872 1113873

ln1

(J + c)dH

Ncd1113872 1113873

d

HNcd1113872 1113873

1113944

N

J1

ln(J + c)

(J + c)d

+ ln HNcd1113872 1113873 (74)

Finally substituting this qJ 1((J + c)dHNcd) in

Corollary 10 (ii) we get the desired result Corollary 12 Let q and p be ZipfndashMandelbrot law withparameters c1 c2 isin [0infin) d1 d2 gt 0 and let

Journal of Mathematics 13

HNc1 d1

1113936Nσ1 1(σ + c1)

d1 and HNc2 d2

1113936Nσ1 1(σ + c2)

d2 Now using qJ 1(J + c1)

d1HNc1 d1

andpJ 1(J + c2)

d2HNc2 d2

in Corollary 11 (ii) the followingholds

D(q p) 1113944N

J1

1J + c1( 1113857

d1HNc1d1

lnJ + c2( 1113857

d2HNc2 d2

J + c1( 1113857d1H

mc1 d1

⎛⎝ ⎞⎠

minusZ H c1 d1( 1113857 +d2

HNc1 d1

1113944

N

J1

ln J + c2( 1113857

J + c1( 1113857d1

+ ln Hmc2d2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qJ + c1( 1113857

d1HNc1 d1

J + c2( 1113857d2H

Nc2 d2

minusln(middot)⎛⎝ ⎞⎠ minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qJ + c1( 1113857

d1HNc1d1

J + c2( 1113857d2H

Nc2d2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(75)

=e Next Result for Hybrid ZipfndashMandelbrot Entropy Fur-ther generalization of ZipfndashMandelbrot entropy is Hybrid

ZipfndashMandelbrot entropy Let N isin 1 2 cge 0ωgt 0then Hybrid ZipfndashMandelbrot entropy can be given as

1113954ZM Hlowast c dω( 1113857

1Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873 (76)

where

Hlowastcdω 1113944

N

J1

ωJ

(J + c)d (77)

Consider

qJ Ψ(J N c dω) ωJ

(J + c)dHlowastcdω

(78)

which is called Hybrid ZipfndashMandelbrot law =ere is aunified approach maximization of Shannon entropy [44]

that naturally follows the path of generalization from Zipfrsquos toHybrid Zipfrsquos law Extending this idea Jaksetic et al in [45]presented a transition from ZipfndashMandelbrot to HybridZipfndashMandelbrot law by employing maximum entropytechnique with one additional constraint It is interesting thatexamination of its densities provides some new insights ofLerchrsquos transcendent

Theorem 15 Let q be Hybrid ZipfndashMandelbrot law asdefined in (78) with parameters cge 0 dωgt 0 andN isin 1 2 and we have

1113954ZM Hlowast c dω( 1113857 S(q)leN 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠ minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠

⎧⎨

⎩

⎫⎬

⎭

(79)

14 Journal of Mathematics

Proof It is interesting to see that for qJ defined in (78)1113936

NJ1 qJ 1 erefore using above qJ in Shannon en-

tropy (64) we get Hybrid ZipfndashMandelbrot entropy (76)

S(q) minus 1113944N

J1qJ ln qJ minus 1113944

N

J1

ωJ

(J + c)dHlowastcdω

lnωJ

(J + c)dHlowastcdω

minus1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

lnωJ

(J + c)d

1113888 1113889 + ln1

Hlowastcdω

1113888 11138891113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

ln(J + c)

d

ωJ1113888 1113889 + ln H

lowastcdω1113872 11138731113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873

(80)

Finally substituting this qJ ωJ(J + c)dHlowastcdω inCorollary 10 (ii) we get the desired result

Corollary 13 Let q and p be Hybrid ZipfndashMandelbrotlaw with parameters c1 c2 isin [0infin)ω1ω2 d1 d2 gt 0 Nowusing qJ ωJ

1 (J + c1)d1Hlowastc1 d1 ω1

and pJ ωJ2

(J + c2)d2Hlowastc2 d2 ω2

in Corollary 11 (ii) the following holds

D(q p) 1113944N

J1

ωJ1

J + c1( 1113857d1Hlowastc1 d1 ω1

lnωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

ωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

⎛⎝ ⎞⎠

minus1113954ZM Hlowast c1 d1ω1( 1113857 +

1Hlowastc1 d1 ω1

1113944

N

J1

ωJ1

J + c1( 1113857d1ln

J + c2( 1113857d2

ωJ2

⎛⎝ ⎞⎠ + ln Hlowastc2 d2 ω2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎫⎪⎬

⎪⎭

(81)

Remark 9 Similarly we can give results for Shannonentropy KullbackndashLeibler distance ZipfndashMandelbrotentropy and Hybrid ZipfndashMandelbrot entropy by usinggeneralized Giaccardi inequality defined in (54) on thesame steps

4 Concluding Remarks

In this paper we gave generalization of Jensenrsquos inequality aswell as converse of Jensenrsquos inequality by using Montgomeryidentity We also formulate results for other inequalities like

JensenndashSteffensen inequality JensenndashBoas inequality andJensenndashBrunk inequality We can obtain JensenndashSteffenseninequality JensenndashBoas inequality and JensenndashBrunk in-equality by changing the assumption of Jensenrsquos inequalityAt the end we gave applications in information theory forour obtained results especially we gave results for HybridZipfndashMandelbrot entropy for our obtained results [46]

Data Availability

No data were used to support this study

Journal of Mathematics 15

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is research is supported by HEC Pakistan under NRPUProject 7906 and RUDN University Strategic AcademicLeadership Program

References

[1] X L Zhu and G H Yang ldquoJensen inequality approach tostability analysis of discrete-time systems with time-varyingdelayrdquo in Proceedings of the American Control Conferencepp 1644ndash1649 Seattle WA USA June 2008

[2] S I Butt M Klaricic Bakula ETH Pecaric and J PecaricldquoJensen-gruss inequality and its applications for the zipf-mandelbrot lawrdquo Mathematical Methods in the Applied Sci-ences vol 44 no 2 pp 1664ndash1673 2021

[3] M A Khan D Pecaric and J Pecaric ldquoNew refinement of theJensen inequality associated to certain functions with appli-cationsrdquo Journal of Inequalities and Applications vol 762020

[4] S Khan M Adil Khan S I Butt and Y-M Chu ldquoA newbound for the Jensen gap pertaining twice differentiablefunctions with applicationsrdquo Advances in Difference Equa-tions vol 1 2020

[5] H Budak andM Z Sarikaya ldquoOn generalized Ostrowski-typeinequalities for functions whose first derivatives absolutevalues are convexrdquo Turkish Journal of Mathematics vol 40pp 1193ndash1210 2016

[6] P Cerone and S S Dragomir ldquoOn some inequalities arisingfrom Montgomeryrsquos identityrdquo Journal of ComputationalAnalysis and Applications vol 5 no 4 pp 341ndash367 2003

[7] S Bernstein ldquoSur les fonctions absolument monotonesrdquo ActaMathematica vol 52 pp 1ndash66 1929

[8] V Culjak and J Pecaric ldquoInterpolation polynomials andinequalities for convex functions of higher orderrdquo ActaMathematica vol 5 pp 369ndash386 2012

[9] A M Fink ldquoBounds of the deviation of a function from itsaveragesrdquo Czechoslovak Mathematical Journal vol 42no 117 pp 289ndash310 1992

[10] J Pecaric and V Culjak ldquoInterpolation polinomials and in-equalities for convex functions of higher orderrdquo Mathe-matical Inequalities amp Applications vol 5 no 3 pp 369ndash3862002

[11] J Pecaric V Culjak and A M Fink ldquoOn some inequalitiesfor convex function of higher orderrdquo Advanced NonlinearStudies vol 6 no 2 pp 131ndash140 1999

[12] J Pecaric V Culjak andM Rogina ldquoOn some inequalities forconvex function of higher order IIrdquo Advanced NonlinearStudies vol 45 pp 281ndash294 2001

[13] J Pecaric I Peric and M Rodic-Lipanovic ldquoUniformtreatment of Jensen type inequalitiesrdquo Mathematical Reportsvol 16 no 66 pp 183ndash205 2014

[14] J Pecaric and J Peric ldquoImprovement of the Giaccardi and thepetrovic inequality and related stolarsky type meansrdquo Annalsof the University of Craiova - Mathematics and ComputerScience vol 39 no 1 pp 65ndash75 2012

[15] J Pecaric M Praljak and A Witkowski ldquoLinear operatorinequality for nminusconvex functions at a pointrdquo MathematicalInequalities amp Applications Appl vol 18 pp 1201ndash1217 2015

[16] Z Pavic ldquoRefinements of Jensenrsquos inequality for infiniteconvex combinationsrdquo Turkish Journal of Inequalities vol 2no 2 pp 44ndash53 2018

[17] R Bibi A Nosheen and J Pecaric ldquoExtended Jensenrsquos typeinequalities for diamond integrals via Taylors formulardquoTurkish Journal of Inequalities vol 3 no 1 pp 7ndash18 2019

[18] T Niaz K A Khan and J Pecaric ldquoOn refinement of Jensenrsquosinequality for 3minusconvex function at a pointrdquo Turkish Journalof Inequalities vol 4 no 1 pp 70ndash80 2020

[19] A Ekinci A O Akdemir and M E Ozdemir ldquoIntegralinequalities for different kinds of convexity via classical in-equalitiesrdquo Turkish Journal of Science vol 5 no 3 pp 305ndash313 2020

[20] Y M Chu S Talib E Set M U Awan and M A Noorldquo(p q)minus Analysis of Montgomery identity and estimates of(p q)minusbounds with applicationsrdquo Journal of Inequalities andApplications vol 2021 9 pages 2021

[21] M Z Sarikaya H Yaldrz and E Set ldquoOn fractional in-equalities via Montgomery identitiesrdquo International Journalof Open Problems in Complex Analysis vol 6 no 2 pp 36ndash432014

[22] A Bnouhachem ldquoA descent SQP alternating directionmethod for minimizing the sum of three convex functionsrdquoJournal of Nonlinear Functional Analysis vol 4 pp 469ndash4822020

[23] A O Akdemir S I Butt M Nadeem and M A RagusaldquoNew general variants of Chebyshev type inequalities viageneralized fractional integral operatorsrdquo Mathematicsvol 92 2021

[24] S I Butt M Umar S Rashid A O Akdemir and Y MingChu ldquoNew Hermite Jensen Mercer type inequalities viakminusfractional integralsrdquo Advances in Difference Equationsvol 635 2020

[25] S I Butt M Nadeem S Qaisar A O Akdemir andT Abdeljawad ldquoHermite Jensen Mercer type inequalities forconformable integrals and related resultsrdquo Advances in Dif-ference Equations vol 501 2020

[26] A Bnouhachem and X Qin ldquoAn inertial proximal Peaceman-Rachford splittingmethod with SQP regularization for convexprogrammingrdquo Journal of Nonlinear Functional Analysisvol 50 2020

[27] S Hamann ldquoMinimality conditions for convex compositefunctions and efficiency conditions in vector optimizationrdquoApplied Set-Valued Analysis and Optimization vol 1pp 221ndash229 2019

[28] A R Khan J Pecaric and M Praljak ldquoPopoviciu type in-equalities for n-convex functions via extension of Mont-gomery identitynminusconvex functions via extension ofMontgomery identityrdquo Analele Universitatii ldquoOvidiusrdquo Con-stanta-Seria Matematica vol 24 no 3 pp 161ndash188 2016

[29] R P Boas ldquoe Jensen-Steffensen inequalityrdquo ElektrotehnFak Ser Mat Fiz vol 302ndash319 pp 1ndash8 1970

[30] J Pecaric F Proschan and Y L Tong Convex FunctionsPartial Orderings and Statistical Applications Academic PressNew York NY USA 1992

[31] H D Brunk ldquoOn an inequality for convex functionsrdquo Pro-ceedings of the American Mathematical Society vol 7 no 5p 817 1956

[32] P R Beesack and J E Pecaric ldquoOn Jessenrsquos inequality forconvex functionsrdquo Journal of Mathematical Analysis andApplications vol 110 no 2 pp 536ndash552 1985

[33] R Barlow A Marshall and F Proschan ldquoSome inequalitiesfor starshaped and convex functionsrdquo Pacific Journal ofMathematics vol 29 no 1 pp 19ndash42 1969

16 Journal of Mathematics

[34] J E Pecaric ldquoOn an inequality of PMV Janicrdquo Publications delrsquoInstitut Mathematique vol 28 no 42 pp 145ndash149 1980

[35] G Szego ldquoUber eine Verallgemeinerung des DirichletschenIntegralsrdquo Mathematische Zeitschrift vol 52 no 1pp 676ndash685 1950

[36] F Giaccardi ldquoSu alcune disuguaglianierdquo Journal of Researchof the National Bureau of Standards vol 1 no 4 pp 139ndash1531955

[37] D S Mitrinovic J E Pecaric and A M Fink Classical andnew inequalities in analysis Mathematics and its Applications(East European Series) Vol 61 Kluwer Academic PublishersGroup Dordrecht Netherlands 1993

[38] I Csiszar ldquoInformation-type measures of diference ofprobability distributions and indirect observationsrdquo StudiaScience Math Hungar vol 2 pp 299ndash318 1967

[39] L Horvath D Pecaric and J Pecaric ldquoEstimations of f - andrenyi divergences by using a cyclic refinement of the Jensenrsquosinequalityrdquo Bulletin of the Malaysian Mathematical SciencesSociety vol 42 2017

[40] S Kullback Information =eory and Statistics Wiley NewYork NY USA 1959

[41] S T Piantadosi ldquoZipfrsquos word frequency law in natural lan-guage a critical review and future directionsrdquo PsychonomicBulletin and Review vol 21 no 5 pp 1112ndash1130 2014

[42] Z K Silagadze ldquoCitations and the zipfndashmandelbrot lawrdquoComplex Systems no 11 pp 487ndash499 1997

[43] J Jaksetic D Pecaric and J Pecaric ldquoSome properties of zipf-mandelbrot law and hurwitz ζ-functionrdquo Mathematical In-equalities amp Applications vol 21 no 2 pp 575ndash584 2018

[44] M Visser ldquoZipfrsquos law power laws and maximum entropyrdquoMathematical Inequalities amp Applications vol 15 Article ID043021 2013

[45] J Jaksetic D Pecaric and J Pecaric ldquoHybrid zipf-mandelbrotlawrdquo Mathematical Inequalities amp Applications vol 13 no 1pp 275ndash286 2019

[46] D V Widder ldquoCompletely convex functions and Lidstoneseriesrdquo Transactions of the American Mathematical Societyvol 51 p 387 1942

Journal of Mathematics 17

HNc1 d1

1113936Nσ1 1(σ + c1)

d1 and HNc2 d2

1113936Nσ1 1(σ + c2)

d2 Now using qJ 1(J + c1)

d1HNc1 d1

andpJ 1(J + c2)

d2HNc2 d2

in Corollary 11 (ii) the followingholds

D(q p) 1113944N

J1

1J + c1( 1113857

d1HNc1d1

lnJ + c2( 1113857

d2HNc2 d2

J + c1( 1113857d1H

mc1 d1

⎛⎝ ⎞⎠

minusZ H c1 d1( 1113857 +d2

HNc1 d1

1113944

N

J1

ln J + c2( 1113857

J + c1( 1113857d1

+ ln Hmc2d2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qJ + c1( 1113857

d1HNc1 d1

J + c2( 1113857d2H

Nc2 d2

minusln(middot)⎛⎝ ⎞⎠ minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qJ + c1( 1113857

d1HNc1d1

J + c2( 1113857d2H

Nc2d2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(75)

=e Next Result for Hybrid ZipfndashMandelbrot Entropy Fur-ther generalization of ZipfndashMandelbrot entropy is Hybrid

ZipfndashMandelbrot entropy Let N isin 1 2 cge 0ωgt 0then Hybrid ZipfndashMandelbrot entropy can be given as

1113954ZM Hlowast c dω( 1113857

1Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873 (76)

where

Hlowastcdω 1113944

N

J1

ωJ

(J + c)d (77)

Consider

qJ Ψ(J N c dω) ωJ

(J + c)dHlowastcdω

(78)

which is called Hybrid ZipfndashMandelbrot law =ere is aunified approach maximization of Shannon entropy [44]

that naturally follows the path of generalization from Zipfrsquos toHybrid Zipfrsquos law Extending this idea Jaksetic et al in [45]presented a transition from ZipfndashMandelbrot to HybridZipfndashMandelbrot law by employing maximum entropytechnique with one additional constraint It is interesting thatexamination of its densities provides some new insights ofLerchrsquos transcendent

Theorem 15 Let q be Hybrid ZipfndashMandelbrot law asdefined in (78) with parameters cge 0 dωgt 0 andN isin 1 2 and we have

1113954ZM Hlowast c dω( 1113857 S(q)leN 1113944

nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889times

(minus1)ℓ+1ℓ

α(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠ minus

(minus1)ℓ+1ℓ

β(ℓ+1)F q

(J + c)dHlowastcdω

ωJ minusln(middot)⎛⎝ ⎞⎠

⎧⎨

⎩

⎫⎬

⎭

(79)

14 Journal of Mathematics

Proof It is interesting to see that for qJ defined in (78)1113936

NJ1 qJ 1 erefore using above qJ in Shannon en-

tropy (64) we get Hybrid ZipfndashMandelbrot entropy (76)

S(q) minus 1113944N

J1qJ ln qJ minus 1113944

N

J1

ωJ

(J + c)dHlowastcdω

lnωJ

(J + c)dHlowastcdω

minus1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

lnωJ

(J + c)d

1113888 1113889 + ln1

Hlowastcdω

1113888 11138891113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

ln(J + c)

d

ωJ1113888 1113889 + ln H

lowastcdω1113872 11138731113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873

(80)

Finally substituting this qJ ωJ(J + c)dHlowastcdω inCorollary 10 (ii) we get the desired result

Corollary 13 Let q and p be Hybrid ZipfndashMandelbrotlaw with parameters c1 c2 isin [0infin)ω1ω2 d1 d2 gt 0 Nowusing qJ ωJ

1 (J + c1)d1Hlowastc1 d1 ω1

and pJ ωJ2

(J + c2)d2Hlowastc2 d2 ω2

in Corollary 11 (ii) the following holds

D(q p) 1113944N

J1

ωJ1

J + c1( 1113857d1Hlowastc1 d1 ω1

lnωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

ωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

⎛⎝ ⎞⎠

minus1113954ZM Hlowast c1 d1ω1( 1113857 +

1Hlowastc1 d1 ω1

1113944

N

J1

ωJ1

J + c1( 1113857d1ln

J + c2( 1113857d2

ωJ2

⎛⎝ ⎞⎠ + ln Hlowastc2 d2 ω2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎫⎪⎬

⎪⎭

(81)

Remark 9 Similarly we can give results for Shannonentropy KullbackndashLeibler distance ZipfndashMandelbrotentropy and Hybrid ZipfndashMandelbrot entropy by usinggeneralized Giaccardi inequality defined in (54) on thesame steps

4 Concluding Remarks

In this paper we gave generalization of Jensenrsquos inequality aswell as converse of Jensenrsquos inequality by using Montgomeryidentity We also formulate results for other inequalities like

JensenndashSteffensen inequality JensenndashBoas inequality andJensenndashBrunk inequality We can obtain JensenndashSteffenseninequality JensenndashBoas inequality and JensenndashBrunk in-equality by changing the assumption of Jensenrsquos inequalityAt the end we gave applications in information theory forour obtained results especially we gave results for HybridZipfndashMandelbrot entropy for our obtained results [46]

Data Availability

No data were used to support this study

Journal of Mathematics 15

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is research is supported by HEC Pakistan under NRPUProject 7906 and RUDN University Strategic AcademicLeadership Program

References

[1] X L Zhu and G H Yang ldquoJensen inequality approach tostability analysis of discrete-time systems with time-varyingdelayrdquo in Proceedings of the American Control Conferencepp 1644ndash1649 Seattle WA USA June 2008

[2] S I Butt M Klaricic Bakula ETH Pecaric and J PecaricldquoJensen-gruss inequality and its applications for the zipf-mandelbrot lawrdquo Mathematical Methods in the Applied Sci-ences vol 44 no 2 pp 1664ndash1673 2021

[3] M A Khan D Pecaric and J Pecaric ldquoNew refinement of theJensen inequality associated to certain functions with appli-cationsrdquo Journal of Inequalities and Applications vol 762020

[4] S Khan M Adil Khan S I Butt and Y-M Chu ldquoA newbound for the Jensen gap pertaining twice differentiablefunctions with applicationsrdquo Advances in Difference Equa-tions vol 1 2020

[5] H Budak andM Z Sarikaya ldquoOn generalized Ostrowski-typeinequalities for functions whose first derivatives absolutevalues are convexrdquo Turkish Journal of Mathematics vol 40pp 1193ndash1210 2016

[6] P Cerone and S S Dragomir ldquoOn some inequalities arisingfrom Montgomeryrsquos identityrdquo Journal of ComputationalAnalysis and Applications vol 5 no 4 pp 341ndash367 2003

[7] S Bernstein ldquoSur les fonctions absolument monotonesrdquo ActaMathematica vol 52 pp 1ndash66 1929

[8] V Culjak and J Pecaric ldquoInterpolation polynomials andinequalities for convex functions of higher orderrdquo ActaMathematica vol 5 pp 369ndash386 2012

[9] A M Fink ldquoBounds of the deviation of a function from itsaveragesrdquo Czechoslovak Mathematical Journal vol 42no 117 pp 289ndash310 1992

[10] J Pecaric and V Culjak ldquoInterpolation polinomials and in-equalities for convex functions of higher orderrdquo Mathe-matical Inequalities amp Applications vol 5 no 3 pp 369ndash3862002

[11] J Pecaric V Culjak and A M Fink ldquoOn some inequalitiesfor convex function of higher orderrdquo Advanced NonlinearStudies vol 6 no 2 pp 131ndash140 1999

[12] J Pecaric V Culjak andM Rogina ldquoOn some inequalities forconvex function of higher order IIrdquo Advanced NonlinearStudies vol 45 pp 281ndash294 2001

[13] J Pecaric I Peric and M Rodic-Lipanovic ldquoUniformtreatment of Jensen type inequalitiesrdquo Mathematical Reportsvol 16 no 66 pp 183ndash205 2014

[14] J Pecaric and J Peric ldquoImprovement of the Giaccardi and thepetrovic inequality and related stolarsky type meansrdquo Annalsof the University of Craiova - Mathematics and ComputerScience vol 39 no 1 pp 65ndash75 2012

[15] J Pecaric M Praljak and A Witkowski ldquoLinear operatorinequality for nminusconvex functions at a pointrdquo MathematicalInequalities amp Applications Appl vol 18 pp 1201ndash1217 2015

[16] Z Pavic ldquoRefinements of Jensenrsquos inequality for infiniteconvex combinationsrdquo Turkish Journal of Inequalities vol 2no 2 pp 44ndash53 2018

[17] R Bibi A Nosheen and J Pecaric ldquoExtended Jensenrsquos typeinequalities for diamond integrals via Taylors formulardquoTurkish Journal of Inequalities vol 3 no 1 pp 7ndash18 2019

[18] T Niaz K A Khan and J Pecaric ldquoOn refinement of Jensenrsquosinequality for 3minusconvex function at a pointrdquo Turkish Journalof Inequalities vol 4 no 1 pp 70ndash80 2020

[19] A Ekinci A O Akdemir and M E Ozdemir ldquoIntegralinequalities for different kinds of convexity via classical in-equalitiesrdquo Turkish Journal of Science vol 5 no 3 pp 305ndash313 2020

[20] Y M Chu S Talib E Set M U Awan and M A Noorldquo(p q)minus Analysis of Montgomery identity and estimates of(p q)minusbounds with applicationsrdquo Journal of Inequalities andApplications vol 2021 9 pages 2021

[21] M Z Sarikaya H Yaldrz and E Set ldquoOn fractional in-equalities via Montgomery identitiesrdquo International Journalof Open Problems in Complex Analysis vol 6 no 2 pp 36ndash432014

[22] A Bnouhachem ldquoA descent SQP alternating directionmethod for minimizing the sum of three convex functionsrdquoJournal of Nonlinear Functional Analysis vol 4 pp 469ndash4822020

[23] A O Akdemir S I Butt M Nadeem and M A RagusaldquoNew general variants of Chebyshev type inequalities viageneralized fractional integral operatorsrdquo Mathematicsvol 92 2021

[24] S I Butt M Umar S Rashid A O Akdemir and Y MingChu ldquoNew Hermite Jensen Mercer type inequalities viakminusfractional integralsrdquo Advances in Difference Equationsvol 635 2020

[25] S I Butt M Nadeem S Qaisar A O Akdemir andT Abdeljawad ldquoHermite Jensen Mercer type inequalities forconformable integrals and related resultsrdquo Advances in Dif-ference Equations vol 501 2020

[26] A Bnouhachem and X Qin ldquoAn inertial proximal Peaceman-Rachford splittingmethod with SQP regularization for convexprogrammingrdquo Journal of Nonlinear Functional Analysisvol 50 2020

[27] S Hamann ldquoMinimality conditions for convex compositefunctions and efficiency conditions in vector optimizationrdquoApplied Set-Valued Analysis and Optimization vol 1pp 221ndash229 2019

[28] A R Khan J Pecaric and M Praljak ldquoPopoviciu type in-equalities for n-convex functions via extension of Mont-gomery identitynminusconvex functions via extension ofMontgomery identityrdquo Analele Universitatii ldquoOvidiusrdquo Con-stanta-Seria Matematica vol 24 no 3 pp 161ndash188 2016

[29] R P Boas ldquoe Jensen-Steffensen inequalityrdquo ElektrotehnFak Ser Mat Fiz vol 302ndash319 pp 1ndash8 1970

[30] J Pecaric F Proschan and Y L Tong Convex FunctionsPartial Orderings and Statistical Applications Academic PressNew York NY USA 1992

[31] H D Brunk ldquoOn an inequality for convex functionsrdquo Pro-ceedings of the American Mathematical Society vol 7 no 5p 817 1956

[32] P R Beesack and J E Pecaric ldquoOn Jessenrsquos inequality forconvex functionsrdquo Journal of Mathematical Analysis andApplications vol 110 no 2 pp 536ndash552 1985

[33] R Barlow A Marshall and F Proschan ldquoSome inequalitiesfor starshaped and convex functionsrdquo Pacific Journal ofMathematics vol 29 no 1 pp 19ndash42 1969

16 Journal of Mathematics

[34] J E Pecaric ldquoOn an inequality of PMV Janicrdquo Publications delrsquoInstitut Mathematique vol 28 no 42 pp 145ndash149 1980

[35] G Szego ldquoUber eine Verallgemeinerung des DirichletschenIntegralsrdquo Mathematische Zeitschrift vol 52 no 1pp 676ndash685 1950

[36] F Giaccardi ldquoSu alcune disuguaglianierdquo Journal of Researchof the National Bureau of Standards vol 1 no 4 pp 139ndash1531955

[37] D S Mitrinovic J E Pecaric and A M Fink Classical andnew inequalities in analysis Mathematics and its Applications(East European Series) Vol 61 Kluwer Academic PublishersGroup Dordrecht Netherlands 1993

[38] I Csiszar ldquoInformation-type measures of diference ofprobability distributions and indirect observationsrdquo StudiaScience Math Hungar vol 2 pp 299ndash318 1967

[39] L Horvath D Pecaric and J Pecaric ldquoEstimations of f - andrenyi divergences by using a cyclic refinement of the Jensenrsquosinequalityrdquo Bulletin of the Malaysian Mathematical SciencesSociety vol 42 2017

[40] S Kullback Information =eory and Statistics Wiley NewYork NY USA 1959

[41] S T Piantadosi ldquoZipfrsquos word frequency law in natural lan-guage a critical review and future directionsrdquo PsychonomicBulletin and Review vol 21 no 5 pp 1112ndash1130 2014

[42] Z K Silagadze ldquoCitations and the zipfndashmandelbrot lawrdquoComplex Systems no 11 pp 487ndash499 1997

[43] J Jaksetic D Pecaric and J Pecaric ldquoSome properties of zipf-mandelbrot law and hurwitz ζ-functionrdquo Mathematical In-equalities amp Applications vol 21 no 2 pp 575ndash584 2018

[44] M Visser ldquoZipfrsquos law power laws and maximum entropyrdquoMathematical Inequalities amp Applications vol 15 Article ID043021 2013

[45] J Jaksetic D Pecaric and J Pecaric ldquoHybrid zipf-mandelbrotlawrdquo Mathematical Inequalities amp Applications vol 13 no 1pp 275ndash286 2019

[46] D V Widder ldquoCompletely convex functions and Lidstoneseriesrdquo Transactions of the American Mathematical Societyvol 51 p 387 1942

Journal of Mathematics 17

Proof It is interesting to see that for qJ defined in (78)1113936

NJ1 qJ 1 erefore using above qJ in Shannon en-

tropy (64) we get Hybrid ZipfndashMandelbrot entropy (76)

S(q) minus 1113944N

J1qJ ln qJ minus 1113944

N

J1

ωJ

(J + c)dHlowastcdω

lnωJ

(J + c)dHlowastcdω

minus1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

lnωJ

(J + c)d

1113888 1113889 + ln1

Hlowastcdω

1113888 11138891113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)d

ln(J + c)

d

ωJ1113888 1113889 + ln H

lowastcdω1113872 11138731113890 1113891

1

Hlowastcdω

1113944

N

J1

ωJ

(J + c)dln

(J + c)d

ωJ1113888 1113889 + ln H

lowastcdω1113872 1113873

(80)

Finally substituting this qJ ωJ(J + c)dHlowastcdω inCorollary 10 (ii) we get the desired result

Corollary 13 Let q and p be Hybrid ZipfndashMandelbrotlaw with parameters c1 c2 isin [0infin)ω1ω2 d1 d2 gt 0 Nowusing qJ ωJ

1 (J + c1)d1Hlowastc1 d1 ω1

and pJ ωJ2

(J + c2)d2Hlowastc2 d2 ω2

in Corollary 11 (ii) the following holds

D(q p) 1113944N

J1

ωJ1

J + c1( 1113857d1Hlowastc1 d1 ω1

lnωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

ωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

⎛⎝ ⎞⎠

minus1113954ZM Hlowast c1 d1ω1( 1113857 +

1Hlowastc1 d1 ω1

1113944

N

J1

ωJ1

J + c1( 1113857d1ln

J + c2( 1113857d2

ωJ2

⎛⎝ ⎞⎠ + ln Hlowastc2 d2 ω2

1113872 1113873

leN 1113944nminus2

ℓ2

1ℓ(ℓ + 2)(β minus α)

1113888 1113889 times(minus1)

ℓ+1ℓα(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

minus(minus1)

ℓ+1ℓβ(ℓ+1)

F qωJ2 J + c1( 1113857

d1Hlowastc1 d1 ω1

ωJ1 J + c2( 1113857

d2Hlowastc2 d2 ω2

minusln(middot)⎛⎝ ⎞⎠⎫⎪⎬

⎪⎭

(81)

Remark 9 Similarly we can give results for Shannonentropy KullbackndashLeibler distance ZipfndashMandelbrotentropy and Hybrid ZipfndashMandelbrot entropy by usinggeneralized Giaccardi inequality defined in (54) on thesame steps

4 Concluding Remarks

In this paper we gave generalization of Jensenrsquos inequality aswell as converse of Jensenrsquos inequality by using Montgomeryidentity We also formulate results for other inequalities like

JensenndashSteffensen inequality JensenndashBoas inequality andJensenndashBrunk inequality We can obtain JensenndashSteffenseninequality JensenndashBoas inequality and JensenndashBrunk in-equality by changing the assumption of Jensenrsquos inequalityAt the end we gave applications in information theory forour obtained results especially we gave results for HybridZipfndashMandelbrot entropy for our obtained results [46]

Data Availability

No data were used to support this study

Journal of Mathematics 15

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is research is supported by HEC Pakistan under NRPUProject 7906 and RUDN University Strategic AcademicLeadership Program

References

[1] X L Zhu and G H Yang ldquoJensen inequality approach tostability analysis of discrete-time systems with time-varyingdelayrdquo in Proceedings of the American Control Conferencepp 1644ndash1649 Seattle WA USA June 2008

[2] S I Butt M Klaricic Bakula ETH Pecaric and J PecaricldquoJensen-gruss inequality and its applications for the zipf-mandelbrot lawrdquo Mathematical Methods in the Applied Sci-ences vol 44 no 2 pp 1664ndash1673 2021

[3] M A Khan D Pecaric and J Pecaric ldquoNew refinement of theJensen inequality associated to certain functions with appli-cationsrdquo Journal of Inequalities and Applications vol 762020

[4] S Khan M Adil Khan S I Butt and Y-M Chu ldquoA newbound for the Jensen gap pertaining twice differentiablefunctions with applicationsrdquo Advances in Difference Equa-tions vol 1 2020

[5] H Budak andM Z Sarikaya ldquoOn generalized Ostrowski-typeinequalities for functions whose first derivatives absolutevalues are convexrdquo Turkish Journal of Mathematics vol 40pp 1193ndash1210 2016

[6] P Cerone and S S Dragomir ldquoOn some inequalities arisingfrom Montgomeryrsquos identityrdquo Journal of ComputationalAnalysis and Applications vol 5 no 4 pp 341ndash367 2003

[7] S Bernstein ldquoSur les fonctions absolument monotonesrdquo ActaMathematica vol 52 pp 1ndash66 1929

[8] V Culjak and J Pecaric ldquoInterpolation polynomials andinequalities for convex functions of higher orderrdquo ActaMathematica vol 5 pp 369ndash386 2012

[9] A M Fink ldquoBounds of the deviation of a function from itsaveragesrdquo Czechoslovak Mathematical Journal vol 42no 117 pp 289ndash310 1992

[10] J Pecaric and V Culjak ldquoInterpolation polinomials and in-equalities for convex functions of higher orderrdquo Mathe-matical Inequalities amp Applications vol 5 no 3 pp 369ndash3862002

[11] J Pecaric V Culjak and A M Fink ldquoOn some inequalitiesfor convex function of higher orderrdquo Advanced NonlinearStudies vol 6 no 2 pp 131ndash140 1999

[12] J Pecaric V Culjak andM Rogina ldquoOn some inequalities forconvex function of higher order IIrdquo Advanced NonlinearStudies vol 45 pp 281ndash294 2001

[13] J Pecaric I Peric and M Rodic-Lipanovic ldquoUniformtreatment of Jensen type inequalitiesrdquo Mathematical Reportsvol 16 no 66 pp 183ndash205 2014

[14] J Pecaric and J Peric ldquoImprovement of the Giaccardi and thepetrovic inequality and related stolarsky type meansrdquo Annalsof the University of Craiova - Mathematics and ComputerScience vol 39 no 1 pp 65ndash75 2012

[15] J Pecaric M Praljak and A Witkowski ldquoLinear operatorinequality for nminusconvex functions at a pointrdquo MathematicalInequalities amp Applications Appl vol 18 pp 1201ndash1217 2015

[16] Z Pavic ldquoRefinements of Jensenrsquos inequality for infiniteconvex combinationsrdquo Turkish Journal of Inequalities vol 2no 2 pp 44ndash53 2018

[17] R Bibi A Nosheen and J Pecaric ldquoExtended Jensenrsquos typeinequalities for diamond integrals via Taylors formulardquoTurkish Journal of Inequalities vol 3 no 1 pp 7ndash18 2019

[18] T Niaz K A Khan and J Pecaric ldquoOn refinement of Jensenrsquosinequality for 3minusconvex function at a pointrdquo Turkish Journalof Inequalities vol 4 no 1 pp 70ndash80 2020

[19] A Ekinci A O Akdemir and M E Ozdemir ldquoIntegralinequalities for different kinds of convexity via classical in-equalitiesrdquo Turkish Journal of Science vol 5 no 3 pp 305ndash313 2020

[20] Y M Chu S Talib E Set M U Awan and M A Noorldquo(p q)minus Analysis of Montgomery identity and estimates of(p q)minusbounds with applicationsrdquo Journal of Inequalities andApplications vol 2021 9 pages 2021

[21] M Z Sarikaya H Yaldrz and E Set ldquoOn fractional in-equalities via Montgomery identitiesrdquo International Journalof Open Problems in Complex Analysis vol 6 no 2 pp 36ndash432014

[22] A Bnouhachem ldquoA descent SQP alternating directionmethod for minimizing the sum of three convex functionsrdquoJournal of Nonlinear Functional Analysis vol 4 pp 469ndash4822020

[23] A O Akdemir S I Butt M Nadeem and M A RagusaldquoNew general variants of Chebyshev type inequalities viageneralized fractional integral operatorsrdquo Mathematicsvol 92 2021

[24] S I Butt M Umar S Rashid A O Akdemir and Y MingChu ldquoNew Hermite Jensen Mercer type inequalities viakminusfractional integralsrdquo Advances in Difference Equationsvol 635 2020

[25] S I Butt M Nadeem S Qaisar A O Akdemir andT Abdeljawad ldquoHermite Jensen Mercer type inequalities forconformable integrals and related resultsrdquo Advances in Dif-ference Equations vol 501 2020

[26] A Bnouhachem and X Qin ldquoAn inertial proximal Peaceman-Rachford splittingmethod with SQP regularization for convexprogrammingrdquo Journal of Nonlinear Functional Analysisvol 50 2020

[27] S Hamann ldquoMinimality conditions for convex compositefunctions and efficiency conditions in vector optimizationrdquoApplied Set-Valued Analysis and Optimization vol 1pp 221ndash229 2019

[28] A R Khan J Pecaric and M Praljak ldquoPopoviciu type in-equalities for n-convex functions via extension of Mont-gomery identitynminusconvex functions via extension ofMontgomery identityrdquo Analele Universitatii ldquoOvidiusrdquo Con-stanta-Seria Matematica vol 24 no 3 pp 161ndash188 2016

[29] R P Boas ldquoe Jensen-Steffensen inequalityrdquo ElektrotehnFak Ser Mat Fiz vol 302ndash319 pp 1ndash8 1970

[30] J Pecaric F Proschan and Y L Tong Convex FunctionsPartial Orderings and Statistical Applications Academic PressNew York NY USA 1992

[31] H D Brunk ldquoOn an inequality for convex functionsrdquo Pro-ceedings of the American Mathematical Society vol 7 no 5p 817 1956

[32] P R Beesack and J E Pecaric ldquoOn Jessenrsquos inequality forconvex functionsrdquo Journal of Mathematical Analysis andApplications vol 110 no 2 pp 536ndash552 1985

[33] R Barlow A Marshall and F Proschan ldquoSome inequalitiesfor starshaped and convex functionsrdquo Pacific Journal ofMathematics vol 29 no 1 pp 19ndash42 1969

16 Journal of Mathematics

[34] J E Pecaric ldquoOn an inequality of PMV Janicrdquo Publications delrsquoInstitut Mathematique vol 28 no 42 pp 145ndash149 1980

[35] G Szego ldquoUber eine Verallgemeinerung des DirichletschenIntegralsrdquo Mathematische Zeitschrift vol 52 no 1pp 676ndash685 1950

[36] F Giaccardi ldquoSu alcune disuguaglianierdquo Journal of Researchof the National Bureau of Standards vol 1 no 4 pp 139ndash1531955

[37] D S Mitrinovic J E Pecaric and A M Fink Classical andnew inequalities in analysis Mathematics and its Applications(East European Series) Vol 61 Kluwer Academic PublishersGroup Dordrecht Netherlands 1993

[38] I Csiszar ldquoInformation-type measures of diference ofprobability distributions and indirect observationsrdquo StudiaScience Math Hungar vol 2 pp 299ndash318 1967

[39] L Horvath D Pecaric and J Pecaric ldquoEstimations of f - andrenyi divergences by using a cyclic refinement of the Jensenrsquosinequalityrdquo Bulletin of the Malaysian Mathematical SciencesSociety vol 42 2017

[40] S Kullback Information =eory and Statistics Wiley NewYork NY USA 1959

[41] S T Piantadosi ldquoZipfrsquos word frequency law in natural lan-guage a critical review and future directionsrdquo PsychonomicBulletin and Review vol 21 no 5 pp 1112ndash1130 2014

[42] Z K Silagadze ldquoCitations and the zipfndashmandelbrot lawrdquoComplex Systems no 11 pp 487ndash499 1997

[43] J Jaksetic D Pecaric and J Pecaric ldquoSome properties of zipf-mandelbrot law and hurwitz ζ-functionrdquo Mathematical In-equalities amp Applications vol 21 no 2 pp 575ndash584 2018

[44] M Visser ldquoZipfrsquos law power laws and maximum entropyrdquoMathematical Inequalities amp Applications vol 15 Article ID043021 2013

[45] J Jaksetic D Pecaric and J Pecaric ldquoHybrid zipf-mandelbrotlawrdquo Mathematical Inequalities amp Applications vol 13 no 1pp 275ndash286 2019

[46] D V Widder ldquoCompletely convex functions and Lidstoneseriesrdquo Transactions of the American Mathematical Societyvol 51 p 387 1942

Journal of Mathematics 17

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is research is supported by HEC Pakistan under NRPUProject 7906 and RUDN University Strategic AcademicLeadership Program

References

[1] X L Zhu and G H Yang ldquoJensen inequality approach tostability analysis of discrete-time systems with time-varyingdelayrdquo in Proceedings of the American Control Conferencepp 1644ndash1649 Seattle WA USA June 2008

[2] S I Butt M Klaricic Bakula ETH Pecaric and J PecaricldquoJensen-gruss inequality and its applications for the zipf-mandelbrot lawrdquo Mathematical Methods in the Applied Sci-ences vol 44 no 2 pp 1664ndash1673 2021

[3] M A Khan D Pecaric and J Pecaric ldquoNew refinement of theJensen inequality associated to certain functions with appli-cationsrdquo Journal of Inequalities and Applications vol 762020

[4] S Khan M Adil Khan S I Butt and Y-M Chu ldquoA newbound for the Jensen gap pertaining twice differentiablefunctions with applicationsrdquo Advances in Difference Equa-tions vol 1 2020

[5] H Budak andM Z Sarikaya ldquoOn generalized Ostrowski-typeinequalities for functions whose first derivatives absolutevalues are convexrdquo Turkish Journal of Mathematics vol 40pp 1193ndash1210 2016

[6] P Cerone and S S Dragomir ldquoOn some inequalities arisingfrom Montgomeryrsquos identityrdquo Journal of ComputationalAnalysis and Applications vol 5 no 4 pp 341ndash367 2003

[7] S Bernstein ldquoSur les fonctions absolument monotonesrdquo ActaMathematica vol 52 pp 1ndash66 1929

[8] V Culjak and J Pecaric ldquoInterpolation polynomials andinequalities for convex functions of higher orderrdquo ActaMathematica vol 5 pp 369ndash386 2012

[9] A M Fink ldquoBounds of the deviation of a function from itsaveragesrdquo Czechoslovak Mathematical Journal vol 42no 117 pp 289ndash310 1992

[10] J Pecaric and V Culjak ldquoInterpolation polinomials and in-equalities for convex functions of higher orderrdquo Mathe-matical Inequalities amp Applications vol 5 no 3 pp 369ndash3862002

[11] J Pecaric V Culjak and A M Fink ldquoOn some inequalitiesfor convex function of higher orderrdquo Advanced NonlinearStudies vol 6 no 2 pp 131ndash140 1999

[12] J Pecaric V Culjak andM Rogina ldquoOn some inequalities forconvex function of higher order IIrdquo Advanced NonlinearStudies vol 45 pp 281ndash294 2001

[13] J Pecaric I Peric and M Rodic-Lipanovic ldquoUniformtreatment of Jensen type inequalitiesrdquo Mathematical Reportsvol 16 no 66 pp 183ndash205 2014

[14] J Pecaric and J Peric ldquoImprovement of the Giaccardi and thepetrovic inequality and related stolarsky type meansrdquo Annalsof the University of Craiova - Mathematics and ComputerScience vol 39 no 1 pp 65ndash75 2012

[15] J Pecaric M Praljak and A Witkowski ldquoLinear operatorinequality for nminusconvex functions at a pointrdquo MathematicalInequalities amp Applications Appl vol 18 pp 1201ndash1217 2015

[16] Z Pavic ldquoRefinements of Jensenrsquos inequality for infiniteconvex combinationsrdquo Turkish Journal of Inequalities vol 2no 2 pp 44ndash53 2018

[17] R Bibi A Nosheen and J Pecaric ldquoExtended Jensenrsquos typeinequalities for diamond integrals via Taylors formulardquoTurkish Journal of Inequalities vol 3 no 1 pp 7ndash18 2019

[18] T Niaz K A Khan and J Pecaric ldquoOn refinement of Jensenrsquosinequality for 3minusconvex function at a pointrdquo Turkish Journalof Inequalities vol 4 no 1 pp 70ndash80 2020

[19] A Ekinci A O Akdemir and M E Ozdemir ldquoIntegralinequalities for different kinds of convexity via classical in-equalitiesrdquo Turkish Journal of Science vol 5 no 3 pp 305ndash313 2020

[20] Y M Chu S Talib E Set M U Awan and M A Noorldquo(p q)minus Analysis of Montgomery identity and estimates of(p q)minusbounds with applicationsrdquo Journal of Inequalities andApplications vol 2021 9 pages 2021

[21] M Z Sarikaya H Yaldrz and E Set ldquoOn fractional in-equalities via Montgomery identitiesrdquo International Journalof Open Problems in Complex Analysis vol 6 no 2 pp 36ndash432014

[22] A Bnouhachem ldquoA descent SQP alternating directionmethod for minimizing the sum of three convex functionsrdquoJournal of Nonlinear Functional Analysis vol 4 pp 469ndash4822020

[23] A O Akdemir S I Butt M Nadeem and M A RagusaldquoNew general variants of Chebyshev type inequalities viageneralized fractional integral operatorsrdquo Mathematicsvol 92 2021

[24] S I Butt M Umar S Rashid A O Akdemir and Y MingChu ldquoNew Hermite Jensen Mercer type inequalities viakminusfractional integralsrdquo Advances in Difference Equationsvol 635 2020

[25] S I Butt M Nadeem S Qaisar A O Akdemir andT Abdeljawad ldquoHermite Jensen Mercer type inequalities forconformable integrals and related resultsrdquo Advances in Dif-ference Equations vol 501 2020

[26] A Bnouhachem and X Qin ldquoAn inertial proximal Peaceman-Rachford splittingmethod with SQP regularization for convexprogrammingrdquo Journal of Nonlinear Functional Analysisvol 50 2020

[27] S Hamann ldquoMinimality conditions for convex compositefunctions and efficiency conditions in vector optimizationrdquoApplied Set-Valued Analysis and Optimization vol 1pp 221ndash229 2019

[28] A R Khan J Pecaric and M Praljak ldquoPopoviciu type in-equalities for n-convex functions via extension of Mont-gomery identitynminusconvex functions via extension ofMontgomery identityrdquo Analele Universitatii ldquoOvidiusrdquo Con-stanta-Seria Matematica vol 24 no 3 pp 161ndash188 2016

[29] R P Boas ldquoe Jensen-Steffensen inequalityrdquo ElektrotehnFak Ser Mat Fiz vol 302ndash319 pp 1ndash8 1970

[30] J Pecaric F Proschan and Y L Tong Convex FunctionsPartial Orderings and Statistical Applications Academic PressNew York NY USA 1992

[31] H D Brunk ldquoOn an inequality for convex functionsrdquo Pro-ceedings of the American Mathematical Society vol 7 no 5p 817 1956

[32] P R Beesack and J E Pecaric ldquoOn Jessenrsquos inequality forconvex functionsrdquo Journal of Mathematical Analysis andApplications vol 110 no 2 pp 536ndash552 1985

[33] R Barlow A Marshall and F Proschan ldquoSome inequalitiesfor starshaped and convex functionsrdquo Pacific Journal ofMathematics vol 29 no 1 pp 19ndash42 1969

16 Journal of Mathematics

[34] J E Pecaric ldquoOn an inequality of PMV Janicrdquo Publications delrsquoInstitut Mathematique vol 28 no 42 pp 145ndash149 1980

[35] G Szego ldquoUber eine Verallgemeinerung des DirichletschenIntegralsrdquo Mathematische Zeitschrift vol 52 no 1pp 676ndash685 1950

[36] F Giaccardi ldquoSu alcune disuguaglianierdquo Journal of Researchof the National Bureau of Standards vol 1 no 4 pp 139ndash1531955

[37] D S Mitrinovic J E Pecaric and A M Fink Classical andnew inequalities in analysis Mathematics and its Applications(East European Series) Vol 61 Kluwer Academic PublishersGroup Dordrecht Netherlands 1993

[38] I Csiszar ldquoInformation-type measures of diference ofprobability distributions and indirect observationsrdquo StudiaScience Math Hungar vol 2 pp 299ndash318 1967

[39] L Horvath D Pecaric and J Pecaric ldquoEstimations of f - andrenyi divergences by using a cyclic refinement of the Jensenrsquosinequalityrdquo Bulletin of the Malaysian Mathematical SciencesSociety vol 42 2017

[40] S Kullback Information =eory and Statistics Wiley NewYork NY USA 1959

[41] S T Piantadosi ldquoZipfrsquos word frequency law in natural lan-guage a critical review and future directionsrdquo PsychonomicBulletin and Review vol 21 no 5 pp 1112ndash1130 2014

[42] Z K Silagadze ldquoCitations and the zipfndashmandelbrot lawrdquoComplex Systems no 11 pp 487ndash499 1997

[43] J Jaksetic D Pecaric and J Pecaric ldquoSome properties of zipf-mandelbrot law and hurwitz ζ-functionrdquo Mathematical In-equalities amp Applications vol 21 no 2 pp 575ndash584 2018

[44] M Visser ldquoZipfrsquos law power laws and maximum entropyrdquoMathematical Inequalities amp Applications vol 15 Article ID043021 2013

[45] J Jaksetic D Pecaric and J Pecaric ldquoHybrid zipf-mandelbrotlawrdquo Mathematical Inequalities amp Applications vol 13 no 1pp 275ndash286 2019

[46] D V Widder ldquoCompletely convex functions and Lidstoneseriesrdquo Transactions of the American Mathematical Societyvol 51 p 387 1942

Journal of Mathematics 17

[34] J E Pecaric ldquoOn an inequality of PMV Janicrdquo Publications delrsquoInstitut Mathematique vol 28 no 42 pp 145ndash149 1980

[35] G Szego ldquoUber eine Verallgemeinerung des DirichletschenIntegralsrdquo Mathematische Zeitschrift vol 52 no 1pp 676ndash685 1950

[36] F Giaccardi ldquoSu alcune disuguaglianierdquo Journal of Researchof the National Bureau of Standards vol 1 no 4 pp 139ndash1531955

[37] D S Mitrinovic J E Pecaric and A M Fink Classical andnew inequalities in analysis Mathematics and its Applications(East European Series) Vol 61 Kluwer Academic PublishersGroup Dordrecht Netherlands 1993

[38] I Csiszar ldquoInformation-type measures of diference ofprobability distributions and indirect observationsrdquo StudiaScience Math Hungar vol 2 pp 299ndash318 1967

[39] L Horvath D Pecaric and J Pecaric ldquoEstimations of f - andrenyi divergences by using a cyclic refinement of the Jensenrsquosinequalityrdquo Bulletin of the Malaysian Mathematical SciencesSociety vol 42 2017

[40] S Kullback Information =eory and Statistics Wiley NewYork NY USA 1959

[41] S T Piantadosi ldquoZipfrsquos word frequency law in natural lan-guage a critical review and future directionsrdquo PsychonomicBulletin and Review vol 21 no 5 pp 1112ndash1130 2014

[42] Z K Silagadze ldquoCitations and the zipfndashmandelbrot lawrdquoComplex Systems no 11 pp 487ndash499 1997

[43] J Jaksetic D Pecaric and J Pecaric ldquoSome properties of zipf-mandelbrot law and hurwitz ζ-functionrdquo Mathematical In-equalities amp Applications vol 21 no 2 pp 575ndash584 2018

[44] M Visser ldquoZipfrsquos law power laws and maximum entropyrdquoMathematical Inequalities amp Applications vol 15 Article ID043021 2013

[45] J Jaksetic D Pecaric and J Pecaric ldquoHybrid zipf-mandelbrotlawrdquo Mathematical Inequalities amp Applications vol 13 no 1pp 275ndash286 2019

[46] D V Widder ldquoCompletely convex functions and Lidstoneseriesrdquo Transactions of the American Mathematical Societyvol 51 p 387 1942

Journal of Mathematics 17