ON AN INEQUALITY OF G. H. HARDYprr.hec.gov.pk/jspui/bitstream/123456789/213/1/1685S.pdf · On an...

On an Inequality of G. H. Hardy

Name : Sajid Iqbal

Year of Admission : 2008

Registration No. : 108-GCU-PHD-SMS-08

Abdus Salam School of Mathematical Sciences

GC University Lahore, Pakistan

Since 1864

i

On an Inequality of G. H. Hardy

Submitted to



in the partial fulfillment of the requirements for the award of degree of

Doctor of Philosophy

in

Mathematics

By

Name : Sajid Iqbal

Year of Admission : 2008

Registration No. : 108-GCU-PHD-SMS-08



ii

DECLARATION

I, Mr. Sajid Iqbal Registration No. 108-GCU-PHD-SMS-08 student at Abdus

Salam School of Mathematical Sciences GC University in the subject of

Mathematics, Year of Admission (2008), hereby declare that the matter printed

in this thesis titled

“On an Inequality of G. H. Hardy”

is my own work and that

(i) I am not registered for the similar degree elsewhere contemporaneously.

(ii) No direct major work had already been done by me or anybody else on

this topic; I worked on, for the Ph. D. degree.

(iii) The work, I am submitting for the Ph. D. degree has not already been

submitted elsewhere and shall not in future be submitted by me for

obtaining similar degree from any other institution.

Dated: ------------------------- ------------------------------------

Signature

iii

RESEARCH COMPLETION CERTIFICATE

Certified that the research work contained in this thesis titled

“On an Inequality of G. H. Hardy”

has been carried out and completed by Mr. Sajid Iqbal Registration No. 108-GCU-

PHD-SMS-08 under my supervision.

----------------------------- -------------------------------

Date Supervisor

Josip Pečarić

Submitted Through

Prof. Dr. A. D. Raza Choudary --------------------------------

Director General Controller of Examination

Abdus Salam School of Mathematical Sciences GC University, Lahore

GC University, Lahore, Pakistan Pakistan

Dedicated

to

my beloved parents

Naseem Akhtar and Muhammad Iqbal

iv

Table of Contents

Table of Contents v

Abstract vii

Acknowledgements x

1 Introduction 1

1.1 Convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Fractional integrals and fractional derivatives . . . . . . . . . . . . . . 4

2 Inequality of G. H. Hardy and its improvements 12

2.1 On an inequality of G. H. Hardy . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 New inequalities involving fractional integrals and fractional derivatives 23

2.3 Improvement of an an inequality of G. H. Hardy and Cauchy means . 28

2.3.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.2 Mean value theorems and Cauchy means . . . . . . . . . . . . 38

2.4 Further results on an inequality of G. H. Hardy . . . . . . . . . . . . 40

2.5 Inequality of G. H. Hardy and superquadratic function . . . . . . . . 46

2.5.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.6 Refinements of an inequality of G. H. Hardy . . . . . . . . . . . . . . 53

2.6.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3 G. H. Hardy-type integral inequalities 61

3.1 New class of the G. H. Hardy-type inequalities with kernels . . . . . 61

3.2 Generalized G. H. Hardy-type inequalities . . . . . . . . . . . . . . . 66

3.2.1 G. H. Hardy-type inequalities for fractional integrals . . . . . 67

3.2.2 G. H. Hardy-type inequalities for fractional derivatives . . . . 71

v

4 Generalization of Hardy’s and related inequalities involving mono-

tone convex functions 75

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2.1 Hardy-Hilbert and Godunova’s inequalities . . . . . . . . . . . 83

4.2.2 G. H. Hardy-type inequalities for fractional integrals involving

monotone convex function . . . . . . . . . . . . . . . . . . . . 88

4.2.3 G. H. Hardy-type inequalities for fractional derivatives involv-

ing monotone convex function . . . . . . . . . . . . . . . . . . 92

5 Generalization of an inequality for integral transforms with some

results related to exponential convexity 94

5.1 Generalization of an inequality for integral transforms with kernel and

related results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.1.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.1.3 Exponential Convexity . . . . . . . . . . . . . . . . . . . . . . 104

5.2 n-Exponential convexity of Hardy-type functionals . . . . . . . . . . . 107

5.2.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Bibliography 114

vi

Abstract

Mathematical inequalities play very important role in development of all branches of

mathematics. A huge effort has been made to discover the new types of inequalities

and the basic work published in 1934 by Hardy, Littlewood and Polya [36]. Later

on Beckenbach and Bellman in 1961 in their book “Inequalities”[13], and the book

“Analytic inequalities”by Mitronovic [53] published in 1970 made considerable con-

tribution in this field. The mathematical inequalities are useful because these are

used as major tool in the development of modern analysis. A wide range of prob-

lems in various branches of mathematics are studied by well known Jensen, Hilbert,

Hadamard, Hardy, Poincare, Opial, Sobolev, Levin and Lyapunov inequalities. In

1992, J. Pecaric, F. Proschan and Y. L. Tong play their vital role in this field and

they published famous book “Convex Functions, Partial Orderings and Statistical

Application”which is considered as a brightening star in this field.

On the other hand, the applications of fractional calculus in mathematical in-

equalities have great importance. Hardy-type inequalities are very famous and play

fundamental role in mathematical inequalities. Many mathematicians gave general-

izations, improvements and application in the development of the Hardy’s inequalities

and they use fractional integrals and fractional derivatives to establish new integral

inequalities. Further details concerning the rich history of the integral inequalities

can be found in [58]–[64], [73]–[75] and the references given therein.

Cizmesija, Krulic, Pecaric and Persson establish some new refined Hardy-type

inequalities with kernels in their recent papers [4], [25], [28], [29], [34], [52] (also see

vii

viii

[15]– [23]). Inequalities lies in the heart of the mathematical analysis and numerous

mathematicians are attracted by these famous Hardy-type inequalities and discover

new inequalities with kernels and applications of different fractional integrals and

fractional derivatives, (see [25], [28], [38], [50], [52], [65]).

In this Ph.D thesis an integral operator with general non-negative kernel on mea-

sure spaces with positive σ-finite measure is considered. Our aim is to give the

inequality of G. H. Hardy and its improvements for Riemann-Liouville fractional in-

tegrals, Canavati-type fractional derivative, Caputo fractional derivative, fractional

integral of a function with respect to an increasing function, Hadamard-type frac-

tional integrals and Erdelyi-Kober fractional integrals with respect to the convex and

superquadratic functions. We will use different weights in this construction to obtain

new inequalities of G. H. Hardy. Such type of results are widely discussed in [38](see

also [28]). Also, we generalize and refine some inequalities of classical Hardy-Hilbert-

type, classical Hardy-Littlewood-Polya-type and Godunova-type inequalities [55] for

monotone convex function.

The first chapter contains the basic concepts and notions from theory of convex

functions and superquadratic functions. Some useful lemmas related to fractional

integrals and fractional derivatives are given which we frequently use in next chapters

to prove our results.

In the second chapter, we state, prove and discuss new general inequality for

convex and increasing functions. Continuing the extension of our general result, we

obtain new results involving different fractional integrals and fractional derivatives.

We give improvements of an inequality of G. H. Hardy for convex and superquadratic

functions as well.

In the third chapter, we give the new class of the G. H. Hardy-type integral inequal-

ities with applications. We provide some generalized G. H. Hardy-type inequalities

for fractional integrals and fractional derivatives.

In fourth chapter, we present generalized Hardy’s and related inequalities involving

monotone convex function. We generalize and refine some inequalities of classical

ix

Polya-Knopp’s, Hardy-Hilbert, classical Hardy-Littlewood-Polya, Hardy-Hilber-type

and Godunova’s. We also give some new fractional inequalities as refinements.

In the fifth chapter, we establish a generalization of the inequality introduced by

D. S. Mitrinovic and J. Pecaric in 1988. We prove mean value theorems of Cauchy

type and discuss the exponential convexity, logarithmic convexity and monotonicity

of the means. Also, we produce the n-exponential convexity of the linear functionals

obtained by taking the non-negative difference of Hardy-type inequalities. At the

end, some related examples are given.

Acknowledgements

All praise unto Allah, Who is the most Gracious and Merciful, Lord of all the worlds.

First of all, I would like to express my heartiest and sincere gratitude to my very

dear and respected supervisor Professor Dr. Josip Pecaric for his quality supervision,

constant support, guidance and encouragement during this research. He has been a

great role model for me as a human being and researcher. I thank to Professor Dr. A.

D. R. Choudary (Director General ASSMS) from the core of my heart for his kindness

and providing me this great opportunity and excellent environment for the research.

I deeply thank to Dr. Kristina Krulic Himmelreich for the scientific cooperation.

She really help me at different stages during my research and her valuable sugges-

tions improve the quality of my research. My thanks are also for the Professor Ivan

Peric for his valueable suggestions during my research. I am thankful to the whole

Mathematical Inequality and Application (MIA) group Pakistan and Croatia for the

scientific cooperation and help throughout this research. I am especially thankful to

my colleagues in Batch VI for giving me the friendly environment for the research.

I am thankful to my family, specially my parents, my brother Hafiz Majid Iqbal,

my sisters and my very cute niece Sana Kainat for their prayers, help and love. Their

prayers make possible for me to complete this research. Here, I want to say my special

thanks to my dear uncle Dr. Muhammd Ishaq for so many things.

I wish to acknowledge the financial support by Higher Education Commission

(HEC) and Government of Punjab throughout the process of completion of this thesis.

Finally, I wish to thank the following: All the foreign professors at ASSMS and

all the staff members; especially Mr. Awais and Mr. Shaukat (for so many things).

Lahore, Pakistan Sajid Iqbal

2012

x

Chapter 1

Introduction

1.1 Convex functions

Convex functions has great importance in the theory of inequalities and they are

widely discuss in the classical book of Hardy, Littlewood and Polya [36] (see also

[57]). In this section we give some of the results concerning convex functions.

Definition 1.1.1. Let I be an interval in R. A function Φ: I → R is called convex

if

Φ(λx+ (1− λ)y) ≤ λΦ(x) + (1− λ)Φ(y) (1.1.1)

for all points x, y ∈ I and all λ ∈ [0, 1]. It is called strictly convex if the inequality

(1.1.1) holds strictly whenever x 6= y and λ ∈ (0, 1).

In the following lemma the equivalent definition of convex functions is given.

Lemma 1.1.1. If Φ : I → R is convex on an interval I ⊆ R, then

(x3 − x2)Φ(x1) + (x1 − x3)Φ(x2) + (x2 − x1)Φ(x3) ≥ 0

holds for every x1, x2, x3 ∈ I such that x1 < x2 < x3.

Definition 1.1.2. Let I be an interval in R. A function Φ : I → R is called convex

in the Jensen sense, or J-convex on I (midconvex, midpoint convex) if for all points

x, y ∈ I the inequality

Φ

(x+ y

2

)≤ Φ(x) + Φ(y)

2(1.1.2)

holds. A J-convex function is said to be strictly J-convex if for all pairs of points

(x, y), x 6= y, strict inequality holds in (1.1.2).

1

2

In the context of continuity the following criteria of equivalence of (1.1.1) and

(1.1.2) is valid.

Theorem 1.1.2. Let Φ : I → R be a continuous function. Then Φ is convex function

if and only if Φ is J-convex function.

Now, we introduce some necessary notations and recall some basic facts about

convex functions, log-convex functions (see e.g. [50], [57], [65]) as well as exponentially

convex functions (see e.g [12], [54], [56]).

In 1929, S. N. Bernstein introduced the notion of exponentially convex function

in [12]. Later on D.V. Widder in [71] introduced these functions as a sub-class of

convex function in a given interval (a, b) (for details see [71], [72]).

Definition 1.1.3. A positive function Φ is said to be logarithmically convex on an

interval I ⊆ R if log Φ is a convex function on I, or equivalently if for all x, y ∈ I and

all λ ∈ [0, 1]

Φ(λx+ (1− λ)y) ≤ Φλ(x)Φ1−λ(y).

For such function Φ, we shortly say Φ is log-convex. A positive function Φ is log-convex

in the Jensen sense if for each x, y ∈ I

Φ2

(x+ y

2

)≤ Φ(x)Φ(y)

holds, i.e., if log Φ is convex in the Jensen sense.

Remark 1.1.1. A function Φ is log-convex on an interval I, if and only if for all

x1, x2, x3 ∈ I, x1 < x2 < x3, it holds

[Φ(x2)]x3−x1 ≤ [Φ(x1)]x3−x2 [Φ(x3)]x2−x1 .

Furthermore, if x1, x2, y1, y2 ∈ I are such that x1 ≤ y1, x2 ≤ y2, x1 6= x2, y1 6= y2,

then (Φ(x2)

Φ(x1)

) 1x2−x1

≤(

Φ(y2)

Φ(y1)

) 1y2−y1

. (1.1.3)

Inequality (1.1.3) is known as Galvani’s theorem for log-convex functions Φ: I → R.

We continue with the definition of exponentially convex function as originally

given in [12] by Berstein (see also [6], [54], [56]).

3

Definition 1.1.4. A function Φ: (a, b)→ R is exponentially convex if it is continuous

andn∑

i,j=1

titjΦ(xi + xj) ≥ 0 (1.1.4)

holds for every n ∈ N and all sequences (tn)n∈N and (xn)n∈N of real numbers, such

that xi + xj ∈ (a, b), 1 ≤ i, j ≤ n.

We continue this section by recalling some notions of our special interest about

n-exponential convexity given in [66].

Definition 1.1.5. A function Φ : I → R is n-exponentially convex in the Jensen

sense on I ifn∑

i,j=1

titjΦ

(xi + xj

2

)≥ 0

holds for all choices of ti ∈ R, xi ∈ I, i = 1, . . . , n.

A function Φ : I → R is n-exponentially convex on I if it is n-exponentially convex

in the Jensen sense and continuous on I.

Remark 1.1.2. It is clear from the definition that 1-exponentially convex functions

in the Jensen sense are in fact non-negative functions. Also, n-exponentially convex

functions in the Jensen sense are k-exponentially convex in the Jensen sense for every

k ∈ N, k ≤ n.

By definition of positive semi-definite matrices and some basic linear algebra we

have the following proposition.

Proposition 1.1.3. Let I be an open interval in R. If Φ is n-exponentially convex

in the Jensen sense on J, then the matrix[Φ(xi+xj

2

)]ki,j=1

is positive semi-definite

matrix for all k ∈ N, k ≤ n. Particularly

det

[Φ

(xi + xj

2

)]ki,j=1

≥ 0, for all k ∈ N, k ≤ n.

Definition 1.1.6. Let I be an open interval in R. A function Φ : I → R is expo-

nentially convex in the Jensen sense on I if it is n-exponentially convex in the Jensen

sense on I for n ∈ N.A function Φ : I → R is exponentially convex if it is exponentially convex in the

Jensen sense and continuous.

4

Remark 1.1.3. It follows that a function is log-convex in the Jensen sense if and only

if it is 2-exponentially convex in the Jensen sense.

Also, using basic convexity theory it follows that a function is log-convex if and only

if it is 2-exponentially convex.

Now, we continue with derivative of convex functions. The derivative of convex

functions is best studied in terms of the left and right derivatives defined by

Φ′

−(x) = limyx

Φ(y)− Φ(x)

y − x, Φ

′

+(x) = limyx

Φ(y)− Φ(x)

y − x.

Definition 1.1.7. Let Φ : I −→ R be a convex function, then the sub-differential of

Φ at x, denoted by ∂Φ(x), is defined as

∂Φ(x) = α ∈ R : Φ(y)− Φ(x)− α(y − x) ≥ 0, y ∈ I.

Many further information on convex and concave functions can be found e.g. in

the monographs [57] and [65] and in references cited therein.

The concept of superquadratic and subquadratic functions is introduced by Abramovich,

Jameson and Sinnamon in [2] (see also [1], [3]).

Definition 1.1.8. [2, Definition 2.1] A function ϕ : [0,∞) → R is superquadratic

provided that for all x ≥ 0 there exists a constant Cx ∈ R such that

ϕ(y)− ϕ(x)− ϕ (|y − x|) ≥ Cx (y − x) , (1.1.5)

for all y ≥ 0. We say that ϕ is subquadratic if −ϕ is superquadratic.

1.2 Fractional integrals and fractional derivatives

Here we give some definitions and useful lemmas, which we frequently use in next

chapters to give the proof of our results. First, let us recall some facts about fractional

derivatives needed in the sequel, for more details see e.g. [7], [33].

Let 0 < a < b ≤ ∞. By Cm[a, b] we denote the space of all functions on [a, b]

which have continuous derivatives up to order m, and AC[a, b] is the space of all

absolutely continuous functions on [a, b]. By ACm[a, b] we denote the space of all

functions g ∈ Cm−1[a, b] with g(m−1) ∈ AC[a, b]. For any α ∈ R we denote by [α] the

integral part of α (the integer k satisfying k ≤ α < k + 1) and dαe is the ceiling of α

(minn ∈ N, n ≥ α). By L1(a, b) we denote the space of all functions integrable on

5

the interval (a, b), and by L∞(a, b) the set of all functions measurable and essentially

bounded on (a, b). Clearly, L∞(a, b) ⊂ L1(a, b).

Now, we give well known definitions of Riemann-Liouville fractional integrals, see

[51]. Let [a, b] be a finite interval on real axis R. The Riemann-Liouville fractional

integrals Iαa+f and Iαb−f of order α > 0 are defined by

Iαa+f(x) =1

Γ(α)

x∫a

(x− y)α−1f(y)dy, x > a

and

Iαb−f(x) =1

Γ(α)

b∫x

(y − x)α−1f(y)dy, x < b.

Here Γ(α) is the Gamma function. These integrals are called the left-sided and right-

sided fractional integrals respectively.

Let us recall the definition, for details see [7, p. 448].

The generalized Riemann–Liouville fractional derivative of f of order α > 0 is given

by

Dαa f(x) =

1

Γ(n− α)

(d

dx

)n ∫ x

a

(x− y)n−α−1f(y)dy,

where n = [α] + 1, x ∈ [a, b].

For a, b ∈ R, we say that f ∈ L1(a, b) has an L∞ fractional derivative Dαa f (α > 0)

in [a, b], iff

1) Dα−ka f ∈ C[a, b], k = 1, . . . , n = [α] + 1

2) Dα−1a f ∈ AC[a, b],

3) Dαa f ∈ L∞(a, b).

The following lemma is given in [7, p.449] (see also [33]).

Lemma 1.2.1. Let β > α ≥ 0, let f ∈ L1(a, b) have an L∞ fractional derivative Dβaf

in [a, b] and let Dβ−ka f(a) = 0, k = 1, . . . , [β] + 1. Then

Dαa f(x) =

1

Γ(β − α)

x∫a

(x− y)β−α−1Dβaf(y)dy

for all a ≤ x ≤ b.

6

Let α > 0 and n = [α] + 1, where [·] is the integral part and we define the

generalized Riemann-Liouville fractional derivative of f of order α. In addition, we

stipulate

D0af := f =: I0

af, I−αa f := Dαa f if α > 0.

If α ∈ N, then Dαa f = dαf

dxα, the ordinary α-order derivative.

The space Iαa (L(a, b)) is defined as the set of all functions f on [a, b] of the form

f = Iαa ϕ for some ϕ ∈ L(a, b), [68, Chapter 1, Definition 2.3]. According to Theorem

2.3 in [68, p. 43], the latter characterization is equivalent to the condition

In−αa f ∈ ACn[a, b] , (1.2.1)

dj

dxjIn−αa f(a) = 0 , j = 0, 1, . . . , n− 1 .

A function f ∈ L(a, b) satisfying (1.2.1) is said to have an integrable fractional deriva-

tive Dαa f, [68, Chapter1, Definition 2.4].

The following lemma summarizes conditions in identity for generalized Riemann-

Liouville fractional derivative. For details see [9].

Lemma 1.2.2. Let β > α ≥ 0, n = [β] + 1, m = [α] + 1. Identity

Dαa f(x) =

1

Γ(β − α)

x∫a

(x− y)β−α−1Dβaf(y) dy , x ∈ [a, b],

is valid if one of the following conditions holds:

(i) f ∈ Iβa (L(a, b)).

(ii) In−βa f ∈ ACn[a, b] and Dβ−ka f(a) = 0 for k = 1, . . . n.

(iii) Dβ−ka f ∈ C[a, b] for k = 1, . . . , n, Dβ−1

a f ∈ AC[a, b] and Dβ−ka f(a) = 0 for

k = 1, . . . n.

(iv) f ∈ ACn[a, b], Dβaf ∈ L(a, b), Dα

a f ∈ L(a, b), β − α /∈ N, Dβ−ka f(a) = 0 for

k = 1, . . . , n and Dα−ka f(a) = 0 for k = 1, . . . ,m.

(v) f ∈ ACn[a, b], Dβaf ∈ L(a, b), Dα

a f ∈ L(a, b), β − α = l ∈ N, Dβ−ka f(a) = 0 for

k = 1, . . . , l.

(vi) f ∈ ACn[a, b], Dβaf ∈ L(a, b), Dα

a f ∈ L(a, b) and f(a) = f ′(a) = · · · =

f (n−2)(a) = 0.

7

(vii) f ∈ ACn[a, b], Dβaf ∈ L(a, b), Dα

a f ∈ L(a, b), β /∈ N and Dβ−1a f is bounded in

a neighborhood of t = a.

Next, we recall Canavati-type fractional derivative (ν−fractional derivative of f),

for details see [7, p. 446]. We consider

Cν [a, b] = f ∈ Cn[a, b] : In−ν+1a+ f (n) ∈ C1[a, b],

ν > 0, n = [ν]. Let f ∈ Cν [a, b]. We define the generalized ν−fractional derivative of

f over [a, b] as

Dνaf = (In−ν+1

a+ f (n))′,

the derivative with respect to x.

Lemma 1.2.3. Let ν ≥ γ+1, where γ ≥ 0 and f ∈ Cν [a, b]. Assume f (i)(a) = 0, i =

0, 1, . . . , [ν]− 1. Then

(Dγaf)(x) =

1

Γ(ν − γ)

x∫a

(x− t)ν−γ−1(Dνaf)(t)dt,

for all x ∈ [a, b].

The definition of Canavati-type fractional derivative is given in [7] but we will

consider the Canavati-type fractional derivative given in [10] with some new conditions

in our results. Now we recall the Canavati-type fractional derivative (ν−fractional

derivative of f). We consider

Cν [0, 1] = f ∈ Cn[0, 1] : I1−νf(n) ∈ C1[0, 1],

ν > 0, n = [ν], [.] is the integral part, and ν = ν − n, 0 ≤ ν < 1.

For f ∈ Cν [0, 1], the Canavati-ν fractional derivative of f is defined by

Dνf = DI1−νf(n),

where D = d/dx.

Lemma 1.2.4. Let ν > γ ≥ 0, n = [ν], m = [γ]. Let f ∈ Cν [0, 1], be such that

f (i)(0) = 0, i = m,m+ 1, . . . , n− 1. Then

(i) f ∈ Cγ[0, 1],

(ii) (Dγaf)(x) = 1

Γ(ν−γ)

x∫a

(x− t)ν−γ−1(Dνaf)(t)dt,

8

for every x ∈ [a, b].

Now we recall Caputo fractional derivative, for details see [7, p. 449].

Let ν ≥ 0, n = dνe, f ∈ ACn[a, b]. The Caputo fractional derivative is given by

Dν∗af(x) =

1

Γ(n− ν)

x∫a

f (n)(y)

(x− y)ν−n+1dy,

for all x ∈ [a, b]. The above function exists almost everywhere for x ∈ [a, b].

We continue with the following lemma that is given in [11].

Lemma 1.2.5. Let ν > γ ≥ 0, n = [ν] + 1, m = [γ] + 1 and f ∈ ACn[a, b]. Suppose

that one of the following conditions hold:

(a) ν, γ 6∈ N0 and f i(a) = 0 for i = m, . . . , n− 1.

(b) ν ∈ N0, γ 6∈ N0 and f i(a) = 0 for i = m, . . . , n− 2.

(c) ν 6∈ N0, γ ∈ N0 and f i(a) = 0 for i = m− 1, . . . , n− 1.

(d) ν ∈ N0, γ ∈ N0 and f i(a) = 0 for i = m− 1, . . . , n− 2.

Then

Dγ∗af(x) =

1

Γ(ν − γ)

x∫a

(x− y)ν−γ−1Dν∗af(y)dy


The following result is given in [7, p. 450].

Lemma 1.2.6. Let α ≥ γ + 1, γ > 0 and n = dαe. Assume f ∈ ACn[a, b] such that

f (k)(a) = 0, k = 0, 1, . . . , n− 1, and Dα∗af ∈ L∞(a, b). Then Dγ

∗af ∈ C[a, b], and

Dγ∗af(x) =

1

Γ(α− γ)

x∫a

(x− y)α−γ−1Dα∗af(y)dy


9

Now, we recall the definition of Hadamard-type fractional integrals.

Let [a, b] be finite or infinite interval of R+ and α > 0. The left and right-sided

Hadamard-type fractional integrals of order α > 0 are given by

(Jαa+f)(x) =

1

Γ(α)

x∫a

(log

x

y

)α−1

f(y)dy

y, x > a

and

(Jαb−f)(x) =1

Γ(α)

b∫x

(log

y

x

)α−1

f(y)dy

y, x < b

respectively.

We continue with definitions and some properties of the fractional integrals of a

function f with respect to given function g. For details see e.g. [51, p. 99]:

Let (a, b), −∞ ≤ a 0. Also let g be an increasing function on (a, b] and g′ be a continuous

function on (a, b). The left- and right-sided fractional integrals of a function f with

respect to another function g in [a, b] are given by

(Iαa+;gf)(x) =1

Γ(α)

x∫a

g′(t)f(t)dt

[g(x)− g(t)]1−α, x > a

and

(Iαb−;gf)(x) =1

Γ(α)

b∫x

g′(t)f(t)dt

[g(t)− g(x)]1−α, x < b

respectively.

Remark 1.2.1. If g(x) = x, then Iαa+;xf reduces to Iαa+f and Iαb−;xf reduces to Iαb−f ,

that is to Riemann–Liouville fractional integrals. Notice that Hadamard fractional

integrals of order α are special case of the left- and right-sided fractional integrals

of a function f with respect to another function g(x) = log(x), x ∈ (a, b) where

0 ≤ a < b ≤ ∞.

We also recall the definition of Erdelyi-Kober type fractional integrals. For details

see [68] (also see [27, p, 154]).

10

Let (a, b), (0 ≤ a 0, σ > 0,

and η ∈ R. The left and right-sided Erdelyi-Kober type fractional integral of order

α > 0 are defined by

(Iαa+;σ;ηf)(x) =σx−σ(α+η)

Γ(α)

x∫a

tση+σ−1f(t)dt

(xσ − tσ)1−α , x > a

and

(Iαb−;σ;ηf)(x) =σxση

Γ(α)

b∫x

tσ(1−η−α)−1f(t)dt

(tσ − xσ)1−α , x < b

respectively.

We recall multidimensional fractional integrals. Such type of fractional integrals

are usually generalization of the corresponding one-dimensional fractional integral

and fractional derivative.

For x = (x1, . . . , xn) ∈ Rn and α = (α1, . . . , αn), we use the following notations:

Γ(α) = (Γ(α1) . . .Γ(αn)), [a,b] = [a1, b1]× . . .× [an, bn],

and by x > a we mean x1 > a1, . . . , xn > an.

We define the mixed Riemann-Liouville fractional integrals of order α > 0 as

(Iαa+f)(x) =

1

Γ(α)

x1∫a1

. . .

xn∫an

f(t)(x− t)α−1dt, x > a

and

(Iαb−f)(x) =1

Γ(α)

b1∫x1

. . .

bn∫xn

f(t)(t− x)α−1dt, x < b.

Conventions. All measures are assumed to be positive, all functions are assumed

to be measurable, and expressions of the form 0 · ∞, 00, a∞ (a ∈ R), and ∞

∞ are taken

to be equal to zero. For a real parameter 0 6= p 6= 1, by p′ we denote its conjugate

exponent p′ = pp−1

, that is, 1p

+ 1p′

= 1. Also, by a weight function (shortly: a

weight) we mean a non-negative measurable function on the actual set. Further, we

set Nk = 1, 2, . . . , k for k ∈ N. An interval I in R is any convex subset of R, while

11

by Int I we denote its interior. By R+ we denote the set of all positive real numbers

i.e. R+ = (0,∞). B( · ; · , · ) denotes the incomplete Beta function, defined by

B(x; a, b) =

x∫0

ta−1(1− t)b−1 dt, x ∈ [0, 1], a, b > 0. (1.2.2)

As usual, B(a, b) = B(1; a, b) stands for the standard Beta function and 2F1(a, b; c; z)

denotes the Euler type of hypergeometric function, defined by

2F1(a, b; c; z) =1

B(b, c− b)

1∫0

tb−1(1− t)c−b−1(1− tz)−adt, Re(c) > Re(b) > 0

provided |z| < 1 or |z| = 1.

Throughout this thesis, we denote

2F1(x) = 2F1

(−η, α;α + 1; 1−

(ax

)σ),

and

2F1(y) = 2F1

(η, α;α + 1; 1−

(b

y

)σ).

Chapter 2

Inequality of G. H. Hardy and itsimprovements

In this chapter, we state, prove and discuss new general inequality for convex and

increasing functions. As a special case of that general result, we obtain new fractional

inequalities involving fractional integrals and derivatives of Riemann–Liouville type.

Consequently, we get the inequality of G. H. Hardy from 1918. We also obtain new

results involving fractional derivatives of Canavati and Caputo as well as fractional

integrals of a function with respect to another function. We apply our result to

multidimensional setting to obtain new results involving mixed Riemman-Liouville

fractional integrals. We use the convex and superquadratic functions to get new

improvements of inequalitity of G. H. Hardy. We also obtain means of Cauchy type

and prove their monotonicity. The results given in this chapter can be seen in [38]–

[43].

2.1 On an inequality of G. H. Hardy

First we denote some properties of the fractional integral operators Iαa+f and Iαb−f of

order α > 0, see also [68]. It is known that the fractional integral operators Iαa+f and

Iαb−f are bounded in Lp(a, b), 1 ≤ p ≤ ∞, that is

‖Iαa+f‖p ≤ K‖f‖p, ‖Iαb−f‖p ≤ K‖f‖p, (2.1.1)

where

K =(b− a)α

Γ(α + 1).

12

13

Inequality (2.1.1), that is the result involving the left-sided fractional integral, was

proved by G. H. Hardy in one of his initial papers, see [35]. He did not write down the

constant, but the calculation of the constant was hidden inside his proof. Inequality

(2.1.1) refer to an inequality of G. H. Hardy. Some recent results involving Riemann-

Liouville fractional integrals can be seen in [38].

Let (Ω1,Σ1, µ1) and (Ω2,Σ2, µ2) be measure spaces with positive σ-finite measures,

k : Ω1 × Ω2 → R be a non-negative function, and

K(x) =

∫Ω2

k(x, y) dµ2(y), x ∈ Ω1. (2.1.2)

Throughout this thesis we suppose K(x) > 0 a.e. on Ω1.

Let U(k) denote the class of measurable functions g : Ω1 → R with the represen-

tation

g(x) =

∫Ω2

k(x, y)f(y)dµ2(y),

where f : Ω2 → R is a measurable function.

2.1.1 Main results

Our first result is given in the following theorem.

Theorem 2.1.1. Let u be a weight function on Ω1, k a non-negative measurable

function on Ω1 × Ω2, and K be defined on Ω1 by (2.1.2). Assume that the function

x 7→ u(x)k(x,y)K(x)

is integrable on Ω1 for each fixed y ∈ Ω2. Define v on Ω2 by

v(y) :=

∫Ω1

u(x)k(x, y)

K(x)dµ1(x) <∞. (2.1.3)

If Φ : (0,∞)→ R is a convex and increasing function, then the inequality∫Ω1

u(x)Φ

(∣∣∣∣ g(x)

K(x)

∣∣∣∣) dµ1(x) ≤∫Ω2

v(y)Φ(|f(y)|)dµ2(y) (2.1.4)

holds for all measurable functions f : Ω2 → R and for all functions g ∈ U(k).

14

Proof. By using Jensen’s inequality and the Fubini theorem, since Φ is an increasing

function, we find that

∫Ω1

u(x)Φ

(∣∣∣∣ g(x)

K(x)

∣∣∣∣) dµ1(x) =

∫Ω1

u(x)Φ

∣∣∣∣∣∣ 1

K(x)

∫Ω2

k(x, y)f(y)dµ2(y)

∣∣∣∣∣∣ dµ1(x)

≤∫Ω1

u(x)

K(x)

∫Ω2

k(x, y)Φ(|f(y)|)dµ2(y)

dµ1(x)

=

∫Ω2

Φ(|f(y)|)

∫Ω1

u(x)k(x, y)

K(x)dµ1(x)

dµ2(y)

=

∫Ω2

v(y)Φ(|f(y)|)dµ2(y)

and the proof is complete.

As a special case of Theorem 2.1.1 we get the following result.

Corollary 2.1.2. Let u be a weight function on (a, b) and α > 0. Iαa+f denotes the

left-sided Riemann-Liouville fractional integral of f . Define v on (a, b) by

v(y) := α

b∫y

u(x)(x− y)α−1

(x− a)αdx <∞.

If Φ : (0,∞)→ R is a convex and increasing function, then the inequality

b∫a

u(x)Φ

(Γ(α + 1)

(x− a)α|Iαa+f(x)|

)dx ≤

b∫a

v(y)Φ(|f(y)|) dy (2.1.5)

holds.

Proof. Applying Theorem 2.1.1 with Ω1 = Ω2 = (a, b), dµ1(x) = dx, dµ2(y) = dy,

k(x, y) =

(x−y)α−1

Γ(α), a < y ≤ x ;

0, x < y ≤ b,(2.1.6)

we get that K(x) = (x−a)α

Γ(α+1)and g(x) = Iαa+f(x), so (2.1.5) follows.

15

Remark 2.1.1. In particular for the weight u(x) = (x − a)α, x ∈ (a, b) in Corollary

2.1.2, we obtain the inequality

b∫a

(x− a)αΦ

(Γ(α + 1)


)dx ≤

b∫a

(b− y)αΦ(|f(y)|) dy. (2.1.7)

Although (2.1.4) holds for all convex and increasing functions, some choices of Φ are

of particular interest. Namely, we shall consider power function. Let q > 1 and the

function Φ : R+ → R be defined by Φ(x) = xq, then (2.1.7) reduces to

b∫a

(x− a)α(

Γ(α + 1)


)qdx ≤

b∫a

(b− y)α|f(y)|q dy. (2.1.8)

Since x ∈ (a, b) and α(1− q) < 0, then we obtain that the left-hand side of (2.1.8)

b∫a

(x− a)α(

Γ(α + 1)


)qdx

≥ (b− a)α(1−q)(Γ(α + 1))qb∫

a

|Iαa+f(x)|q dx (2.1.9)

and the right-hand side of (2.1.8)

b∫a

(b− y)α|f(y)|q dy ≤ (b− a)αb∫

a

|f(y)|q dy. (2.1.10)

Combining (2.1.9) and (2.1.10) we get

b∫a

|Iαa+f(x)|q dx ≤(

(b− a)α

Γ(α + 1)

)q b∫a

|f(y)|q dy. (2.1.11)

Taking power 1q

on both sides we obtain (2.1.1).

We can give the similar results like Corollary 2.1.2 and Remark 2.1.1 for the

right-sided Riemann-Liouville fractional integral but here we omit the details.

17

Remark 2.1.2. For α ≥ 1 inequalities (2.1.12) and (2.1.13) are refinements of (2.1.1)

since

qα(p(α− 1) + 1)q−1 ≥ qαq > αq, so C <

((b− a)α

αΓ(α)

)q.

We proved that Theorem 2.1.3 is a refinement of (2.1.1) and Corollary 2.1.2 are

generalizations of (2.1.1).

Corollary 2.1.4. Let u be a weight function on (a, b) and let assumptions in Lemma

1.2.1 be satisfied. Define v on (a, b) by

v(y) := (β − α)

b∫y

u(x)(x− y)β−α−1

(x− a)β−αdx <∞.


b∫a

u(x)Φ

(Γ(β − α + 1)

(x− a)β−α|Dα

a f(x)|)dx ≤

b∫a

v(y)Φ(|Dβaf(y)|) dy (2.1.14)

holds.


k(x, y) =

(x−y)β−α−1

Γ(β−α), a < y ≤ x ;

0, x < y ≤ b,(2.1.15)

we get that K(x) = (x−a)β−α

Γ(β−α+1). Replace f by Dβ

af . Then, by Lemma 1.2.1 g(x) =

Dαa f(x) and we get (2.1.14).

Remark 2.1.3. In particular for the weight function u(x) = (x− a)β−α, x ∈ (a, b) and

Φ(x) = xq, q > 1, x ∈ R+ in Corollary 2.1.4 after short calculations we obtain the

following inequality

b∫a

|Dαa f(x)|q dx ≤

((b− a)(β−α)

Γ(β − α + 1)

)q b∫a

|Dβaf(y)|q dy.

Corollary 2.1.5. Let u be a weight function on (a, b) and let the assumptions in

Lemma 1.2.3 be satisfied. Define v on (a, b) by

v(y) := (ν − γ)

b∫y

u(x)(x− y)ν−γ−1

(x− a)ν−γdx <∞.

18


b∫a

u(x)Φ

(Γ(ν − γ + 1)

(x− a)ν−γ|Dγ

af(x)|)dx ≤

b∫a

v(y)Φ(|Dνaf(y)|) dy

holds.

Proof. Similar to Corollary 2.1.4.

Remark 2.1.4. In particular for the weight function u(x) = (x − a)ν−γ, x ∈ (a, b)

and Φ(x) = xq, q > 1, then after some calculations like Remark 2.1.1, we obtain the

inequality

‖(Dγaf(x)‖q ≤

(b− a)(ν−γ)

Γ(ν − γ + 1)‖Dν

af(y)‖q.

When γ = 0, we find

‖f‖q ≤(b− a)ν

Γ(ν + 1)‖Dν

af(y)‖q.

In the next corollary we give results for Caputo fractional derivative.

Corollary 2.1.6. Let u be a weight function on (a, b) and α > 0. Dα∗af denotes the

Caputo fractional derivative of f . Define v on (a, b) by

v(y) := (n− α)

b∫y

u(x)(x− y)n−α−1

(x− a)n−αdx <∞.


b∫a

u(x)Φ

(Γ(n− α + 1)

(x− a)n−α|Dα∗af(x)|

)dx ≤

b∫a

v(y)Φ(|f (n)(y)|) dy

holds.


Remark 2.1.5. For particular weight function u(x) = (x − a)n−α, x ∈ (a, b), and

Φ(x) = xq, q > 1 in Corollary 2.1.6, after some calculation like Remark 2.1.1, we

obtain the inequality

‖Dα∗af(x)‖q ≤

(b− a)(n−α)

Γ(n− α + 1)‖f (n)(y)‖q.

19


+ 1q

= 1, n − α > 1q, Dα

∗af(x) denotes the Caputo

fractional derivative of f . Then the following inequality

b∫a

|Dα∗af(x)|q dx ≤ (b− a)q(n−α)

(Γ(n− α))q(p(n− α− 1) + 1)qp q(n− α)

b∫a

|f (n)(y)|q dy

holds.

Proof. Similar to Theorem 2.1.3.

Corollary 2.1.8. Let u be a weight function on (a, b) and α > 0. Dα∗af denotes the

Caputo fractional derivative of f and let the assumptions in Lemma 1.2.6 be satisfied.

Define v on (a, b) by

v(y) := (α− γ)

b∫y

u(x)(x− y)α−γ−1

(x− a)α−γdx <∞.


b∫a

u(x)Φ

(Γ(α− γ + 1)

(x− a)α−γ|Dγ∗af(x)|

)dx ≤

b∫a

v(y)Φ(|Dα∗af(y)|) dy

holds.


Remark 2.1.6. In particular for the weight function u(x) = (x − a)α−γ, x ∈ (a, b)

and Φ(x) = xq, q > 1 in Corollary 2.1.8 after some calculations like Remark 2.1.1, we

obtain the following inequality

b∫a

|Dγ∗af(x)|q dx ≤

((b− a)(α−γ)

Γ(α− γ + 1)

)q b∫a

|Dα∗af(y)|q dy.

For γ = 0, we obtain

b∫a

|f(x)|q dx ≤(

(b− a)α

Γ(α + 1)

)q b∫a

|Dα∗af(y)|q dy.

20

Corollary 2.1.9. Let u be a weight function on (a, b), g be an increasing function

on (a, b] such that g′ be a continuous function on (a, b) and α > 0. Iαa+;gf denotes

the left-sided fractional integral of a function f with respect to another function g in

[a, b]. Define v on (a, b) by

v(y) := αg′(y)

b∫y

u(x)(g(x)− g(y))α−1

(g(x)− g(a))αdx <∞.


b∫a

u(x)Φ

(Γ(α + 1)

(g(x)− g(a))α|Iαa+;gf(x)|

)dx ≤

b∫a

v(y)Φ(|f(y)|) dy

holds.


Remark 2.1.7. In particular for the weight function u(x) = g′(x)(g(x) − g(a))α, x ∈(a, b) in Corollary 2.1.9, we obtain the inequality

b∫a

g′(x)(g(x)− g(a))αΦ

(Γ(α + 1)

(g(x)− g(a))α|Iαa+;gf(x)|

)dx

≤b∫

a

g′(y)(g(b)− g(y))αΦ(|f(y)|) dy. (2.1.16)

Let q > 1 and Φ : R+ → R be defined by Φ(x) = xq, then (2.1.16) reduces to

(Γ(α + 1))qb∫

a

g′(x)(g(x)− g(a))α(1−q)|Iαa+;gf(x)|q dx

≤b∫

a

g′(y)(g(b)− g(y))α|f(y)|q dy.

Since x ∈ (a, b) and α(1− q) < 0, g is increasing, then (g(x)− g(a))α(1−q) > (g(b)−g(a))α(1−q) and (g(b)− g(y))α < (g(b)− g(a))α, we obtain

b∫a

g′(x)|Iαa+;gf(x)|q dx ≤(

(g(b)− g(a))α

Γ(α + 1)

)q b∫a

g′(y)|f(y)|q dy. (2.1.17)

21

Remark 2.1.8. If g(x) = x, then Iαa+;xf(x) reduces to Iαa+f(x) Riemann–Liouville

fractional integral and (2.1.17) becomes (2.1.11).

Analogous to the Corollary 2.1.9, we obtain the similar result for right-sided frac-

tional integral of a function f with respect to another function g.

The refinements of (2.1.17) and its analogous case for left-sided fractional integral

of a function f with respect to another function g for α > 1q

are given in the following

theorem.


+ 1q

= 1, α > 1q, Iαa+;gf denotes the left-sided

fractional integral of a function f with respect to another function g in [a, b]. Then

the following inequality

b∫a

|Iαa+;gf(x)|qg′(x) dx ≤ (g(b)− g(a))αq

αq(Γ(α))q(p(α− 1) + 1)qp

b∫a

|f(y)|qg′(y) dy

hold.

If g(x) = log(x), x ∈ (a, b) where 0 ≤ a 1q

b∫a

|(Jαa+f)(x)|q dxx≤

(log ba)qα

qα(Γ(α))q(p(α− 1) + 1)qp

b∫a

|f(y)|q dyy.

Some results involving Hadamard-type fractional integrals are given in [51, p.

110]. Here we mention the following result that can not be compared with our result.

Let α > 0, 1 ≤ p ≤ ∞ and 0 ≤ a < b ≤ ∞. Then the operators Jαa+f and Jαb−f

are bounded in Lp(a, b) as follows:

‖Jαa+f‖p ≤ K1‖f‖p and ‖Jαb−f‖p ≤ K2‖f‖p,

where

K1 =1

Γ(α)

log(b/a)∫0

tα−1etpdt and K2 =

1

Γ(α)

log(b/a)∫0

tα−1e−tpdt.

22

Corollary 2.1.11. Let u be a weight function on (a, b), 2F1(a, b; c; z) denotes the hy-

pergeometric function and Iαa+;σ;ηf denotes the Erdelyi–Kober-type left-sided fractional

integral. Define v by

v(y) := ασyση+σ−1

b∫y

u(x)x−ση(xσ − yσ)α−1

(xσ − aσ)α 2F1(−η, α;α + 1; 1−(ax

)σ)dx <∞.


b∫a

u(x)Φ

(Γ(α + 1)(

1−(ax

)σ)α2F1(−η, α;α + 1; 1−

(ax

)σ)|Iαa+;σ;ηf(x)|

)dx

≤b∫

a

v(y)Φ(|f(y)|) dy

holds.


Remark 2.1.9. In particular for the weight function u(x) = xσ−1(xσ − aσ)α 2F1(x) in

Corollary 2.1.11 we obtain the inequality

b∫a

xσ−1(xσ − aσ)α 2F1(x)Φ

(Γ(α + 1)(

1−(ax

)σ)α2F1(x)

|Iαa+;σ;ηf(x)|

)dx

≤b∫

a

yσ−1(bσ − yσ)α 2F1(y)Φ(|f(y)|) dy.

In the previous corollaries we derived only inequalities over some subsets of R.

However, Theorem 2.1.1 covers much more general situations. We give result for

multidimensional fractional integrals. Such operations of fractional integration in

the n-dimensional Euclidean space Rn, (n ∈ N) are natural generalizations of the

corresponding one-dimensional fractional integrals and fractional derivatives, being

taken with respect to one or several variables.

Corollary 2.1.12. Let u be a weight function on (a,b) and α > 0. Iαa+f denotes the

mixed Riemann-Liouville fractional integral of f . Define v on (a,b) by

v(y) := α

b1∫y1

· · ·bn∫

yn

u(x)(x− y)α−1

(x− a)αdx <∞.

23


b1∫a1

· · ·bn∫

an

u(x)Φ

(Γ(α + 1)


)dx ≤

b1∫a1

· · ·bn∫

an

v(y)Φ(|f(y)|) dy

holds for all measurable functions f : (a,b)→ R.


Remark 2.1.10. Analogous to Remark 2.1.1 we obtain multidimensional version of

inequality (2.1.1) for q > 1:

b1∫a1

· · ·bn∫

an

|Iαa+f(x)|q dx ≤(

(b− a)α

Γ(α + 1)

)q b1∫a1

· · ·bn∫

an

|f(y)|q dy.

2.2 New inequalities involving fractional integrals

and fractional derivatives

If we substitute k(x, y) by k(x, y)f2(y) and f byf1

f2

, where fi : Ω2 → R, (i = 1, 2) are

measurable functions in Theorem 2.1.1 we obtain the following result.

Theorem 2.2.1. Let fi : Ω2 → R be measurable functions, gi ∈ U(k), where g2(x) >

0 for every x ∈ Ω1. Let u be a weight function on Ω1, k be a non-negative measurable

function on Ω1×Ω2. Assume that the function x 7→ u(x)f2(y)k(x,y)g2(x)

is integrable on Ω1

for each fixed y ∈ Ω2. Define v on Ω2 by

v(y) := f2(y)

∫Ω1

u(x)k(x, y)

g2(x)dµ1(x) <∞. (2.2.1)

If Φ : (0,∞)→ R is a convex and increasing function, then the inequality∫Ω1

u(x)Φ

(∣∣∣∣g1(x)

g2(x)

∣∣∣∣) dµ1(x) ≤∫Ω2

v(y)Φ

(∣∣∣∣f1(y)

f2(y)

∣∣∣∣) dµ2(y)

holds.

Remark 2.2.1. If Φ is strictly convex and f1(x)f2(x)

is non-constant, then in Theorem 2.2.1

the inequality is strict.

24

Remark 2.2.2. As a special case of Theorem 2.2.1 for Ω1 = Ω2 = [a, b] and dµ1(x) =

dx, dµ2(y) = dy we obtain the result in [55] (see also [65, p. 236]).

As a special case of Theorem 2.2.1 we obtain the following result.



v(y) =Dνaf2(y)

Γ(ν − γ)

b∫y


Dγaf2(x)

dx <∞.

If Φ : (0,∞)→ R is a convex and increasing, then the inequality

b∫a

u(x)Φ

(∣∣∣∣Dγaf1(x)

Dγaf2(x)

∣∣∣∣) dx ≤b∫

a

v(y)Φ

(∣∣∣∣Dνaf1(y)

Dνaf2(y)

∣∣∣∣) dy (2.2.2)

holds.


k(x, y) =

(x−y)ν−γ−1

Γ(ν−γ), a < y ≤ x;

0, x < y ≤ b,(2.2.3)

and replacing fi by Dνafi, i = 1, 2 we obtain (2.2.2).



v(y) =Dβaf2(y)

Γ(β − α)

b∫y

u(x)(x− y)β−α−1

Dαa f2(x)

dx <∞.


b∫a

u(x)Φ

(∣∣∣∣Dαa f1(x)

Dαa f2(x)

∣∣∣∣) dx ≤b∫

a

v(y)Φ

(∣∣∣∣Dβaf1(y)

Dβaf2(y)

∣∣∣∣) dyholds.

Proof. Similar to the Corollary 2.2.2.

25

Corollary 2.2.4. Let u be a weight function on (a, b) and α ≥ 0. Dα∗af denotes the

Caputo fractional derivative of f . Define v on (a, b) by

v(y) =f

(n)2 (y)

Γ(n− α)

b∫y

u(x)(x− y)n−α−1

Dα∗af2(x)

dx <∞.


b∫a

u(x)Φ

(∣∣∣∣Dα∗af1(x)

Dα∗af2(x)

∣∣∣∣) dx ≤b∫

a

v(y)Φ

(∣∣∣∣∣f (n)1 (y)

f(n)2 (y)

∣∣∣∣∣)dy

holds.




v(y) =Dν∗af2(y)

Γ(ν − γ)

b∫y


Dγ∗af2(x)

dx <∞.


b∫a

u(x)Φ

(∣∣∣∣Dγ∗af1(x)

Dγ∗af2(x)

∣∣∣∣) dx ≤b∫

a

v(y)Φ

(∣∣∣∣Dν∗af1(y)

Dν∗af2(y)



Now we will show some new inequalities for fractional integrals.

Corollary 2.2.6. Let u be a weight function on (a, b) and α > 0. Iαa+f denotes the

left-sided Riemann-Liouville fractional integral of f . Define v on (a, b) by

v(y) =f2(y)

Γ(α)

b∫y

u(x)(x− y)α−1

Iαa+f2(x)dx <∞.

26


b∫a

u(x)Φ

(∣∣∣∣Iαa+f1(x)

Iαa+f2(x)

∣∣∣∣) dx ≤b∫

a

v(y)Φ

(∣∣∣∣f1(y)

f2(y)


Proof. Similar to the Corollary 2.2.2.

Corollary 2.2.7. Let u be a weight function, α > 0 and Jαa+f denotes the left-sided

Hadamard-type fractional integral. Define v on (a, b) by

v(y) =f2(y)

yΓ(α)

b∫y

u(x)

(log

x

y

)α−11

(Jαa+f2)(x)dx <∞.


b∫a

u(x)Φ

(∣∣∣∣∣Jαa+f1(x)

Jαa+f2(x)

∣∣∣∣∣)dx ≤

b∫a

v(y)Φ

(∣∣∣∣f1(y)

f2(y)



Corollary 2.2.8. Let u be a weight function, Iαa+;σ;ηf denotes the left-sided Erdelyi-

Kober-type fractional integral of function f of order α > 0. Define v on (a, b) by

v(y) =σf2(y)

Γ(α)

b∫y

u(x)x−σ(α+η)yση+σ−1

(xσ − yσ)1−α(Iαa+;σ;ηf2)(x)dx <∞.


b∫a

u(x)Φ

(∣∣∣∣Iαa+;σ;ηf1(x)

Iαa+;σ;ηf2(x)

∣∣∣∣) dx ≤b∫

a

v(y)Φ

(∣∣∣∣f1(y)

f2(y)

27

Corollary 2.2.9. Let u be a weight function on (a,b) and α > 0. Iαa+f denotes the

mixed Riemann-Liouville fractional integral of f . Define v on (a,b) by

v(y) :=f2(y)

Γ(α)

b1∫y1

· · ·bn∫

yn

u(x)(x− y)α−1

(Iαa+f2)(x)dx <∞.


b1∫a1

...

b1∫a1

u(x)Φ

(∣∣∣∣Iαa+f1(x)

Iαa+f2(x)

∣∣∣∣) dx ≤b1∫

a1

...

b1∫a1

v(y)Φ

(∣∣∣∣f1(y)

f2(y)



Note that Theorem 2.2.1 can be generalized for convex functions of several vari-

ables.

Theorem 2.2.10. Let gi ∈ U(k), (i = 1, 2, 3), where g2(x) > 0 for every x ∈ Ω1. Let

u be a weight function on Ω1, k be a non-negative measurable function on Ω1 × Ω2.

Let v be defined by (2.2.1). If Φ : (0,∞) × (0,∞) → R is a convex and increasing

function, then the inequality∫Ω1

u(x)Φ

(∣∣∣∣g1(x)

g2(x)

∣∣∣∣ , ∣∣∣∣g3(x)

g2(x)

∣∣∣∣) dµ1(x) ≤∫Ω2

v(y)Φ

(∣∣∣∣f1(y)

f2(y)

∣∣∣∣ , ∣∣∣∣f3(y)

f2(y)

∣∣∣∣) dµ2(y) (2.2.4)

holds.

Remark 2.2.3. Apply Theorem 2.2.10 with Ω1 = Ω2 = [a, b] and dµ1(x) = dx, dµ2(y) =

dy. Then

v(y) = f2(y)

b∫a

u(x)k(x, y)

g2(x)dx

and (2.2.4) reduces to

b∫a

u(x)Φ

(∣∣∣∣g1(x)

g2(x)

∣∣∣∣ , ∣∣∣∣g3(x)

g2(x)

∣∣∣∣) dx ≤b∫

a

v(y)Φ

(∣∣∣∣f1(y)

f2(y)

∣∣∣∣ , ∣∣∣∣f3(y)

f2(y)

∣∣∣∣) dy.This result is given in [55] (see also [65, p. 236]).

28

2.3 Improvement of an an inequality of G. H. Hardy

and Cauchy means

In this section, we give the improvement of an inequality of G. H. Hardy using frac-

tional integrals and fractional derivatives. We obtain means of Cauchy type and

prove their monotonicity. Also, using the concept of exponential convexity and log-

convexity we establish some new inequalities.

Let (Ω1,Σ1, µ1) and (Ω2,Σ2, µ2) be measure spaces with σ-finite measures and Akbe an integral operator defined by

Akf(x) :=1

K(x)

∫Ω2

k(x, y)f(y)dµ2(y), (2.3.1)

where k : Ω1 ×Ω2 → R is measurable and non-negative kernel, f is measurable func-

tion on Ω2 and K be defined by (2.1.2).

The following theorem is give in [52](see also [28]).

Theorem 2.3.1. Let (Ω1,Σ1, µ1) and (Ω2,Σ2, µ2) be measure spaces with σ-finite

measures, u be a weight function on Ω1, k be a non-negative measurable function on

Ω1×Ω2 and K be defined on Ω1 by (2.1.2). Suppose that the function x 7→ u(x)k(x,y)K(x)

is integrable on Ω1 for each fixed y ∈ Ω2 and that v is defined on Ω2 by (2.1.3). If Φ

is a convex function on the interval I ⊆ R, then the inequality∫Ω1

u(x)Φ (Akf(x)) dµ1(x) ≤∫Ω2

v(y)Φ (f(y)) dµ2(y) (2.3.2)

holds for all measurable functions f : Ω2 → R, such that Imf ⊆ I, where Ak is

defined by (2.3.1).

2.3.1 Main results

Lemma 2.3.2. Let s ∈ R, ϕs : R+ → R be a function defined as

ϕs(x) :=

xs

s(s−1), s 6= 0, 1,

− log x, s = 0,

x log x, s = 1.

(2.3.3)

Then ϕs is strictly convex on R+ for each s ∈ R.

29

The following theorem is given in [28].

Theorem 2.3.3. Let the assumption in the Theorem 2.3.1 be satisfied and ϕs be

defined by (2.3.3). Let f be a positive function. Then the function ξ : R→ [0,∞) is

defined by

ξ(s) =

∫Ω2

v(y)ϕs(f(y))dµ2(y)−∫Ω1

u(x)ϕs(Akf(x))dµ1(x) (2.3.4)

is exponentially convex.

The function ξ being exponentially convex is also log-convex function. Then by

Remark 1.1.1(c) and Remark 1.1.1 the following inequality holds true:

[ξ(p)]q−r ≤ [ξ(q)]p−r[ξ(r)]q−p (2.3.5)

for every choice r, p, q ∈ R, such that r 1, α > 0, g be increasing function on (a, b] such that g′ be

continuous on (a, b) and ξ1 : R→ [0,∞). Then the following inequality holds:

ξ1(s) ≤ H1(s),

where

ξ1(s) =1

s(s− 1)

b∫a

g′(y)(g(b)− g(y))αf s(y)dy

−(Γ(α + 1))sb∫

a

g′(x)(g(x)− g(a))α(1−s) (Iαa+;gf(x))sdx

,and

H1(s) =(g(b)− g(a))α(1−s)

s(s− 1)

(g(b)− g(a))αsb∫

a

f s(y)g′(y)dy

− (Γ(α + 1))sb∫

a

(Iαa+;gf(x))sg′(x)dx

.

30


k(x, y) =

g′(y)

Γ(α)(g(x)−g(y))1−α , a < y ≤ x;

0, x < y ≤ b,(2.3.6)

we get that K(x) = 1Γ(α+1)

(g(x)− g(a))α, then (2.3.4) becomes

ξ1(s) =

b∫a

v(y)Φs(f(y))dy −b∫

a

u(x)Φs

(Γ(α + 1)

(g(x)− g(a))αIαa+;gf(x)

)dx.

For the particular weight function u(x) = g′(x)(g(x) − g(a))α, we obtain v(y) =

g′(y)(g(b)− g(y))α and take Φs(x) = xs

s(s−1), x ∈ R+, we get

ξ1(s) =1

s(s− 1)

b∫a


−b∫

a

g′(x)(g(x)− g(a))α(

Γ(α + 1)


)sdx

.Since x ∈ (a, b) and α(1− s) < 0, g is increasing, then (g(x)− g(a))α(1−s) > (g(b)−g(a))α(1−s) and (g(b)− g(y))α < (g(b)− g(a))α, we obtain

ξ1(s) ≤ (g(b)− g(a))α(1−s)

s(s− 1)


a

f s(y)g′(y)dy

− (Γ(α + 1))sb∫

a


= H1(s).

This complete the proof.

If g(x) = x, then Iαa+;xf(x) reduces to Iαa+f(x) left-sided Riemann–Liouville frac-

tional integral and next result follows.

Corollary 2.3.5. Let s > 1, α > 0 and ξ2 : R→ [0,∞). Then the following inequality

holds true:

ξ2(s) ≤ H2(s),

31

where

ξ2(s) =1

s(s− 1)

b∫a

(b− y)αf s(y)dy − (Γ(α + 1))sb∫

a

(x− a)α(1−s) (Iαa+f(x))s dx

,and

H2(s) =(b− a)α(1−s)

s(s− 1)

(b− a)αsb∫

a

f s(y)dy − (Γ(α + 1))sb∫

a

(Iαa+f(x))sdx

.Notice that Hadamard fractional integrals of order α are special case of the left-

and right-sided fractional integrals of a function f with respect to another function

g(x) = log(x), x ∈ (a, b) where 0 ≤ a 1, α > 0 and ξ3 : R → [0,∞). Then the following

inequality holds:

ξ3(s) ≤ H3(s),

where

ξ3(s) =1

s(s− 1)

b∫a

(log b− log y)αf s(y)dy

y

−(Γ(α + 1))sb∫

a

(log x− log a)α(1−s) (Jαa+f(x)

)s dxx

,and

H3(s) =(log b− log a)α(1−s)

s(s− 1)

(log b− log a)αsb∫

a

f s(y)dy

y

− (Γ(α + 1))sb∫

a

(Jαa+f(x))s

dx

x

.Next we give result with respect to the generalized Riemann-Liouville fractional

derivative for details see [7, p. 448].

32

Theorem 2.3.7. Let s > 1, β > α ≥ 0, let f ∈ L1(a, b) have an L∞ fractional

derivative Dβaf in [a, b] and ξ4 : R→ [0,∞). Then the following inequality holds true:

ξ4(s) ≤ H4(s),

where

ξ4(s) =1

s(s− 1)

b∫a

(b− y)β−α(Dβaf(y))sdy

−(Γ(β − α + 1))sb∫

a

(x− a)(β−α)(1−s) (Dαa f(x))s dx

,and

H4(s) =(b− a)(β−α)(1−s)

s(s− 1)

(b− a)(β−α)s

b∫a

(Dβaf(y))sdy

− (Γ(β − α + 1))sb∫

a

(Dαa f(x))sdx

.Proof. Applying Theorem 2.3.3 with Ω1 = Ω2 = (a, b), dµ1(x) = dx, dµ2(y) = dy and

k(x, y) is given in (2.3.6). Replace f by Dβaf, then (2.3.4) becomes

ξ4(s) =

b∫a

v(y)Φs(Dβaf(y))dy −

b∫a

u(x)Φs

(Γ(β − α + 1)

(x− a)β−αDαa f(x)

)dx. (2.3.7)

For the particular weight function u(x) = (x − a)β−α, we get v(y) = (b − y)β−α and

we choose Φs(x) = xs

s(s−1), x ∈ R+, then (2.3.7) becomes

ξ4(s) =1

s(s− 1)

b∫a


−b∫

a

(x− a)β−α(

Γ(β − α + 1)

(x− a)β−α(Dα

a f(x))

)sdx

33

≤ 1

s(s− 1)

(b− a)β−αb∫

a

(Dβaf(y))sdy

−(b− a)(β−α)(1−s)(Γ(β − α + 1))sb∫

a

(Dαa f(x))sdx

= H4(s).

In the following theorem, we will construct new inequality for the Canavati-type

fractional derivative.

Theorem 2.3.8. Let s > 1, let the assumptions in Lemma 1.2.4 be satisfied and

ξ5 : R→ [0,∞). Then the following inequality holds:

ξ5(s) ≤ H5(s),

where

ξ5(s) =1

s(s− 1)

b∫a

(b− y)ν−γ(Dνaf(y))sdy

−(Γ(ν − γ + 1))sb∫

a

(x− a)(ν−γ)(1−s) (Dγaf(x))s dx

,and

H5(s) =(b− a)(ν−γ)(1−s)

s(s− 1)

(b− a)(ν−γ)s

b∫a

(Dνaf(y))sdy

−(Γ(ν − γ + 1))sb∫

a

(Dγaf(x))sdx

.Proof. Similar to Theorem 2.3.7.

As a special case of Theorem 2.3.3 to construct new inequality for the Caputo

fractional derivative in the upcoming theorem.

34

Theorem 2.3.9. Let s > 1, α ≥ 0, n = dαe, f ∈ ACn[a, b] and ξ6 : R → [0,∞).

Then the following inequality holds true:

ξ6(s) ≤ H6(s),

where

ξ6(s) =1

s(s− 1)

b∫a

(b− y)n−α(f (n)(y))sdy

−(Γ(n− α + 1))sb∫

a

(x− a)(n−α)(1−s) (Dα∗af(x))s dx

,and

H6(s) =(b− a)(n−α)(1−s)

s(s− 1)

(b− a)(n−α)s

b∫a

(f (n)(y))sdy

− (Γ(n− α + 1))sb∫

a

(Dα∗af(x))sdx


Theorem 2.3.10. Let s > 1, let the assumptions in Lemma 1.2.6 be satisfied and


ξ7(s) ≤ H7(s),

where

ξ7(s) =1

s(s− 1)

b∫a

(b− y)α−γ(Dα∗af(y))sdy

−(Γ(α− γ + 1))sb∫

a

(x− a)(α−γ)(1−s) (Dγ∗af(x))s dx

,

35

and

H7(s) =(b− a)(α−γ)(1−s)

s(s− 1)

(b− a)(α−γ)s

b∫a

(Dα∗af(y))sdy

− (Γ(α− γ + 1))sb∫

a

(Dγ∗af(x))sdx


Now, we give the following result for Erdelyi–Kober type fractional integral of f.

Theorem 2.3.11. Let s > 1, 2F1(a, b; c; z) denotes the hypergeometric function and


ξ8(s) ≤ H8(s),

where

ξ8(s) =1

s(s− 1)

b∫a

yσ−1(bσ − yσ)α 2F1(y)f s(y)dy

−b∫

a

xσ−1(xσ − aσ)α 2F1(x)

(Γ(α + 1)(

1−(ax

)σ)α2F1(x)

Iαa+;σ;ηf(x)

)s

dx

,and

H8(s) =(bσ − aσ)α(1−s)

s(s− 1)

(bσ − aσ)αsbσ−1

b∫a

2F1(y)f s(y)dy

− aσ−1+ασs(Γ(α + 1))sb∫

a

( (2F1(x))1−sIαa+;σ;ηf(x))sdx

.Proof. Applying Theorem 2.3.3 with Ω1 = Ω2 = (a, b), dµ1(x) = dx, dµ2(y) = dy,

k(x, y) =

1Γ(α)

σx−σ(α+η)

(xσ−yσ)1−αyση+σ−1, a < y ≤ x ;

0, x < y ≤ b,(2.3.8)

36

we get that K(x) = 1Γ(α+1)

(1−

(ax

)σ)α2F1(−η, α;α+1; 1−

(ax

)σ) and (2.3.4) becomes

ξ8(s) =

b∫a

v(y)Φs(f(y))dy−b∫

a

u(x)Φs

(Γ(α + 1)(

1−(ax

)σ)α2F1(x)

Iαa+;σ;ηf(x)

)dx. (2.3.9)

For the particular weight function u(x) = xσ−1(xσ − aσ)α 2F1(x), we obtain v(y) =

yσ−1(bσ − yσ)α 2F1(y), and if we choose Φs(x) = xs

s(s−1), x ∈ R+, then (2.3.9) becomes

ξ8(s) =1

s(s− 1)

b∫a


−b∫

a


(Γ(α + 1)(

1−(ax

)σ)α2F1(x)

Iαa+;σ;ηf(x)

)s

dx

≤ 1

s(s− 1)

bσ−1(bσ − aσ)αb∫

a

2F1(y)f s(y)dy

−aσ−1+ασs(bσ − aσ)α(1−s)(Γ(α + 1))sb∫

a

(2F1(x))1−s(Iαa+;σ;ηf(x))sdx

= H8(s).


In the following theorem we prove the three different cases for the above results.

Theorem 2.3.12. For i = 1, . . . , 8 the following inequalities hold true:

(i). [ξi(p)]q−rq−p [ξi(q)]

r−pq−p ≤ Hi(r) (2.3.10)

(ii). [ξi(r)]p−qp−r [ξi(p)]

q−rp−r ≤ Hi(q) (2.3.11)

(iii). ξi(p) ≤ [Hi(r)]q−pq−r [Hi(q)]

p−rq−r (2.3.12)

for every choice p, q, r ∈ R, such that 1 < r < p < q.

Proof. We will prove this theorem just in case i = 1, since all other case are proved

analogous.

37

(i). Since the function ξ1 is exponentially convex, it is also log-convex, then for

1 < r < p < q, r, p, q ∈ R, (2.3.5) can be written as:

[ξ1(p)]q−r[ξ1(q)]r−p ≤ [ξ1(r)]q−p,

this implies that

[ξ1(p)]q−rq−p [ξ1(q)]

r−pq−p ≤ (g(b)− g(a))α(1−r)

r(r − 1)

(g(b)− g(a))αrb∫

a

f r(y)g′(y)dy

−(Γ(α + 1))rb∫

a

(Iαa+f(x))rg′(x)dx

= H1(r).

It follows (2.3.10).

(ii). Now (2.3.5) can be written as:

[ξ1(r)]p−q[ξ1(p)]q−r ≤ [ξ1(q)]p−r,

this implies that

[ξ1(r)]p−qp−r [ξ1(p)]

q−rp−r ≤ (g(b)− g(a))α(1−q)

q(q − 1)

(g(b)− g(a))αqb∫

a

f q(y)g′(y)dy

−(Γ(α + 1))qb∫

a

(Iαa+f(x))qg′(x)dx

= H1(q).


(iii). The (2.3.5) can be written as:

[ξ1(p)]q−rp−r ≤ [ξ1(r)]

q−pp−r ξ1(q),

[ξ1(p)]q−rp−r ≤ [ξ1(r)]

q−pp−rH1(q),

this implies that

ξ1(p) ≤ [H1(r)]q−pq−r [H1(q)]

p−rq−r .

38

2.3.2 Mean value theorems and Cauchy means

Now we will give the mean value theorems and means of Cauchy type for different

fractional integrals and fractional derivatives. For this purpose we define a notation

ξi(s) := ξi(v,Φs(f(y)); u,Φs(Akf(x)), for (i = 1, . . . , 8) (2.3.13)

where Akf and v are defined by (2.3.1) and (2.1.3) respectively.

The following theorems are given in [28]. Such type of results are also given in

[67] and [70].

Theorem 2.3.13. Let (Ω1,Σ1, µ1), (Ω2,Σ2, µ2) be measure spaces with σ-finite mea-

sures and u : Ω1 → R be a weight function. Let I be compact interval of R, h ∈ C2(I),

and f : Ω2 → R be a measurable function such that Imf ⊆ I. Then there exists η ∈ Isuch that ∫

Ω2

v(y)h(f(y))dµ2(y)−∫Ω1

u(x)h(Akf(x))dµ1(x)

=h′′(η)

2

∫Ω2

v(y)f 2(y)dµ2(y)−∫Ω1

u(x)(Akf(x))2dµ1(x)

,where Akf and v are defined by (2.3.1) and (2.1.3) respectively.

Theorem 2.3.14. Assume that all conditions of Theorem 2.3.13 are satisfied. Let I

be a compact interval in R and g, h ∈ C2(I) such that h′′(x) 6= 0 for every x ∈ I. Let

f : Ω2 → R be a measurable function such that Imf ⊆ I and∫Ω2


u(x)h(Akf(x))dµ1(x) 6= 0.

Then there exists η ∈ I such that it holds

g′′(η)

h′′(η)=

∫Ω2

v(y)g(f(y))dµ2(y)−∫Ω1

u(x)g(Akf(x))dµ1(x)∫Ω2


u(x)h(Akf(x))dµ1(x).

We will give some special cases of Theorems 2.3.13 and 2.3.14 for different frac-

tional integrals and fractional derivatives in upcoming results.

39

Theorem 2.3.15. Let u be weight functions on (a, b), Akf(x) be defined by (2.3.1)

and v be defined by (2.1.3). Let I be a compact interval of R, h ∈ C2(I) and ξi : R→[0,∞). Then there exists ηi ∈ I such that

ξi(v, h(f(y)); u, h(Akf(x))) =h′′(ηi)

2ξi(v, (f(y))2; u, (Akf(x))2), (i = 1, . . . , 8).

Theorem 2.3.16. Let u be weight functions on (a, b), Akf(x) be defined by (2.3.1)

and v be defined by (2.1.3). Let I be a compact interval of R, g, h ∈ C2(I) such that

h′′(x) 6= 0 for every x ∈ I, ξi : R→ [0,∞) and

ξi(v, h(f(y)); u, h(Akf(x))) 6= 0.

Then there exists ηi ∈ I such that

g′′(ηi)

h′′(ηi)=ξi(v, g(f(y)); u, g(Akf(x)))

ξi(v, h(f(y)); u, h(Akf(x))), (i = 1, . . . , 8).

If we apply Theorem 2.3.16 with g(x) = xp

p(p−1), h(x) = xs

s(s−1), p 6= s, p, s 6= 0, 1,

we get the following result.

Corollary 2.3.17. Let u be weight functions on (a, b), Akf(x) be defined by (2.3.1)

and v be defined by (2.1.3). Let I be a compact interval of R+, ξi : R→ [0,∞), (i =

1, . . . , 8), then for p 6= s, p, s 6= 1, there exist ηi ∈ I such that

ηp−si =ξi(p)

ξi(s)=s(s− 1)

p(p− 1)

ξi(v, fp(y); u, (Akf(x))p)

ξi(v, f s(y); u, (Akf(x))s). (2.3.14)

Remark 2.3.1. Since g′′(x) = xp−2 and h

′′(x) = xs−2, g

′′

h′′are invertible. Then from

(2.3.14), we obtain

inft∈[a,b]

f(t) ≤(ξi(p)

ξi(s)

) 1p−s

≤ supt∈[a,b]

f(t).

So,

Mp,si (v, ϕs(f(y)); u, ϕs(Akf(x))) =

(ξi(p)

ξi(s)

) 1p−s

and for simplicity we denote

Mp,si := Mp,s

i (v, ϕs(f(y)); u, ϕs(Akf(x))

40

p 6= s, p, s 6= 0, 1 are means. Moreover we can extend these means to excluded cases.

Taking a limit we can define

Mp,si =

(ξi(v,ϕp(f(y));u,ϕp(Akf(x)))

ξi(v,ϕs(f(y));u,ϕs(Akf(x)))

) 1p−s

, p 6= s,

exp(

1−2ss(s−1)

− ξi(v,ϕs(f(y))ϕ0(f(y));u,ϕs(Akf(x))ϕ0(Akf(x)))ξi(v,ϕs(f(y));u,ϕs(Akf(x)))

), p = s 6= 0, 1,

exp(−ξi(v,ϕ1(f(y))(ϕ0(f(y)+2)));u,ϕ1(Akf(x))(ϕ0(Akf(x)+2))

2ξi(v,f(y)+ϕ1(f(y));u,Akf(x)+ϕ1(Akf(x)))

), p = s = 1,

exp(ξi(v,(2ϕ0(f(y))−ϕ2

0(f(y)));u, (2ϕ0(Akf(x))−ϕ20(Akf(x))))

2ξi(v,1+ϕ0f(y);u,1+ϕ0(Akf(x)))

), p = s = 0.

In the following theorem, we prove the monotonicity of means.

Theorem 2.3.18. Let r ≤ s, l ≤ p, then the following inequality is valid,

M l,ri ≤Mp,s

i , (i = 1, . . . , 8), (2.3.15)

that is, the means Mp,si are monotonic.

Proof. Since ξi are exponentially convex is also log-convex, we can apply Remark

1.1.1 (Galvani’s theorem) and get (2.3.15). For r = s, l = p, we get the result by

taking limit in (2.3.15).

2.4 Further results on an inequality of G. H. Hardy

In [52], K. Krulic et. al. gave the new generalization of the Theorem 2.3.1 and they

prove the following theorem.



Ω1×Ω2 and K be defined on Ω1 by (2.1.2). Let 0 < p ≤ q <∞ and that the function

x 7→ u(x)(k(x,y)K(x)

) qp

is integrable on Ω1 for each fixed y ∈ Ω2 and that v is defined on

Ω2 by

v(y) :=

∫Ω1

u(x)

(k(x, y)

K(x)

) qp

dµ1(x)

pq

<∞. (2.4.1)

41

If Φ is a non-negative convex function on the interval I ⊆ R, then the inequality ∫Ω1

u(x)[Φ (Akf(x))]qpdµ1(x)

1q

≤

∫Ω2

v(y)Φ (f(y)) dµ2(y)

1p

, (2.4.2)


defined by (2.3.1).

Using Theorem 2.4.1, we will give some special cases for different fractional inte-

grals and fractional derivatives to establish new Hardy-type inequalities as a gener-

alization of the results given in [40].

Our next result involving fractional integral of f with respect to another increasing

function g is given in the following theorem.

Theorem 2.4.2. Let 0 < p ≤ q <∞, u be a weight function on (a, b), g be increasing

function on (a, b] such that g′ be continuous on (a, b). Let v be defined on (a, b) by

v(y) := αg′(y)

b∫y

u(x)

((g(x)− g(y))α−1

(g(x)− g(a))α

) qp

dx

pq

<∞. (2.4.3)

If Φ is a non-negative convex function on the interval I ⊆ R, then the inequality b∫a

u(x)

[Φ

(Γ(α + 1)


)] qp

dx

1q

≤

b∫a

v(y)Φ (f(y)) dy

1p

(2.4.4)

holds for all measurable functions f : (a, b)→ R, such that Imf ⊆ I.


and k(x, y) is given in (2.3.6). Then Akf(x) = Γ(α+1)(g(x)−g(a))α

Iαa+;gf(x) and the inequality

in (2.4.2) reduces to (2.4.4) with v defined by (2.4.3).

Corollary 2.4.3. Let 0 1− pq, g be increasing function on

(a, b] such that g′ be continuous on (a, b). Then the inequality b∫a

g′(x)(Iαa+;gf(x))sqp dx

1q

≤ α1p (g(b)− g(a))

q(αs−1)+ppq

((α− 1) qp

+ 1)1q (Γ(α + 1))

sp

b∫a

g′(y)f s(y)dy

1p

(2.4.5)

holds.

42

Proof. For particular convex function Φ : R+ → R+, Φ(x) = xs, s ≥ 1 and weight

function u(x) = g′(x)(g(x)−g(a))αqp , x ∈ (a, b) in (2.4.4), we get v(y) = (αg′(y)(g(b)−

g(y))α−1+ pq )/(((α− 1) q

p+ 1)

pq ) and (2.4.4) becomes

b∫a

g′(x)(g(x)− g(a))αqp

(1−s)(Iαa+;gf(x))sqp dx

1q

≤ α1p

((α− 1) qp

+ 1)1q (Γ(α + 1))

sp

b∫a

g′(y)(g(b)− g(y))α−1+ pq f s(y)dy

1p

.

Since (g(x) − g(a))αqp

(1−s) ≥ (g(b) − g(a))αqp

(1−s) and (g(b) − g(y))α−1+ pq ≤ (g(b) −

g(a))α−1+ pq , α > 1− p

qwe obtain (2.4.5).

If g(x) = x, then Iαa+;xf(x) reduces to Iαa+f(x) left-sided Riemann-Liouville frac-

tional integral, so the following result follows.

Corollary 2.4.4. Let 0 < p ≤ q < ∞, u be a weight function on (a, b) and let v be

defined on (a, b) by

v(y) := α

b∫y

u(x)

((x− y)α−1

(x− a)α

) qp

dx

pq

<∞.


u(x)

[Φ

(Γ(α + 1)

(x− a)αIαa+f(x)

)] qp

dx

1q

≤

b∫a

v(y)Φ (f(y)) dy

1p


Corollary 2.4.5. Let 0 1− pq. Then the inequality

b∫a

(Iαa+f(x))sqp dx

1q

≤ α1p (b− a)

q(αs−1)+ppq

((α− 1) qp

+ 1)1q (Γ(α + 1))

sp

b∫a

f s(y)dy

1p

holds.

43

If g(x) = log(x), x ∈ (a, b) where 0 ≤ a 1 − pq. Then the following

inequality holds b∫a

(Jαa+f(x))sqpdx

x

1q

≤ α1p (log b− log a)

q(αs−1)+ppq

((α− 1) qp

+ 1)1q (Γ(α + 1))

sp

b∫a

f s(y)dy

y

1p

.

Next we give result with respect to the generalized Riemann–Liouville fractional

derivative.

Theorem 2.4.7. Let 0 α ≥ 0

and let the assumptions in Lemma 1.2.2 be satisfied. Let v be defined on (a, b) by

v(y) := (β − α)

b∫y

u(x)

((x− y)β−α−1

(x− a)β−α

) qp

dx

pq

<∞.


u(x)

[Φ

(Γ(β − α + 1)


)] qp

dx

1q

≤

b∫a

v(y)Φ(Dβaf(y)

)dy

1p

(2.4.6)



and k(x, y) is given in (2.1.15). Replace f by Dβaf . Then Akf(x) = Γ(β−α+1)

(x−a)β−αDαa f(x)

and the inequality given in (2.4.2) reduces to (2.4.6).

If we take Φ(x) = xs, s ≥ 1 and u(x) = (x − a)(β−α)q

p , x ∈ (a, b), similar to

Corollary 2.4.3 we obtain the following result.

Corollary 2.4.8. Let 0 1 − pq

and let assumption in

Lemma 1.2.2 be satisfied.. Then the following inequality holds b∫a

(Dαa f(x))

sqp dx

1q

≤ (β − α)1p (b− a)

q((β−α)s−1)+ppq

((β − α− 1) qp

+ 1)1q (Γ(β − α + 1))

sp

b∫a

(Dβaf(y))sdy

1p

.

In the following theorem, we will construct new inequality for the Canavati-type

44

Theorem 2.4.9. Let 0 γ > 0, u be a weight function on (a, b)

and assumptions in Lemma 1.2.4 be satisfied. Let v be defined on (a, b) by

v(y) := (ν − γ)

b∫y

u(x)

((x− y)ν−γ−1

(x− a)ν−γ

) qp

dx

pq

<∞.


u(x)

[Φ

(Γ(ν − γ + 1)

(x− a)ν−γDγaf(x)

)] qp

dx

1q

≤

b∫a

v(y)Φ (Dνaf(y)) dy

1p

(2.4.7)



Example 2.4.1. If we take Φ(x) = xs, s ≥ 1, ν − γ > 1 − pq

and weight function

u(x) = (x− a)(ν−γ)q

p , x ∈ (a, b) in (2.4.7), after some calculations we obtain b∫a

(Dγaf(x))

sqp dx

1q

≤ (ν − γ)1p (b− a)

q((ν−γ)s−1)+ppq

((ν − γ − 1) qp

+ 1)1q (Γ(ν − γ + 1))

sp

b∫a

(Dνaf(y))sdy

1p

.

Using the definition of Caputo fractional derivative [7, p. 449], we will prove the

following result.

Theorem 2.4.10. Let 0 < p ≤ q < ∞, u be a weight function on (a, b) and Dα∗af

denotes the Caputo fractional derivative of f . Let v be defined on (a, b) by

v(y) := (n− α)

b∫y

u(x)

((x− y)n−α−1

(x− a)n−α

) qp

dx

pq

<∞.


u(x)

[Φ

(Γ(n− α + 1)

(x− a)n−αDα∗af(x)

)] qp

dx

1q

≤

b∫a

v(y)Φ(f (n)(y)

)dy

1p

(2.4.8)

45


Example 2.4.2. If we take Φ(x) = xs, s ≥ 1, n − α > 1 − pq

and weight function

u(x) = (x− a)(n−α)q

p , x ∈ (a, b), in (2.4.8), after some calculations we obtain b∫a

(Dα∗af(x))

sqp dx

1q

≤ (n− α)1p (b− a)

q((n−α)s−1)+ppq

((n− α− 1) qp

+ 1)1q (Γ(n− α + 1))

sp

b∫a

(f (n)(y))sdy

1p

.

Now, we give the following result for Erdelyi–Kober type fractional integral.

Theorem 2.4.11. Let 0 < p ≤ q < ∞, u be a weight function on (a, b) and

2F1(a, b; c; z) denotes the hypergeometric function. Let v be defined on (a, b) by

v(y) := ασ

b∫y

u(x)

(x−σηyση+σ−1

(xσ − yσ)1−α(xσ − aσ)α

) qp

dx

pq

<∞.


u(x)

[Φ

(Γ(α + 1)(

1−(ax

)σ)α2F1(x)

Iαa+;σ;ηf(x)

)] qp

dx

1q

≤

b∫a

v(y)Φ (f(y)) dy

1p

(2.4.9)



Example 2.4.3. If we take Φ(x) = xs, s ≥ 1, u(x) = xσ−1 ((xσ − aσ)α 2F1(x))qp ,

x ∈ (a, b) in (2.4.9), after some calculations we obtain b∫a

(2F1(x))qp

(1−s) (Iαa+;σ;ηf(x)) sqp dx

1q

≤ C

b∫a

(2F1(y))f s(y)dy

1p

,

where

C =α

1pσ

q−ppq b

σ−1p (bσ − aσ)

q(αs−1)+ppq

apσ−p+qsσα

pq ((α− 1) qp

+ 1)1q (Γ(α + 1))

sp

.

Remark 2.4.1. For p = q = 1 all above results reduce to results given in [38, 40].

46

2.5 Inequality of G. H. Hardy and superquadratic

function

In [52] (also see [28], [30]) K. Krulic et. al. study some new weighted Hardy-type

inequalities. But here we give the improvements of an inequality of G. H. Hardy via

superquadratic function. We establish new inequalities of Hardy-type using different

fractional integrals and fractional derivatives.

In [4] the following refined Hardy-type inequality is given:

Theorem 2.5.1. Let u be a weight function, k(x, y) ≥ 0. Assume that k(x,y)K(x)

u(x) is

locally integrable on Ω1 for each fixed y ∈ Ω2. Define v by (2.1.3). Suppose I = [0, c),

c ≤ ∞, ϕ : I → R. If ϕ is a superquadratic function, then the inequality∫Ω1

ϕ(Akf(x))u(x)dµ1(x) +

∫Ω2

∫Ω1

ϕ (|f(y)− Akf(x)|) u(x)k(x, y)

K(x)dµ1(x) dµ2(y)

≤∫Ω2

ϕ(f(y))v(y)dµ2(y) (2.5.1)


defined by (2.3.1).

If ϕ is subquadratic, then the inequality sign in (2.5.1) is reversed.

Let us continue by defining a linear functional as a difference between the right-

hand side and the left-hand side of the refined Hardy-type inequality (2.5.1) as:

A(ϕ) =

∫Ω2

ϕ(f(y))v(y)dµ2(y)−∫Ω1

ϕ(Akf(x))u(x)dµ1(x)

−∫Ω2

∫Ω1

ϕ (|f(y)− Akf(x)|) u(x)k(x, y)

K(x)dµ1(x)dµ2(y) (2.5.2)

It is clear, that if ϕ is superquadratic function, then A(ϕ) ≥ 0.

Lemma 2.5.2. Consider the function ϕs for s > 0 defined as

ϕs(x) =

xs

s(s−2), s 6= 2,

x2

2log x, s = 2.

(2.5.3)

Then, with the convention 0 log 0 = 0, it is superquadratic.

For linear functional A defined by (2.5.2) we have A(ϕs) ≥ 0 for all s > 0.

47

2.5.1 Main results

Our first result is for Riemann-Liouville fractional integral.

Theorem 2.5.3. Let s > 2, α > 0. Then the function A1 : R→ [0,∞) defined by

A1(s) =1

s(s− 2)

b∫a

f s(y)(b− y)αdy − (Γ(α + 1))sb∫

a


−αb∫

a

b∫y

(∣∣∣∣f(y)− Γ(α + 1)

(x− a)αIαa+f(x)

∣∣∣∣)s (x− y)α−1dx dy

(2.5.4)

is exponentially convex.

Proof. Applying Theorem 2.5.1 with Ω1 = Ω2 = (a, b), dµ1(x) = dx, dµ2(y) = dy and

k(x, y) is given in (2.1.6), then (2.5.2) reduces to

A1(s) =

b∫a

ϕs(f(y))v(y)dy −b∫

a

ϕs

(Γ(α + 1)

(x− a)αIαa+f(x)

)u(x)dx

− αb∫

a

b∫y

ϕs

(∣∣∣∣f(y)− Γ(α + 1)

(x− a)αIαa+f(x)

∣∣∣∣) u(x)(x− y)α−1

(x− a)αdxdy, (2.5.5)

where ϕs is defined by (2.5.3). Function A1 is exponentially convex. Applying (2.5.5)

with particular weight function u(x) = (x− a)α, x ∈ (a, b) we get (2.5.4).

Remark 2.5.1. Notice that for s > 2

A1(s) =1

s(s− 2)

b∫a

f s(y)(b− y)αdy −b∫

a

(Γ(α + 1)

(x− a)αIαa+f(x)

)s(x− a)αdx

−αb∫

a

b∫y

(∣∣∣∣f(y)− Γ(α + 1)

(x− a)αIαa+f(x)

∣∣∣∣)s (x− y)α−1dx dy

≤ 1

s(s− 2)

(b− a)αb∫

a

f s(y)dy−(b− a)α(1−s)(Γ(α + 1))sb∫

a

(Iαa+f(x))sdx

=

(b− a)α(1−s)

s(s− 2)

(b− a)αsb∫

a

f s(y)dy − (Γ(α + 1))sb∫

a

(Iαa+f(x))s dx

,

48

so the following inequality

A1(s) ≤ H1(s),

holds, where

H1(s) =(b− a)α(1−s)

s(s− 2)

(b− a)αsb∫

a

f s(y)dy − (Γ(α + 1))sb∫

a

(Iαa+f(x))sdx

.Next we give results with respect to the generalized Riemann–Liouville fractional

derivative.

Theorem 2.5.4. Let s > 2, α ≥ 0 and let the assumptions in Lemma 1.2.2 be satis-

fied. Then the function A2 : R→ [0,∞) defined by

A2(s) =1

s(s− 2)

b∫a


−(Γ(β − α + 1))sb∫

a


−(β − α)

b∫a

b∫y

(∣∣∣∣Dαf(y)− Γ(β − α + 1)


∣∣∣∣)s (x− y)β−α−1dx dy

is exponentially convex and the following inequality holds true:

A2(s) ≤ H2(s),

where

H2(s) =(b− a)(β−α)(1−s)

s(s− 2)

(b− a)(β−α)s

b∫a

(Dβaf(y))sdy

− (Γ(β − α + 1))sb∫

a

(Dαa f(x))sdx

.Proof. Similar to Theorem 2.5.3 and Theorem 2.3.7.

In the following theorem, new inequality for the Canavati-type fractional deriva-

tive is given.

49

Theorem 2.5.5. Let s > 2 and let the assumptions in Lemma 1.2.4 be satisfied.

Then the function A3 : R→ [0,∞) defined by

A3(s) =1

s(s− 2)

b∫a


−(Γ(ν − γ + 1))

b∫a


−(ν − γ)

b∫a

b∫y

(∣∣∣∣Dνaf(y)− Γ(ν − γ + 1)


∣∣∣∣)s (x− y)ν−γ−1dx dy


A3(s) ≤ H3(s),

where

H3(s) =(b− a)(ν−γ)(1−s)

s(s− 2)

(b− a)(ν−γ)s

b∫a

(Dνaf(y))sdy

−(Γ(ν − γ + 1))sb∫

a

(Dγaf(x))sdx


Next, new inequalities for the Caputo fractional derivative are given.

Theorem 2.5.6. Let s > 2, ν ≥ 0. Then the function A4 : R→ [0,∞) defined by

A4(s) =1

s(s− 2)

[ b∫a

(b− y)n−ν(f (n)(y))sdy

−(Γ(n− ν + 1))sb∫

a

(x− a)(n−ν)(1−s) (Dν∗af(x))s dx

−(n− ν)

b∫a

b∫y

(∣∣∣∣f (n)(y)−Γ(n− ν + 1)

(x−a)n−νDν∗af(x)

∣∣∣∣)s (x− y)n−ν−1dxdy

]

50

is exponentially convex and the following inequality holds:

A4(s) ≤ H4(s),

where

H4(s) =(b− a)(n−ν)(1−s)

s(s− 2)

(b− a)(n−ν)s

b∫a

(f (n)(y))sdy

− (Γ(n− ν + 1))sb∫

a

(Dν∗af(x))sdx


Theorem 2.5.7. Let s > 2 and let the assumptions in Lemma 1.2.5 be satisfied.


A5(s) =1

s(s− 2)

[ b∫a

(b− y)ν−γ(Dν∗af(y))sdy

−(Γ(ν − γ + 1))sb∫

a

(x− a)(ν−γ)(1−s) (Dγ∗af(x))s dx

−(ν − γ)

b∫a

b∫y

(∣∣∣∣Dν∗af(y)− Γ(ν − γ + 1)

(x− a)ν−γDγ∗af(x)

∣∣∣∣)s (x− y)ν−γ−1dxdy

]


A5(s) ≤ H5(s),

where

H5(s) =(b− a)(ν−γ)(1−s)

s(s− 2)

(b− a)(ν−γ)s

b∫a

(Dν∗af(y))sdy

− (Γ(ν − γ + 1))sb∫

a

(Dγ∗af(x))sdx

.

51

Proof. Similar to Theorem 2.5.3 and Theorem 2.3.7.

Now, we give the following result.

Theorem 2.5.8. Let s > 2 and 2F1(a, b; c; z) denotes the hypergeometric function.


A6(s) =1

s(s− 2)

[ b∫a


− (Γ(α + 1))sb∫

a

xσ+σαs−1(xσ − aσ)α(s−1) (2F1(x))1−s (Iαa+;σ;ηf(x))sdx

−ασb∫

a

b∫y

(∣∣∣∣∣f(y)− Γ(α + 1)(1−

(ax

)σ)α2F1(x)

Iαa+;σ;ηf(x)

∣∣∣∣∣)s

×x−ση+σ−1yση+σ−1(xσ − yσ)α−1dx dy

]


A6(s) ≤ H6(s),

where

H6(s) =(bσ − aσ)α(1−s)

s(s− 2)


b∫a

2F1(y)f s(y)dy

− aσ−1+ασs(Γ(α + 1))sb∫

a

(2F1(x))1−s(Iαa+;σ;ηf(x))sdx


The following result is about Hadamard-type fractional integrals.

52

Theorem 2.5.9. Let s > 2, α > 0. Then the function A7 : R→ [0,∞) defined by

A7(s) =1

s(s− 2)

b∫a

(log b− log y)α

yf s(y)dy

−(Γ(α + 1))sb∫

a

(log x− log a)α(1−s)

x(Jαa+

f(x))sdx

−αb∫

a

b∫y

(∣∣∣∣f(y)− Γ(α + 1)

(log x− log a)αJαa+

f(x)

∣∣∣∣)s (log x− log y)α−1

xydxdy

is exponentially convex and the following inequality holds:

A7(s) ≤ H7(s),

where

H7(s) =1

s(s− 2)

(log b− log a)α(1−s)

ab

b(log b− log a)αsb∫

a

f s(y)dy

− a(Γ(α + 1))sb∫

a

(Jαa+f(x))sdx


In the following theorem we will discuss three cases for results given in Theorems

2.5.3–2.5.9.

Theorem 2.5.10. For i = 1, . . . , 7 the following inequalities hold true:

(i). [Ai(p)]q−rq−p [Ai(q)]

r−pq−p ≤ Hi(r)

(ii). [Ai(r)]p−qp−r [Ai(p)]

q−rp−r ≤ Hi(q)

(iii). Ai(p) ≤ [Hi(r)]q−pq−r [Hi(q)]

p−rq−r

for every choice r, p, q ∈ R+, such that 2 < r < p < q.

53

2.6 Refinements of an inequality of G. H. Hardy

Here we are ready to give the applications of refined Hardy-type inequality for frac-

tional integrals and fractional derivatives for arbitrary convex function. K. Krulic et.

al. gave the refinement of Theorem 2.3.1 and they proved the following theorem in

[25].



Ω1×Ω2 and K be defined on Ω1 by (2.1.2). Suppose that the function x 7→ u(x)k(x,y)K(x)

is integrable on Ω1 for each fixed y ∈ Ω2 and that v is defined on Ω2 by (2.1.3). If Φ

is a convex function on the interval I ⊆ R and ϕ : I → R is any function, such that

ϕ(x) ∈ ∂Φ(x) for all x ∈ Int I, then the inequality∫Ω2

v(y)Φ (f(y)) dµ2(y)−∫Ω1

u(x)Φ (Akf(x)) dµ1(x)

≥∫Ω1

u(x)

K(x)

∫Ω2

k(x, y) | |Φ(f(y))− Φ(Akf(x))|

−|ϕ(Akf(x))|.|f(y)− Akf(x)| | dµ2(y) dµ1(x) (2.6.1)

holds for all measurable functions f : Ω2 → R, such that f(y) ∈ I, for all fixed y ∈ Ω2

where Ak is defined by (2.3.1).

2.6.1 Main results

Let us continue by taking a non-negative difference between the left-hand side and

the right-hand side of refined Hardy-type inequality given in (2.6.1).

ψ(Φ) =

∫Ω2

v(y)Φ (f(y)) dµ2(y)−∫Ω1


−∫Ω1

u(x)

K(x)

∫Ω2

k(x, y) | |Φ(f(y))− Φ(Akf(x))|

−|ϕ(Akf(x))|.|f(y)− Akf(x)| | dµ2(y) dµ1(x). (2.6.2)

We will show some new inequalities for different fractional integrals and fractional

derivatives by using the non-negative difference given in (2.6.2).

Our first result is given in the following theorem.

54

Theorem 2.6.2. Let s ≥ 1, α > 0, f ≥ 0, g be increasing function on (a, b] such

that g′ be continuous on (a, b), ψ1 : R→ [0,∞). Then the following inequality holds:

ψ1(s) ≤ H1(s),

where

ψ1(s) =

b∫a


−(Γ(α + 1))sb∫

a

g′(x)(g(x)− g(a))α(1−s) (Iαa+;gf(x))sdx

−αb∫

a

x∫a

g′(x)g′(y)(g(x)− g(y))α−1

∣∣∣∣ ∣∣∣∣f s(y)−(

Γ(α + 1)


)s∣∣∣∣−s∣∣∣∣ Γ(α + 1)


∣∣∣∣s−1

.

∣∣∣∣f(y)− Γ(α + 1)


∣∣∣∣∣∣∣∣∣ dydx(2.6.3)

and

H1(s) = (g(b)− g(a))α(1−s)


a

f s(y)g′(y)dy

− (Γ(α + 1))sb∫

a


.Proof. Rewrite (2.6.2) with Ω1 = Ω2 = (a, b), dµ1(x) = dx, dµ2(y) = dy and k(x, y)

is given in (2.3.6). Then Akf(x) = Γ(α+1)(g(x)−g(a))α

Iαa+;gf(x). For the particular weight

function u(x) = g′(x)(g(x)− g(a))α, we obtain v(y) = g′(y)(g(b)− g(y))α. If we take

Φ(x) = xs, x ∈ R+, after some calculations we get (2.6.3). Since

α

b∫a

x∫a

g′(x)g′(y)(g(x)− g(y))α−1

∣∣∣∣ ∣∣∣∣f s(y)−(

Γ(α + 1)


)s∣∣∣∣−s∣∣∣∣ Γ(α + 1)


∣∣∣∣s−1

.

∣∣∣∣f(y)− Γ(α + 1)


∣∣∣∣∣∣∣∣∣ dy dx ≥ 0.

55

Then

ψ1(s) ≤b∫

a


−b∫

a

g′(x)(g(x)− g(a))α(

Γ(α + 1)


)sdx

≤ (g(b)− g(a))α(1−s)


a

f s(y)g′(y)dy

− (Γ(α + 1))sb∫

a


= H1(s).

This complete the proof. If g(x) = x, then Iαa+;xf(x) reduces to Iαa+f(x) left-sided Riemann–Liouville frac-

tional integral and the following result follows.

Corollary 2.6.3. Let s ≥ 1, α > 0, f ≥ 0 and ψ2 : R→ [0,∞). Then the following

inequality holds true:

ψ2(s) ≤ H2(s),

where

ψ2(s) =

b∫a

(b− y)αf s(y)dy − (Γ(α + 1))sb∫

a


−αb∫

a

x∫a

(x− y)α−1

∣∣∣∣ ∣∣∣∣f s(y)−(

Γ(α + 1)

(x− a)αIαa+f(x)

)s∣∣∣∣−s∣∣∣∣Γ(α + 1)

(x− a)αIαa+f(x)

∣∣∣∣s−1

.

∣∣∣∣f(y)− Γ(α + 1)

(x− a)αIαa+f(x)

∣∣∣∣∣∣∣∣∣ dy dx,

and

H2(s) = (b− a)α(1−s)

(b− a)αsb∫

a

f s(y)dy − (Γ(α + 1))sb∫

a

(Iαa+f(x))sdx

.

56

If we take g(x) = log x, the following result is obtained.

Corollary 2.6.4. Let s ≥ 1, α > 0, f ≥ 0 and ψ3 : R→ [0,∞). Then the following

inequality holds

ψ3(s) ≤ H3(s),

where

ψ3(s) =

b∫a

(log b− log y)αf s(y)dy

y

−(Γ(α + 1))sb∫

a

(log x− log a)α(1−s) (Jαa+f(x)

)s dxx

−αb∫

a

x∫a

(log x− log y)α−1

∣∣∣∣ ∣∣∣∣f s(y)−(

Γ(α + 1)


f(x)

)s∣∣∣∣−s∣∣∣∣ Γ(α + 1)


f(x)

∣∣∣∣s−1

.

∣∣∣∣f(y)− Γ(α + 1)


f(x)

∣∣∣∣∣∣∣∣∣ dyy dx

x,

and

H3(s) = (log b− log a)α(1−s)

(log b− log a)αsb∫

a

f s(y)dy

y

−(Γ(α + 1))sb∫

a

(Jαa+f(x))s

dx

x

.Theorem 2.6.5. Let s ≥ 1 and let the assumptions in Lemma 1.2.2 be satisfied. Let

ψ4 : R → [0,∞). Then for non-negative functions f , Dβaf and Dα

a f the following


ψ4(s) ≤ H4(s),

where

ψ4(s) =

b∫a


−(Γ(β − α + 1))sb∫

a


57

−(β − α)

b∫a

x∫a

(x− y)β−α−1

∣∣∣∣ ∣∣∣∣(Dβaf(y))s −

(Γ(β − α + 1)


)s∣∣∣∣−s∣∣∣∣Γ(β − α + 1)


∣∣∣∣s−1

.

∣∣∣∣Dβaf(y)− Γ(β − α + 1)


∣∣∣∣∣∣∣∣∣ dy dx

and

H4(s) = (b− a)(β−α)(1−s)

(b− a)(β−α)s

b∫a

(Dβaf(y))sdy

− (Γ(β − α + 1))sb∫

a

(Dαa f(x))sdx


In the following theorem, we give result for the Canavati-type fractional derivative.

Theorem 2.6.6. Let s ≥ 1, let the assumptions in Lemma 1.2.4 be satisfied and

ψ5 : R → [0,∞). Then for non-negative functions f , Dνaf and Dγ

af the following

inequality holds:

ψ5(s) ≤ H5(s), (2.6.4)

where

ψ5(s) =

b∫a


−(Γ(ν − γ + 1))sb∫

a


−(ν − γ)

b∫a

x∫a

(x− y)ν−γ−1

∣∣∣∣ ∣∣∣∣(Dνaf(y))s −

(Γ(ν − γ + 1)


)s∣∣∣∣−s∣∣∣∣Γ(ν − γ + 1)


∣∣∣∣s−1

.

∣∣∣∣Dνaf(y)− Γ(ν − γ + 1)


∣∣∣∣∣∣∣∣∣ dy dx,

58

and

H5(s) = (b− a)(ν−γ)(1−s)

(b− a)(ν−γ)s

b∫a

(Dνaf(y))sdy

− (Γ(ν − γ + 1))sb∫

a

(Dγaf(x))sdx


As a special case of Theorem 2.6.1, we construct new inequality for the Caputo


Theorem 2.6.7. Let s ≥ 1, α ≥ 0, f (n) ≥ 0 and ψ6 : R→ [0,∞). Then the following


ψ6(s) ≤ H6(s),

where

ψ6(s) =

b∫a

(b− y)n−α(f (n)(y))sdy

−(Γ(n− α + 1))sb∫

a

(x− a)(n−α)(1−s) (Dα∗af(x))s dx

−(n− α)

b∫a

x∫a

(x− y)n−α−1

∣∣∣∣ ∣∣∣∣(f (n)(y))s −(

Γ(n− α + 1)


)s∣∣∣∣−s∣∣∣∣Γ(n− α + 1)


∣∣∣∣s−1

.

∣∣∣∣f (n)(y)− Γ(n− α + 1)


∣∣∣∣∣∣∣∣∣ dy dx,

and

H6(s) = (b− a)(n−α)(1−s)

(b− a)(n−α)s

b∫a

(f (n)(y))sdy

− (Γ(n− α + 1))sb∫

a

(Dα∗af(x))sdx

.

59

Proof. Similar to Theorem 2.6.2 and Theorem 2.3.7.

Theorem 2.6.8. Let s ≥ 1, let the assumptions in Lemma 1.2.6 be satisfied and

ψ7 : R → [0,∞). Then for non-negative functions f, Dν∗af and Dγ

∗af the following

inequality holds:

ψ7(s) ≤ H7(s),

where

ψ7(s) =

b∫a

(b− y)ν−γ(Dν∗af(y))sdy

−(Γ(ν − γ + 1))sb∫

a

(x− a)(ν−γ)(1−s) (Dγ∗af(x))s dx

−(ν − γ)

b∫a

x∫a

(x− y)ν−γ−1

∣∣∣∣ ∣∣∣∣(Dν∗af(y))s −

(Γ(ν − γ + 1)


)s∣∣∣∣−s∣∣∣∣Γ(ν − γ + 1)


∣∣∣∣s−1

.

∣∣∣∣Dν∗af(y)− Γ(ν − γ + 1)


∣∣∣∣∣∣∣∣∣ dy dx,

and

H7(s) = (b− a)(ν−γ)(1−s)

(b− a)(ν−γ)s

b∫a

(Dν∗af(y))sdy

− (Γ(ν − γ + 1))sb∫

a

(Dγ∗af(x))sdx



Theorem 2.6.9. Let s ≥ 1, α > 0, f ≥ 0, 2F1(a, b; c; z) denotes the hypergeometric

function and ψ8 : R→ [0,∞). Then the following inequality holds:

ψ8(s) ≤ H8(s),

where

60

ψ8(s) =

b∫a


−b∫

a


(Γ(α + 1)(

1−(ax

)σ)α2F1(x)

Iαa+;σ;ηf(x)

)s

dx

−ασb∫

a

x∫a

(yx

)ση (xy)σ−1

(xσ − yσ)1−α

∣∣∣∣∣∣∣∣∣∣f s(y)−

(Γ(α + 1)(

1−(ax

)σ)α2F1(x)

Iαa+;σ;ηf(x)

)s∣∣∣∣∣−s

∣∣∣∣∣ Γ(α + 1)(1−

(ax

)σ)α2F1(x)

Iαa+;σ;ηf(x)

∣∣∣∣∣s−1∣∣∣∣∣f(y)− Γ(α + 1)(

1−(ax

)σ)α2F1(x)

Iαa+;σ;ηf(x)

∣∣∣∣∣∣∣∣∣∣∣ dy dx

and

H8(s) = (bσ − aσ)α(1−s)


b∫a

2F1(y)f s(y)dy

− aσ−1+ασs(Γ(α + 1))sb∫

a

( (2F1(x))1−sIαa+;σ;ηf(x))sdx

,Proof. Similar to Theorem 2.6.2 and Theorem 2.3.11.

Chapter 3

G. H. Hardy-type integralinequalities

In this chapter, we show new class of general G. H. Hardy-type integral inequalities

with fractional integrals and fractional derivatives. Also we give some improvements

of generalize G. H. Hardy-type inequalities. The results given in this chapter can be

found in [44], [45] ( see also [48]).

3.1 New class of the G. H. Hardy-type inequalities

with kernels

We establish some new integral inequalities of G. H. Hardy-type for fractional integrals

and fractional derivatives using the following theorem given in [26].

Theorem 3.1.1. Let 0 < p ≤ q < ∞. Let (Ω1,Σ1, µ1) and (Ω2,Σ2, µ2) be measure

spaces with σ-finite measures, u be a weight function on Ω1, w be a µ2− a.e. positive

function on Ω2, k be a non-negative measurable function on Ω1×Ω2 and K be defined

on Ω1 by (2.1.2). Suppose that K(x) > 0 for all x ∈ Ω1 and that the function

x 7→ u(x)(k(x,y)K(x)

) qp

is integrable on Ω1 for each fixed y ∈ Ω2. Let Φ be a non-negative

convex function on an interval I ⊆ R. If

A = supy∈Ω2

w−1p (y)

∫Ω1

u(x)

(k(x, y)

K(x)

) qp

dµ1(x)

1q

<∞,

61

62

then there exists a positive constant C, such that the inequality∫Ω1


1q

≤ C

∫Ω2

w(y)Φ (f(y)) dµ2(y)

1p

(3.1.1)

holds for all measurable function f : Ω2 → R with values in I and Akf be defined by

(2.3.1). Moreover, if C is smallest constant for (3.1.1) to hold, then C ≤ A.

The upcoming corollary is given in [26].

Corollary 3.1.2. Let −∞ < q ≤ p < 0 and let the assumption in Theorem 3.1.1 be

satisfied with a positive convex function Φ. If

B = infy∈Ω2

w−1p (y)

∫Ω1

u(x)

(k(x, y)

K(x)

) qp

dµ1(x)

1q

<∞,

then there exists a positive real constant C, such that the inequality∫Ω1


1q

≥ C

∫Ω2

w(y)Φ (f(y)) dµ2(y)

1p

(3.1.2)

holds for all measurable function f : Ω2 → R with values in Ω2. Moreover, if C is

smallest constant for (3.1.2) to hold, then C ≥ B.

We will give some applications of Theorem 3.1.1 for different type of fractional

integrals and fractional derivatives. Our first result deals with fractional integral of

f with respect to another increasing function g.

Theorem 3.1.3. Let 0 0, u be a weight function on (a, b),

w be a.e. positive function on (a, b), g be increasing function on (a, b] such that g′ be

continuous on (a, b). If

A = supy∈(a,b)

w−1p (y)

b∫y

u(x)

(α g′(y)(g(x)− g(y))α−1

(g(x)− g(a))α

) qp

dx

1q

<∞,

then there exists a positive constant C, such that the inequality b∫a

u(x)

(Γ(α + 1)


) sqp

dx

1q

≤ C

b∫a

w(y)f s(y)dy

1p

(3.1.3)

holds. Moreover, if C is the smallest constant for (3.1.3) to hold, then C ≤ A.

63


k(x, y) is given in (2.3.6) and Akf(x) = Γ(α+1)(g(x)−g(a))α

Iαa+;gf(x). Then the inequality given

in (3.1.1) takes the form b∫a

u(x)

[Φ

(Γ(α + 1)


)] qp

dx

1q

≤ C

b∫a

w(y)Φ(f(y))dy

1p

.

(3.1.4)

If we choose the function Φ : R+ → R defined by Φ(x) = xs, s ≥ 1, then (3.1.4)

becomes (3.1.3).


tional integral, so the following result follows.

Corollary 3.1.4. Let 0 0, s ≥ 1, u be a weight function on (a, b),

w be a.e. positive function on (a, b). If

A = supy∈(a,b)

w−1p (y)

b∫y

u(x)

(α (x− y)α−1

(x− a)α

) qp

dx

1q

<∞,


u(x)

(Γ(α + 1)

(x− a)αIαa+f(x)

) sqp

dx

1q

≤ C

b∫a

w(y)f s(y)dy

1p

(3.1.5)


If g(x) = log x, x ∈ (a, b) where 0 ≤ a 0, u be a weight function on

(a, b), w be a.e. positive function on (a, b). If

A = supy∈(a,b)

w−1p (y)

b∫y

u(x)

(α (log x− log y)α−1

y(log x− log a)α

) qp

dx

1q

<∞,


u(x)

(Γ(α + 1)


f(x)

) sqp

dx

1q

≤ C

b∫a

w(y)f s(y)dy

1p

(3.1.6)

64

Corollary 3.1.6. Let 0 α ≥ 0, u be a weight function

on (a, b), w be a.e. positive function on (a, b) and let the assumptions in the Lemma

1.2.2 be satisfied. If

A = supy∈(a,b)

w−1p (y)

b∫y

u(x)

((β − α)(x− y)β−α−1

(x− a)β−α

) qp

dx

1q

<∞,


u(x)

(Γ(β − α + 1)


) sqp

dx

1q

≤ C

b∫a

w(y)(Dβaf(y))sdy

1p

(3.1.7)


Proof. Applying Theorem 3.1.1 with Ω1 = Ω2 = (a, b), dµ1(x) = dx, dµ2(y) = dy and

k(x, y) is given in (2.1.15). Replace f by Dβaf and Akf(x) = Γ(β−α+1)

(x−a)β−αDαa f(x). Then

the inequality given in (3.1.1) takes the form b∫a

u(x)

[Φ

(Γ(β − α + 1)


)] qp

dx

1q

≤ C

b∫a

w(y)Φ(Dβaf(y)

)dy

1p

.

(3.1.8)

For s ≥ 1, Φ : R+ → R be defined by Φ(x) = xs, then (3.1.8) becomes (3.1.7).

Corollary 3.1.7. Let 0 < p ≤ q < ∞, s ≥ 1, u be a weight function on (a, b), w be

a.e. positive function on (a, b) and assumptions in Lemma 1.2.4 be satisfied. If

A = supy∈(a,b)

w−1p (y)

b∫y

u(x)

((ν − γ)(x− y)ν−γ−1

(x− a)ν−γ

) qp

dx

1q

<∞,


u(x)

(Γ(ν − γ + 1)


) sqp

dx

1q

≤ C

b∫a

w(y)(Dνaf(y))sdy

1p

(3.1.9)

65


Now, we construct new inequality for the Caputo fractional derivative.


a.e. positive function on (a, b). If

A = supy∈(a,b)

w−1p (y)

b∫y

u(x)

((n− α)(x− y)n−α−1

(x− a)n−α

) qp

dx

1q

<∞,


u(x)

(Γ(n− α + 1)


) sqp

dx

1q

≤ C

b∫a

w(y)(f (n)(y))sdy

1p

(3.1.10)




a.e. positive function on (a, b) and let the assumptions in Lemma 1.2.6 be satisfied.

If

A = supy∈(a,b)

w−1p (y)

b∫y

u(x)

((ν − γ)(x− y)ν−γ−1

(x− a)ν−γ

) qp

dx

1q

<∞,


u(x)

(Γ(n− α + 1)


) sqp

dx

1q

≤ C

b∫a

w(y)(Dν∗af(y))sdy

1p

(3.1.11)

66

Corollary 3.1.10. Let 0 < p ≤ q <∞, s ≥ 1, u be a weight function on (a, b), w be

a.e. positive function on (a, b), 2F1(a, b; c; z) denotes the hypergeometric function. If

A = supy∈(a,b)

w−1p (y)

b∫y

u(x)

(ασ x−σηyση+σ−1(xσ − yσ)α−1

(xσ − aσ)α 2F1(x)

) qp

dx

1q

<∞,


u(x)

(Γ(α + 1)(

1−(ax

)σ)α2F1(x)

Iαa+;σ;ηf(x)

) sqp

dx

1q

≤ C

b∫a

w(y)f s(y)dy

1p

.

(3.1.12)



Remark 3.1.1. Analogous to the Corollary 3.1.2, we can obtain the results with infi-

mum, but here we omit the details.

3.2 Generalized G. H. Hardy-type inequalities

This section is dedicated to the generalized G. H. Hardy-type inequalities for arbitrary

convex functions. In [52], K. Krulic prove the Hardy-type inequality with general

kernel in the following theorem.

Theorem 3.2.1. Let 0 < p ≤ q < ∞, or −∞ < q ≤ p < 0, (Ω1,Σ1, µ1) and

(Ω2,Σ2, µ2) be measure spaces with σ-finite measures, u be a weight function on Ω1, k

be a non-negative measurable function on Ω1×Ω2 and K be defined on Ω1 by (2.1.2),

and that the function x 7→ u(x)(k(x,y)K(x)

) qp

is integrable on Ω1 for each fixed y ∈ Ω2

and that v is defined on Ω2 by (2.4.1). If Φ is a non-negative convex function on the

interval I ⊆ R and ϕ : I → R is any function, such that ϕ ∈ ∂Φ(x) for all x ∈ IntI,then the inequality ∫

Ω2


qp

−∫Ω1


≥ q

p

∫Ω1

u(x)

K(x)Φqp−1(Akf(x))

∫Ω2

k(x, y)r(x, y)dµ2(y)dµ1(x) (3.2.1)

67

holds for all measurable functions f : Ω2 → R, such that f(y) ∈ I for all y ∈ Ω2,

where Ak defined by (2.3.1) and r : Ω1 × Ω2 → R is non-negative function defined by

r(x, y) = | |Φ(f(y))− Φ(Akf(x))| − |ϕ(Akf(x))| |f(y)− Akf(x)| |. (3.2.2)

Remark 3.2.1. For p = q, the Theorem 3.2.1 becomes [25, Theorem 2.1] and convex

function Φ need not to be non-negative.

3.2.1 G. H. Hardy-type inequalities for fractional integrals

Let us continue by taking the non-negative difference of the left- and the right-hand

side of the inequality given in (3.2.1) by taking Φ : R+ → R+, Φ(x) = xs, s ≥ 1 as:

ρ(s) =

∫Ω2

v(y)f s(y)dµ2(y)

qp

−∫Ω1

u(x)(Akf(x))sqp dµ1(x)

−qp

∫Ω1

u(x)

K(x)(Akf(x))s(

qp−1)

∫Ω2

k(x, y)r(x, y)dµ2(y)dµ1(x), (3.2.3)

where r(x, y) is defined by (3.2.2).

Our first result involving fractional integral of f with respect to another increasing

function g is given.

Theorem 3.2.2. Let 0 1 − pq, f ≥ 0, g be increasing

function on (a, b] such that g′ be continuous on (a, b). Then the following inequality

holds true:

0 ≤ ρ1(s) ≤ H1(s)−M1(s) ≤ H1(s),

where

ρ1(s) =(Γ(α + 1))

qp

(α− 1) qp

+ 1

((Iα+ p

q

a+;g fs)(b)

) qp

− αsqp (Γ(α))

sqp

+1(Iα qp

(1−s)+1

b−;g [(Iαa+;gf)(x)]sqp

)(a)−M1(s),

M1(s) =q(Γ(α + 1))s(

qp−1)+1

p

b∫a

g′(x)(g(x)− g(a))α(q−p)(1−s)

p(Iαa+;gf(x)

)s( qp−1)

× (Iαa+;gr1)(x, ·) dx,

68

r1(x, y) =

∣∣∣∣ ∣∣∣∣f s(y)−(

Γ(α + 1)


)s∣∣∣∣−s∣∣∣∣ Γ(α + 1)


∣∣∣∣s−1

·∣∣∣∣f(y)− Γ(α + 1)


∣∣∣∣∣∣∣∣∣ ,

and

H1(s) = (g(b)− g(a))αqp

(1−s)

α qp (g(b)− g(a))

q(αs−1)+pp

(α− 1) qp

+ 1

b∫a

g′(y)f s(y)dy

qp

− (Γ(α + 1))sqp

b∫a

g′(x)(Iαa+;gf(x))sqp dx

. (3.2.4)

Proof. Applying Theorem 3.2.1 with Ω1 = Ω2 = (a, b), dµ1(x) = dx, dµ2(y) =

dy, k(x, y) is given in (2.3.6) and Akf(x) = Γ(α+1)(g(x)−g(a))α

Iαa+;gf(x). For the particular

weight function u(x) = g′(x)(g(x)− g(a))αqp , x ∈ (a, b), we get v(y) = (αg′(y)(g(b)−

g(y))α−1+ pq )/(((α− 1) q

p+ 1)

pq ), then (3.2.3) takes the form

ρ1(s) =αqp

(α− 1) qp

+ 1

b∫a


qp

− (Γ(α + 1))sqp

b∫a

g′(x)(g(x)− g(a))αq(1−s)

p(Iαa+;gf(x)

) sqp dx−M1(s).

Since αqp

(1− s) ≤ 0, g is increasing and M1(s) ≥ 0, we obtain that

0 ≤ ρ1(s) ≤ αqp (g(b)− g(a))(α−1) q

p+1

(α− 1) qp

+ 1

b∫a

g′(y)f s(y)dy

qp

−(g(b)− g(a))αqp

(1−s)(Γ(α + 1))sqp

b∫a

g′(x)(Iαa+;gf(x))sqp dx−M1(s)

= H1(s)−M1(s)

≤ H1(s).

This completes the proof.

69

If g(x) = x, then Iαa+;xf(x) reduces to Iαa+f(x) left-sided Riemann-Liouville frac-


Corollary 3.2.3. Let 0 1− pq, s ≥ 1, f ≥ 0. Then the following


0 ≤ ρ2(s) ≤ H2(s)−M2(s) ≤ H2(s),

where

ρ2(s) =(Γ(α + 1))

qp

(α− 1) qp

+ 1

((Iα+ p

q

a+ f s)(b)) qp

− αsqp (Γ(α))

sqp

+1(Iα qp

(1−s)+1

b− [(Iαa+f)(x)]sqp

)(a)−M2(s),

M2(s) =q(Γ(α + 1))s(

qp−1)+1

p

b∫a

(x − a)α(q−p)(1−s)

p (Iαa+f(x))s(qp−1) (Iαa+r2)(x, ·) dx,

r2(x, y) =

∣∣∣∣ ∣∣∣∣f s(y)−(

Γ(α + 1)

(x− a)αIαa+f(x)

)s∣∣∣∣−s∣∣∣∣Γ(α + 1)

(x− a)αIαa+

f(x)

∣∣∣∣s−1

·∣∣∣∣f(y)− Γ(α + 1)

(x− a)αIαa+f(x)

∣∣∣∣∣∣∣∣∣ ,

and

H2(s) = (b− a)αqp

(1−s)

α qp (b− a)

q(αs−1)+pp

(α− 1) qp

+ 1

b∫a

f s(y)dy

qp

−(Γ(α + 1))sqp

b∫a

(Iαa+f(x))sqp dx

. (3.2.5)

If g(x) = log x, x ∈ (a, b) where 0 ≤ a 1− pq, f ≥ 0. Then the following

inequality holds:

0 ≤ ρ3(s) ≤ H3(s)−M3(s) ≤ H3(s),

70

where

ρ3(s) =(Γ(α + 1))

qp

(α− 1) qp

+ 1

((J

α+ pq

a+ f s)(b)) qp

− αsqp (Γ(α))

sqp

+1(Jα qp

(1−s)+1

b−[(Jαa+

f)(x)]sqp

)(a)−M3(s),

M3(s) =q(Γ(α + 1))s(

qp−1)+1

p

b∫a

(log x− log a)α(q−p)(1−s)

p(Jαa+

f(x))s( q

p−1)

× (Jαa+r3)(x, ·) dx

x,

r3(x, y) =

∣∣∣∣ ∣∣∣∣f s(y)−(

Γ(α + 1)


f(x)

)s∣∣∣∣−s∣∣∣∣ Γ(α + 1)


f(x)

∣∣∣∣s−1

·∣∣∣∣f(y)− Γ(α + 1)


f(x)

∣∣∣∣∣∣∣∣∣ ,

and

H3(s) = (log b− log a)αqp

(1−s)

α qp (log b− log a)

q(αs−1)+pp

(α− 1) qp

+ 1

b∫a

f s(y)dy

y

qp

− (Γ(α + 1))sqp

b∫a

(Jαa+f(x))

sqpdx

x

. (3.2.6)

Now, we give the following result for Erdelyi-Kober fractional integral.

Theorem 3.2.5. Let 0 1 − pq, f ≥ 0 and 2F1(a, b; c; z)

denotes the hypergeometric function. Then the following inequality holds true:

0 ≤ ρ4(s) ≤ H4(s)−M4(s) ≤ H4(s),

where

ρ4(s) =αqpσ

qp−1

(α− 1) qp

+ 1

b∫a

yσ−12F1(y) (bσ − yσ)α−1+ p

q f s(y)dy

qp

− (Γ(α + 1))sqp

b∫a

xσαsqp

+σ−1 ((xσ − aσ)α 2F1(x))q(1−s)p(Iαa+;σ;ηf(x)

) sqp dx−M4(s),

71

M4(s) =αqσ(Γ(α + 1))

sqp

p

b∫a

xσαs(qp−1)+σ−ση−1 ((xσ − aσ)α 2F1(x))

(q−p)(1−s)p

×(Iαa+;σ;ηf(x)

)s( qp−1)

x∫a

r4(x, y)yση+σ−1

(xσ − yσ)1−α dy dx,

r4(x, y) =

∣∣∣∣∣∣∣∣∣∣f s(y)−

(Γ(α + 1)(

1−(ax

)σ)α2F1(x)

Iαa+;σ;ηf(x)

)s∣∣∣∣∣−s

∣∣∣∣∣ Γ(α + 1)(1−

(ax

)σ)α2F1(x)

Iαa+;σ;ηf(x)

∣∣∣∣∣s−1

·

∣∣∣∣∣f(y)− Γ(α + 1)(1−

(ax

)σ)α2F1(x)

Iαa+;σ;ηf(x)

∣∣∣∣∣∣∣∣∣∣∣ ,

and

H4(s) = (bσ − aσ)αqp

(1−s)

α qpσ

qp−1b(σ−1) q

p (bσ − aσ)q(αs−1)+p

p

(α− 1) qp

+ 1

b∫a

2F1(y)f s(y)dy

qp

−aσαsqp

+σ−1(Γ(α + 1))sqp

b∫a

(2F1(x))qp

(1−s) (Iαa+;σ;ηf(x)) sqp dx

, (3.2.7)


3.2.2 G. H. Hardy-type inequalities for fractional derivatives

In the following theorem, new inequality for the Canavati-type fractional derivative

is given.

Theorem 3.2.6. Let 0 1− pq

and let the assumptions in

Lemma 1.2.4 be satisfied. Then for non-negative functions Dνaf and Dγ

af the following


0 ≤ ρ5(s) ≤ H5(s)−M5(s) ≤ H5(s),

72

where

ρ5(s) =(ν − γ)

qp

(ν − γ − 1) qp

+ 1

b∫a

(b− y)ν−γ−1+ pq (Dν

af(y))sdy

qp

− (Γ(ν − γ + 1))sqp

b∫a

(x− a)(ν−γ)q(1−s)

p (Dγaf(x))

sqp dx−M5(s),

M5(s) =q(ν − γ)(Γ(ν − γ + 1))s(

qp−1)

p

b∫a

(x− a)(ν−γ)(q−p)(1−s)

p (Dγaf(x))s(

qp−1)

×x∫a

r5(x, y)(x− y)ν−γ−1dy dx,

r5(x, y) =

∣∣∣∣ ∣∣∣∣(Dνaf(y))s −

(Γ(ν − γ + 1)

(x− a)γDγaf(x)

)s∣∣∣∣−s∣∣∣∣Γ(ν − γ + 1)


∣∣∣∣s−1 ∣∣∣∣Dνaf(y)− Γ(ν − γ + 1)


∣∣∣∣∣∣∣∣∣ ,

and

H5(s) = (b− a)(ν−γ) qp

(1−s)

(ν − γ)qp (b− a)

q((ν−γ)s−1)+pp

(ν − γ − 1) qp

+ 1

b∫a

(Dνaf(y))sdy

qp

− (Γ(ν − γ + 1))sqp

b∫a

(Dγaf(x))

sqp dx

. (3.2.8)


k(x, y) is given in (2.2.3) and Akf(x) = Γ(ν−γ+1)(x−a)ν−γ

Dγaf(x). Replace f by Dν

af . For

particular weight function u(x) = (x− a)(ν−γ)q

p , x ∈ (a, b) we get v(y) = ((ν − γ)(b−y)ν−γ−1+ p

q )/(((ν − γ − 1) qp

+ 1)pq ), then (3.2.3) takes the form

73

ρ5(s) =(ν − γ)

qp

(ν − γ − 1) qp

+ 1

b∫a


af(y))sdy

qp

− (Γ(ν − γ + 1))sqp

b∫a

(x− a)(ν−γ)q(1−s)

p (Dγaf(x))

sqp dx−M5(s).

Since (ν−γ)qp

(1− s) ≤ 0 and M5(s) ≥ 0, we obtain that

ρ5(s) ≤ (ν − γ)qp (b− a)(ν−γ−1) q

p+1

(ν − γ − 1) qp

+ 1

b∫a

(Dνaf(y))sdy

qp

−(b− a)(ν−γ)q

p(1−s)(Γ(ν − γ + 1))

sqp

b∫a

(Dγaf(x))

sqp dx−M5(s)

= H5(s)−M5(s)

≤ H5(s).


Theorem 3.2.7. Let 0 1 − pq

and assumptions

in Lemma 1.2.6 be satisfied. Then for non-negative functions Dν∗af and Dγ

∗af the

following inequality holds true:

0 ≤ ρ6(s) ≤ H6(s)−M6(s) ≤ H6(s),

where

ρ6(s) =(ν − γ)

qp

(ν − γ − 1) qp

+ 1

b∫a


∗af(y))sdy

qp

− (Γ(ν − γ + 1))sqp

b∫a

(x− a)(ν−γ)q(1−s)

p (Dγ∗af(x))

sqp dx−M6(s),

74

M6(s) =q(ν − γ)(Γ(ν − γ + 1))s(

qp−1)

p

b∫a

(x− a)(ν−γ)(q−p)(1−s)

p (Dγ∗af(x))s(

qp−1)

×x∫a

r6(x, y)(x− y)ν−γ−1dy dx,

r6(x, y) =

∣∣∣∣ ∣∣∣∣(Dν∗af(y))s −

(Γ(ν − γ + 1)

(x− a)γDγ∗af(x)

)s∣∣∣∣−s∣∣∣∣Γ(ν − γ + 1)


∣∣∣∣s−1

·∣∣∣∣Dν∗af(y)− Γ(ν − γ + 1)


∣∣∣∣∣∣∣∣∣ ,

and

H6(s) = (b− a)(ν−γ) qp

(1−s)

(ν − γ)qp (b− a)

q((ν−γ)s−1)+pp

(ν − γ − 1) qp

+ 1

b∫a

(Dν∗af(y))sdy

qp

− (Γ(ν − γ + 1))sqp

b∫a

(Dγ∗af(x))

sqp dx

. (3.2.9)


Chapter 4

Generalization of Hardy’s andrelated inequalities involvingmonotone convex functions

In this chapter, we present generalized Hardy’s and related inequalities involving

monotone convex functions. Also, the generalize and refined inequalities of classical

Polya-Knopp’s, Hardy-Hilbert, classical Hardy-Littlewood-Polya, Hardy-Hilbert-type

and Godunova are given. At the end, some refined Hardy-type inequalities for dif-

ferent kinds of fractional integrals and fractional derivatives are proved. The results

given in this chapter can be seen in [46].

4.1 Introduction

We recall some well-known integral inequalities. First inequality is classical Hardy’s

inequality.∞∫

0

1

x

x∫0

f(t)dt

p

dx ≤(

p

p− 1

)p ∞∫0

fp(x)dx, (4.1.1)

where 1 < p < ∞,R+ = (0,∞), and f ∈ Lp(R+) is a non-negative function. By

rewriting (4.1.1) with the function f1p instead of f and then by letting limit p→∞,

we get the limiting case of Hardy’s inequality known as Polya-Knopp’s inequality,

that is:∞∫

0

exp

1

x

x∫0

ln f(t)dt

dx ≤ e

∞∫0

f(x)dx, (4.1.2)

75

76

which holds for all positive functions f ∈ L1(R+). Two important inequalities related

to (4.1.1) are Hardy-Hilbert’s inequality,

∞∫0

∞∫0

f(x)

(x+ y)dx

p

dy ≤

(π

sin πp

)p ∞∫0

fp(x)dx, (4.1.3)

and the Hardy-Littlewood-Polya inequality

∞∫0

∞∫0

f(y)

maxx, ydx

p

dy ≤ (pp′)p

∞∫0

fp(y)dy, (4.1.4)

which hold for 1 < p < ∞ and non-negative f ∈ Lp(R+). The constants(

pp−1

)p, e,(

πsin π

p

)p, (pp′)p in the above inequalities are the best possible constants. For further

details we refer [14]-[21] (also see [5]) and the references therein.

Godunova in [31] (see also [65]) proved the inequality

∫Rn+

Φ

1

x1, · · ·xn

∫Rn+

l

(y1

x1

, · · ·ynxn

)f(y)dy

dx

x1, · · ·xn≤∫Rn+

Φ(f(y))

x1, · · ·xndx, (4.1.5)

holds for all non-negative measurable functions l : Rn+ → R+, such that

∫Rn+l(x)dx =

1, convex function Φ : [0,∞) → [0,∞), and a non-negative function f on Rn+, such

that the function x→∫Rn+

Φ(f(x))x1,···xn is integrable on Rn

+.

4.2 Main results

In Theorem 3.2.1, the refinement of the inequality (2.4.2) is given (see [26]), but in the

following theorem we provide another refinement of the inequality (2.4.2). Infact this

is more general result related to Hardy-type inequalities involving monotone convex

functions as an extension of the Theorem 3.2.1.

Theorem 4.2.1. Let 0 Akf(x) for y ∈ Ω2′ (Ω2

′ ⊂ Ω2) and ϕ : I → R is any function, such

77

that ϕ(x) ∈ ∂Φ(x) for all x ∈ Int I, then the inequality

∫Ω2


qp

−∫Ω1

u(x)Φqp (Akf(x)) dµ1(x)

≥ q

p

∣∣∣∣∣∫Ω1

u(x)

K(x)Φqp−1 (Akf(x))

∫Ω2

sgn(f(y)− Akf(x))k(x, y)[Φ(f(y))− Φ(Akf(x))

− |ϕ(Akf(x))| · (f(y)− Akf(x))]dµ2(y) dµ1(x)

∣∣∣∣∣ (4.2.1)


where Akf is defined by (2.3.1).

If Φ is a non-negative monotone concave, then the order of terms on the left-hand

side of (4.2.1) is reversed.

Proof. Consider the case, when Φ is non-decreasing on the interval I. Then∫Ω2

k(x, y)|Φ(f(y))− Φ(Akf(x))|dµ2(y)

=

∫Ω′2

k(x, y)[Φ(f(y))− Φ(Akf(x)))]dµ2(y)

+

∫Ω2\Ω

′2

k(x, y)[Φ(Akf(x))− Φ(f(y))]dµ2(y)

=

∫Ω′2

k(x, y)Φ(f(y))dµ2(y)−∫

Ω2\Ω′2

k(x, y)Φ(f(y))dµ2(y)

−Φ(Akf(x))

∫Ω′2

k(x, y)dµ2(y) + Φ(Akf(x))

∫Ω2\Ω

′2

k(x, y)dµ2(y)

=

∫Ω2

sgn(f(y)− Akf(x))k(x, y)[Φ(f(y))− Φ(Akf(x)]dµ2(y). (4.2.2)

78

Similarly, we can write∫Ω2

k(x, y)|f(y)− Akf(x)|dµ2(y)

=

∫Ω2

sgn(f(y)− Akf(x))k(x, y)(f(y)− Akf(x))dµ2(y). (4.2.3)

From (3.2.1), (4.2.2) and (4.2.3), we get (4.2.1).

The case, when Φ is non-increasing can be discussed in the similar way.

For p = q, we get the following result which is in fact the new version of the [25,

Theorem 2.1] involving monotone convex function and the function Φ not need to be

non-negative.

Corollary 4.2.2. Let Ω1,Ω2, µ1, µ2, u, k,K, and v be as in Theorem 3.2.1. If Φ is

a monotone convex on the interval I ⊆ R, f(y) > Akf(x) for y ∈ Ω2′ (Ω2

′ ⊂ Ω2)

and ϕ : I → R is any function, such that ϕ(x) ∈ ∂Φ(x) for all x ∈ Int I, then the

inequality∫Ω2

v(y)Φ (f(y)) dµ2(y)−∫Ω1


≥

∣∣∣∣∣∫Ω1

u(x)

K(x)

∫Ω2

sgn(f(y)− Akf(x))k(x, y)[Φ(f(y))− Φ(Akf(x))

− |ϕ(Akf(x))| · (f(y)− Akf(x))]dµ2(y) dµ1(x)

∣∣∣∣∣ (4.2.4)


where Akf is defined by (2.3.1).

If Φ is a monotone concave, then the order of terms on the left-hand side of (4.2.4)

is reversed.

Although the (4.2.1), holds for non-negative monotone convex functions some

choices of Φ are of our particular interest. Here, we consider the power and exponential

functions. Let the function Φ : R+ → R be defined by Φ(x) = xp. It is non-negative

and monotone function. Obviously, ϕ(x) = Φ′(x) = pxp−1, x ∈ R+, so Φ is convex

for p ∈ R \ [0, 1), concave for p ∈ (0, 1], and affine, that is, both convex and concave

for p = 1.

79

Corollary 4.2.3. Let Ω1,Ω2, µ1, µ2, u, k,K, and v be as in Theorem 3.2.1. Let p ∈ Rbe such that p 6= 0, f : Ω2 → R be a non-negative measurable function (positive for

p < 0), Akf be defined by (2.3.1) and

Mp,kf(x, y) = fp(y)− Apkf(x)− |p| · |Akf(x)|p−1(f(y)− Akf(x)) (4.2.5)

for x ∈ Ω1, y ∈ Ω2. If p ≥ 1 or p < 0, then the inequality∫Ω2

v(y)fp(y)dµ2(y)

qp

−∫Ω1

u(x)Aqkf(x)dµ1(x)

≥ q

p

∣∣∣∣∣∫Ω1

u(x)

K(x)(Akf(x))q−p

∫Ω2

sgn(f(y)− Akf(x))k(x, y)Mp,kf(x, y) dµ2(y) dµ1(x)

∣∣∣∣∣(4.2.6)

holds. If p ∈ (0, 1) relation (4.2.6) holds with

∫Ω1

u(x)Aqkf(x)dµ1(x)−

∫Ω2

v(y)fp(y)dµ2(y)

qp

on its left hand-side.

For the monotone convex function Φ : R+ → R defined by Φ(x) = ex, x ∈ R+ the

following result follows.

Corollary 4.2.4. Let Ω1,Ω2, µ1, µ2, u, k,K and v be defined as in Theorem 3.2.1 and

let p > 0. Let Gkf(x) be defined by

Gkf(x) := exp

1

K(x)

∫Ω2

k(x, y) ln f(y)dµ2(y)

, (4.2.7)

Pp,kf(x, y) = fp(y)−Gpkf(x)− p |Gp

k(x)| ln f(y)

Gkf(x)(4.2.8)

and f : Ω2 → R be a positive measurable function, f(y) > Gkf(x) for y ∈ Ω2′ (Ω2

′ ⊂

80

Ω2). Then the following inequality holds∫Ω2

v(y)fp(y)dµ2(y)

qp

−∫Ω1

u(x)Gqkf(x)dµ1(x)

≥ q

p

∣∣∣∣∣∫Ω1

u(x)

K(x)Gq−pk f(x)

∫Ω2

sgn(f(y)−Gkf(x))k(x, y)Pp,kf(x, y)dµ2(y) dµ1(x)

∣∣∣∣∣.(4.2.9)

Proof. Apply (4.2.1) with Φ : R → R, Φ(x) = ex, and replace the function f with

p ln f . Note that Gkf = exp(Ak(ln f)).

Here we give the results for one dimensional settings, with intervals in R and

Lebesgue measures. Also we give the related dual results.

Theorem 4.2.5. Let 0 < b ≤ ∞ and k : (0, b) × (0, b) → R be a non-negative

measurable function, such that

K(x) =:

x∫0

k(x, y)dy, x ∈ (0, b). (4.2.10)

Let u be a weight function such that the function x 7→ u(x)x·(k(x,y)K(x)

) qp

is integrable on

(y, b) for each fixed y ∈ (0, b), and let the function w : (0, b)→ R be defined by

w(y) = y

b∫y

(k(x, y)

K(x)

) qp

u(x)dx

x

pq

. (4.2.11)

If Φ is a non-negative monotone convex on the interval I ⊆ R and ϕ : I → R is that

ϕ(x) ∈ ∂Φ(x) for all x ∈ IntI, then the following inequality b∫0

w(y)Φ (f(y))dy

y

qp

−b∫

0

u(x)Φqp (Akf(x))

dx

x

≥ q

p

∣∣∣∣∣b∫

0

u(x)


x∫0

sgn(f(y)− Akf(x))k(x, y)[Φ(f(y)− Φ(Akf(x))

− |ϕ(Akf(x))|.(f(y)− Akf(x))]dy

dx

x

∣∣∣∣∣ (4.2.12)

81

holds for all measurable functions f : (0, b) → R, f(y) > Akf(x) for y ∈ I ′(I ′ ⊂(0, b)), such that f(y) ∈ I, for all fixed y ∈ (0, b) where Akf is defined by

Akf(x) :=1

K(x)

x∫0

k(x, y)f(y)dy, x ∈ (0, b). (4.2.13)

Proof. Similar to [25, Theorem 3.1].

By considering the power and exponential functions, we can give the following

results.

Corollary 4.2.6. Let 0 Akf(x) for

y ∈ I ′ (I ′ ⊂ (0, b)), where Akf and Mp,k be defined by (4.2.13) and (4.2.5) respectively.

If p > 1 or p < 0, then the following inequality holds

b∫0

w(y)fp(y)dy

y

qp

−b∫

0

u(x) (Akf(x))qdx

x

≥ q

p

∣∣∣∣∣b∫

0

u(x)

K(x)(Akf(x))q−p

x∫0

sgn(f(y)− Akf)k(x, y)Mp,kf(x, y) dydx

x

∣∣∣∣∣. (4.2.14)

If p ∈ (0, 1), then the order of terms on the left-hand side of relation (4.2.14) is

reversed.

Corollary 4.2.7. Let 0 1 and f be a positive measurable function on (0, b), f(y) > Gkf(x)

for y ∈ I ′ (I ′ ⊂ (0, b)). Then the following inequality holds:

b∫0

w(y)fp(y)dy

y

qp

−b∫

0

u(x)Gqkf(x)

dx

x

≥ q

p

∣∣∣∣∣b∫

0

u(x)

K(x)Gq−pk f(x)

x∫0

sgn(f(y)−Gkf(x))k(x, y)Pp,kf(x, y)dydx

x

∣∣∣∣∣,

82

where Gkf(x) is defined by

Gkf(x) := exp

1

K(x)

b∫0

k(x, y) ln f(y)dµ2(y)

, x ∈ (0, b). (4.2.15)

Now we give the dual results to Theorem 4.2.5 with some related corollaries.

Theorem 4.2.8. Let 0 < b ≤ ∞, let k : (b,∞) × (b,∞) → R be a non-negative

measurable function and K(x) be defined by

K(x) :=

∞∫x

k(x, y)dy, x ∈ (b,∞). (4.2.16)

Let u be a weight function such that the function x 7→ u(x)x·(k(x,y)

K(x)

) qp

is integrable on

(b, y) for each fixed y ∈ (b,∞), and let the function w : (b,∞)→ R be defined by

w(y) = y

y∫b

(k(x, y)

K(x)

) qp

u(x)dx

x

pq

. (4.2.17)

If Φ is a non-negative monotone convex on the interval I ⊆ R, and ϕ : I → R is that

ϕ(x) ∈ ∂Φ(x) for all x ∈ IntI, then the following inequality

∞∫b

w(y)Φ (f(y))dy

y

qp

−∞∫b

u(x)Φqp

(Akf(x)

) dxx

≥ q

p

∣∣∣∣∣∞∫b

u(x)

K(x)Φqp−1(Akf(x)

) ∞∫x

sgn(f(y)− Akf)k(x, y)[Φ(f(y))− Φ(Akf(x))

− |ϕ(Akf(x))|.(f(y)− Akf(x))]dy

dx

x

∣∣∣∣∣ (4.2.18)

holds for all measurable functions f : (b,∞) → R, such that f(y) ∈ I, for all fixed

y ∈ (b,∞), f(y) > Akf for y ∈ I ′′(I ′′ ⊂ (b,∞)), where Akf is defined by

Akf(x) :=1

K(x)

∞∫x

k(x, y)f(y)dy, x ∈ (b,∞). (4.2.19)

83

Proof. Similar to [25, Theorem 3.2].

Corollary 4.2.9. Let 0 1, f be a non-negative measurable function on (b,∞), f(y) > Akf for y ∈ I ′′(I ′′ ⊂(b,∞)), where Akf be defined by (4.2.19). Then the following inequality holds: ∞∫

b

w(y)fp(y)dy

y

qp

−∞∫b

u(x)(Akf(x)

)q dxx

≥ q

p

∣∣∣∣∣∞∫b

u(x)

K(x)

(Akf(x)

)q−p ∞∫x

sgn(f(y)− Akf)k(x, y)[fp(y)− (Akf(x))p

− p|Akf(x)|p−1.(f(y)− Akf(x))]dy

dx

x

∣∣∣∣∣.Corollary 4.2.10. Let 0 Gkf for y ∈ I ′′(I ′′ ⊂ (b,∞)).

Then the following inequality holds: ∞∫b

w(y)fp(y)dy

y

qp

−∞∫b

u(x)Gqkf(x)

dx

x

≥ q

p

∣∣∣∣∣∞∫b

u(x)

K(x)Gq−pk f(x)

∞∫x

sgn(f(y)− Gkf(x))k(x, y)[fp(y)− Gp

kf(x)

− p|Gpkf(x)|.(ln f(y)− ln Gkf(x))

]dy

dx

x

∣∣∣∣∣,where Gkf(x) is defined by

Gkf(x) := exp

1

K(x)

∞∫x

k(x, y) ln f(y)dy

x ∈ (b,∞). (4.2.20)

4.2.1 Hardy-Hilbert and Godunova’s inequalities

In this section, we give some examples for different kernels using Theorem 4.2.1. We

take Ω1 = Ω2 = R+, dµ1(x) = dx, dµ2(y) = dy and function Φ : R+ → R defined by

Φ(x) = xp, where p ∈ R, p 6= 0.

84

In our first example, we generalize and refine the Hardy-Hilbert’s inequality given

in (4.1.3).

Example 4.2.1. Let p, q, s ∈ R be such that qp> 0 and s−2

p, s−2

p> −1, and let

α ∈(− qp

(s−2p

+ 1), qp

(s−2p

+ 1))

. Denote

B1 = B

(q

p

(s− 2

p+ 1

)− α, q

p

(s− 2

p+ 1

)+ α

),

and

B2 = B

(s− 2

p+ 1,

s− 2

p+ 1

),

where B(·, ·) is the usual beta function and define k : R2+ → R by k(x, y) =

(yx

) s−2p (x+

y)−s, we obtain that K(x) = x1−sB2. For particular weight u(x) = xα−1, we get that

v(y) = yαpq−1B

pq

1 B−12 . Finally, let f be a non-negative function on R+ (positive for

p < 0) and Sf its generalized Steiltjes transform,

Sf(x) =

∞∫0

f(y)

(x+ y)sdy, x ∈ R+,

(see [8] and [69] for further details). Rewrite (4.2.1) with the above parameters and

with f(y)y2−sp instead of f(y), we get Ak(f(x)x

2−sp ) = x

s−2p

+1B−12 Sf(x) for 1 ≤ p ≤

q <∞ or −∞ < q ≤ p ≤ 0, the following inequality holds:

B1Bqp

2

∞∫0

yαpq−s+1fp(y)dy

qp

−∞∫

0

xα−1+(s−1)qp

+ qp (Sf(x))q dx

≥ q

p

∣∣∣∣∣Bp−12

∞∫0

xα+q−p+( s−2p )(q−p+1) (Sf(x))q−p

∞∫0

sgn(ys−2p f(y)− x

s−2p

+1B−12 Sf(x)

)

× ys−2p

(x+ y)s

[y2−sfp(y)−

(xs−2p

+1B−12 Sf(x)

)p− p

∣∣∣x s−2p

+1B−12 Sf(x)

∣∣∣p−1

×(y

2−sp f(y)− x

s−2p

+1B−12 Sf(x)

)]dy dx

∣∣∣∣∣. (4.2.21)

In next our example, similarly we generalize and refine Hardy-Littlewood-Polya’s

inequality given in (4.1.4).

85

Example 4.2.2. Let p, q, s, α ∈ R and that the function u and f be same as in the

Example 4.2.1. Define k : R2+ → R by k(x, y) =

(yx

) s−2p maxx, y−s and we get that

K(x) = x1−sL2. For particular weight function xα−1, we obtain v(y) = yαpq−1L

pq

1 L−12

and the transformation is

Lf(x) =

∞∫0

f(y)

maxx, ysdy, x ∈ R+.

Set

L1 =p2pqs

(αpp+ pq + qs− 2q)(pq + qs− αp2 − 2q),

and

L2 =pps

(p+ s− 2)(p+ s− 2).

Consider 1 ≤ p ≤ q < ∞ or −∞ < q ≤ p ≤ 0 and f(y)y2−sp instead of f(y), we

obtain Ak

(f(x)x

2−sp

)= x

s−2p

+1L−12 Lf(x), then the following inequality hold:

L1Lqp

2

∞∫0

yαpq−s+1fp(y)dy

qp

−∞∫

0

xα−1+(s−1)qp

+ qp (Lf(x))q dx

≥ q

p

∣∣∣∣∣Lp−12

∞∫0

xα+q−p+( s−2p )(q−p+1) (Lf(x))q−p

∞∫0

sgn(y

2−sp f(y)− x

s−2p

+1L−12 Lf(x)

)

× ys−2p

maxx, ys

[y2−sfp(y)−

(xs−2p

+1L−12 Lf(x)

)p− p

∣∣∣x s−2p

+1L−12 Lf(x)

∣∣∣p−1

×(y

2−sp f(y)− x

s−2p

+1L−12 Lf(x)

)]dy dx

∣∣∣∣∣.We continue with Hardy-Hilbert-type inequality, making the use of the well-known

reflection formula for the Digamma function ψ,

∞∫0

ln t

t− 1t−αdt = ψ(1− α) + ψ(α) =

π2

sin2 πα, α ∈ (0, 1),

and of the fact that

86

Z(a, b) =

∞∫0

tbe−at(1− e−t)bdt <∞, a ∈ R+, b ≥ 1.

More precisely, Z(a, b) = Γ(b + 1)φ∗b(1, b + 1, a), where φ∗µ is the so called unified

Riemann-Zeta function,

φ∗µ(z, s, a) =1

Γ(s)

∞∫0

ts−1e−at(1− ze−t)−µdt,

where µ ≥ 1, Re a > 0 and either |z| ≤ 1, z 6= 1, Re s > 0 or z = 1 and Re s > µ (for

more details see [32])

In upcoming example, we generalize and refine another Hardy-Hilbert’s inequality.

Example 4.2.3. Suppose that α ∈ (0, 1) and p, q, β ∈ R such that qp≥ 1 and α q

p+β ∈

(−1, qp− 1). Define the kernel k : R2

+ → R by k(x, y) = ln y−lnxy−x

(xy

)αwe get that

K(x) = π2

sin2πα. We also define the weight function u : R+ → R by u(x) = xβ, we

obtain v(y) = yβpq

+ pq−1M

pq

1 M−12 with the transformation

Mf(x) =

∞∫0

ln y − lnx

y − xf(y)dy, x ∈ R+,

where f is a non-negative function on R+ (positive, if p < 0),

M1 =

∞∫0

(ln t

t− 1

) qp

tαqp

+βdt = Z

(αq

p+ β + 1,

q

p

)+ Z

(q

p− αq

p− β − 1,

q

p

),

and

M2 =

∞∫0

ln t

t− 1t−αdt =

π2

sin2 πα.

Rewrite (4.2.1) with the above details and f(y)yα instead of f(y), we get that Ak (f(x)xα) =

xαM−12 Mf(x), then the following inequality holds:

M1Mqp

2

∞∫0

yαp+(β+1) pq−1fp(y)dy

qp

−∞∫

0

xαq+β (Mf(x))q dx

87

≥ q

p

∣∣∣∣∣Mp−12

∞∫0

xα(q−p+1)+β (Mf(x))q−p∞∫

0

sgn(yαf(y)− xαM−1

2 Mf(x))

× y−α ln y − lnx

y − x

[yαpfp(y)−

(xαM−1

2 Mf(x))p − p ∣∣xαM−1

2 Mf(x)∣∣p−1

×(yαf(y)− xαM−1

2 Mf(x)) ]

dy dx

∣∣∣∣∣.We complete this section with the general Godunova’s inequality for multidimen-

sional result. Let us take Ω1 = Ω2 = Rn+, dµ1(x) = dx, dµ2(y) = dy, let x

yand xy be

defined for x,y ∈ Rn+:

x = (x1, · · ·, xn) and y = (y1, ·, ·, ·, yn) and we denote

x

y=

(x1

y1

, · · ·, xnyn

)and xy = xy1

1 , · · ·, xynn .

Especially, x1 =∏n

i=1 xi and x−1 = (∏n

i=1 ui)−1.

Let the kernel k : Rn+ ×Rn

+ → R be of the form k(x,y) = l(yx), where l : Rn

+ → Ris a non-negative measurable function.

In our next result we obtain the generalization and refinement of the Godunova’s

inequality given in (4.1.5) using Theorem 4.2.1.

Theorem 4.2.11. Let 0 < p ≤ q < ∞, or −∞ < q ≤ p < 0, . Let l and u be

non-negative measurable functions on Rn+, such that 0 < L(x) = x1

∫Rn+l(y)dy < ∞

for all x ∈ Rn+, and let the function x → u(x)

(l(y

x)L(x)

) qp

is integrable on Rn+ for each

fixed y ∈ Rn+. Let the function v be defined on Rn

+ by

v(y) =

∫Rn+

u(x)

(l(yx

)L(x)

) qp

pq

dx.

If Φ is a non-negative monotone convex on the interval I ⊆ R, f(y) > Alf(x) for

y ∈ Rn+′ (Rn

+

′ ⊂ Rn+) and ϕ : I → R is any function, such that ϕ(x) ∈ ∂Φ(x) for all

88

x ∈ Int I, then the inequality∫Rn+

v(y)Φ (f(y)) dy

qp

−∫Rn+

u(x)Φqp (Alf(x)) dx

≥ q

p

∣∣∣∣∣∫Rn+

u(x)

L(x)Φqp−1 (Alf(x))

∫Rn+

sgn(f(y)− Alf(x))l(y

x

) [Φ(f(y))− Φ(Alf(x))

− |ϕ(Alf(x))| · (f(y)− Alf(x))]dy dx

∣∣∣∣∣ (4.2.22)

holds for all measurable functions f : Rn+ → R, with the values in I, for all fixed

y ∈ Rn+ where Alf is defined by

Alf(x) =1

L(x)

∫Rn+

l(yx

)L(x)

f(y)dy.

If Φ is a positive monotone concave function, then the order of terms on the left-hand

side of (4.2.22) is reversed.

4.2.2 G. H. Hardy-type inequalities for fractional integrals

involving monotone convex function

We continue our analysis about improvements by taking the non-negative difference

of left-hand side and right-hand side of inequality given in (4.2.1) by taking Φ : R+ →R+, Φ(x) = xs, s ≥ 1 as:

π(s) =

∫Ω2

v(y)f s(y)dµ2(y)

qp

−∫Ω1

u(x) (Akf(x))sqp dµ1(x)

− q

p

∣∣∣∣∣∫Ω1

u(x)


∫Ω2

sgn(f(y)− Akf(x))k(x, y)

[f s(y)− (Akf(x))s

− s|Akf(x)|s−1 · (f(y)− Akf(x))

]dµ2(y) dµ1(x)

∣∣∣∣∣. (4.2.23)

Our first result involve fractional integral of f with respect to another increasing

function g.

89

Theorem 4.2.12. Let 0 1 − pq, g be increasing

function on (a, b) such that g′ be continuous on (a, b], f(y) > Γ(α+1)(g(x)−g(a))α

Iαa+;gf(x) for

y ∈ I(I ⊂ (a, b)). Then the following inequality holds true:

π1(s) ≤ H1(s)−B1(s) ≤ H1(s),

where

π1(s) =αqp

((α− 1) qp

+ 1)

b∫a


qp

− (Γ(α + 1))sqp

b∫a

g′(x)(g(x)− g(a))αqp

(1−s) (Iαa+;gf(x)) sqp dx−B1(s),

B1(s) =αq(Γ(α + 1))s(

qp−1)

p

∣∣∣∣∣b∫

a

x∫a

sgn

(f(y)− Γ(α + 1)


)g′(x)g′(y)

× (g(x)− g(a))α(q−p)(1−s)

p

(g(x)− g(y))1−α

(Iαa+;gf(x)

)s( qp−1)[f s(y)−

(Γ(α + 1)


)s−s∣∣∣∣ Γ(α + 1)


∣∣∣∣s−1 (f(y)− Γ(α + 1)


)]dy dx

∣∣∣∣∣,and H1(s) is defined by (3.2.4).




Corollary 4.2.13. Let 0 1 − pq, s ≥ 1, f ≥ 0, f(y) >

Γ(α+1)(x−a)α

Iαa+f(x), for y ∈ I(I ⊂ (a, b)). Then the following inequality holds true:

π2(s) ≤ H2(s)−B2(s) ≤ H2(s),

90

where

π2(s) =αqp

(α− 1) qp

+ 1

b∫a

(b− y)α−1+ pq f s(y)dy

qp

− (Γ(α + 1))sqp

b∫a

(x− a)αqp

(1−s) (Iαa+f(x))sqp dx−B2(s),

B2(s) =αq(Γ(α + 1))s(

qp−1)

p

∣∣∣∣∣b∫

a

x∫a

sgn

(f(y)− Γ(α + 1)

(x− a)αIαa+f(x)

)(x− a)

α(q−p)(1−s)p

× (Iαa+f(x))s(qp−1) (x− y)α−1

[f s(y)−

(Γ(α + 1)

(x− a)αIαa+f(x)

)s−s∣∣∣∣Γ(α + 1)

(x− a)αIαa+

f(x)

∣∣∣∣s−1 (f(y)− Γ(α + 1)

(x− a)αIαa+f(x)

)]dy dx


Remark 4.2.1. If we take g(x) = log x, then Iαa+;xf(x) reduces to Jαa+f(x) left-sided

Hadamard-type fractional integral that is defined for α > 0 and we obtain the follow-

ing inequality:

π3(s) ≤ H3(s)−B3(s) ≤ H3(s),

where

π3(s) =αqp

(α− 1) qp

+ 1

b∫a

(log b− log y)α−1+ pq f s(y)

dy

y

qp

− (Γ(α + 1))sqp

b∫a

(log x− log a)αqp

(1−s) (Jαa+f(x)

) sqpdx

x−B3(s),

91

B3(s) =αq(Γ(α + 1))s(

qp−1)

p

∣∣∣∣∣b∫

a

x∫a

sgn

(f(y)− Γ(α + 1)


f(x)

)

×(Jαa+

f(x))s( q

p−1) (log x− log a)

α(q−p)(1−s)p

(log x− log y)1−α

[f s(y)−

(Γ(α + 1)


f(x)

)s−s∣∣∣∣ Γ(α + 1)


f(x)

∣∣∣∣s−1 (f(y)− Γ(α + 1)


f(x)

)]dy

y

dx

x



Theorem 4.2.14. Let 0 1 − pq, f ≥ 0, f(y) >

Γ(α+1)

(1−( ax)σ)α

2F1(x)Iαa+;σ;ηf(x) for y ∈ I(I ⊂ (a, b)) and 2F1(a, b; c; z) denotes the hy-

pergeometric function. Then the following inequality holds true:

0 ≤ π4(s) ≤ H4(s)−B4(s) ≤ H4(s),

where

π4(s) =αqpσ

pq−1

(α− 1) qp

+ 1

b∫a

yσ−12F1(y)(bσ − yσ)α−1+ p

q f s(y)dy

qp

− (Γ(α + 1))sqp

b∫a

xσαsqp

+σ−1 ((xσ − aσ)α 2F1(x))qp

(1−s) (Iαa+;σ;ηf(x)) sqp dx−B4(s),

B4(s) =ασq(Γ(α + 1))s(

qp−1)

p

∣∣∣∣∣b∫

a

x∫a

sgn

(f(y)− Γ(α + 1)(

1−(ax

)σ)α2F1(x)

Iαa+;σ;ηf(x)

)

× xσαs(qp−1)−ση+σ−1 ((xσ − aσ)α 2F1(x))( q

p−1)(1−s) (Iαa+;σ;ηf(x)

)s( qp−1)

× yση+σ−1

(xσ − yσ)1−α

[f s(y)−

(Γ(α + 1)(

1−(ax

)σ)α2F1(x)

Iαa+;σ;ηf(x)

)s

−s

∣∣∣∣∣ Γ(α + 1)(1−

(ax

)σ)α2F1(x)

Iαa+;σ;ηf(x)

∣∣∣∣∣s−1 (

f(y)− Γ(α + 1)(1−

(ax

)σ)α2F1(x)

Iαa+;σ;ηf(x)

) dy dx∣∣∣∣∣,and H4(s) is defined by (3.2.7).

92

4.2.3 G. H. Hardy-type inequalities for fractional derivatives

involving monotone convex function

Here we give the improvements for different fractional derivatives.

Theorem 4.2.15. Let 0 1 − pq, Dν

af(y) >Γ(ν−γ+1)(x−a)ν−γ

Dγaf(x) for y ∈ I(I ⊂ (a, b)) and let the assumptions in Lemma 1.2.4 be

satisfied. Then for non-negative functions Dνaf and Dγ

af the following inequality

holds true:

0 ≤ π5(s) ≤ H5(s)−B5(s) ≤ H5(s),

where

π5(s) =(ν − γ)

qp

(ν − γ − 1) qp

+ 1

b∫a


af(y))sdy

qp

− (Γ(ν − γ + 1))sqp

b∫a

(x− a)(ν−γ)q(1−s)

p (Dγaf(x))

sqp dx−B5(s),

B5(s) =q(ν − γ)(Γ(ν − γ + 1))s(

qp−1)

p

∣∣∣∣∣b∫

a

x∫a

sgn

(Dνaf(y)− Γ(ν − γ + 1)


)× (x− a)

(ν−γ)(q−p)(1−s)p (Dγ

af(x))s(qp−1) (x− y)ν−γ−1

×[(Dν

af(y))s −(

Γ(ν − γ + 1)

(x− a)γDγaf(x)

)s−s∣∣∣∣Γ(ν − γ + 1)


∣∣∣∣s−1 (Dνaf(y)− Γ(ν − γ + 1)


)]dy dx



Theorem 4.2.16. Let 0 1−pq, Dν∗af(y) > Γ(ν−γ+1)

(x−a)ν−γDγ∗af(x)

for y ∈ I ⊂ (a, b) and let the assumptions in Lemma 1.2.6 be satisfied. Then for non-

negative functions Dν∗af and Dγ

∗af the following inequality holds true:

π6(s) ≤ H6(s)−B6(s) ≤ H6(s),

93

where

π6(s) =(ν − γ)

qp

(ν − γ − 1) qp

+ 1

b∫a


∗af(y))sdy

qp

− (Γ(ν − γ + 1))sqp

b∫a

(x− a)(ν−γ)q(1−s)

p (Dγ∗af(x))

sqp dx−B6(s),

B6(s) =q(ν − γ)(Γ(ν − γ + 1))s(

qp−1)

p

∣∣∣∣∣b∫

a

x∫a

sgn

(Dν∗af(y)− Γ(ν − γ + 1)


)× (x− a)

(ν−γ)(q−p)(1−s)p (Dγ

∗af(x))s(qp−1) (x− y)ν−γ−1

×[(Dν∗af(y))s −

(Γ(ν − γ + 1)

(x− a)γDγ∗af(x)

)s−s∣∣∣∣Γ(ν − γ + 1)


∣∣∣∣s−1 (Dν∗af(y)− Γ(ν − γ + 1)


)]dy dx



Remark 4.2.2. For the case p = q we can get the similar improvements of the inequal-

ity given in (4.2.4) for different fractional integrals and fractional derivative.

Remark 4.2.3. In Chapter 2, Chapter 3 and Chapter 4, we discussed only the results

for left-sided fractional integrals. Similarly we can give the results for right-sided

fractional integrals but here we omit the details.

Chapter 5

Generalization of an inequality forintegral transforms with someresults related to exponentialconvexity

This chapter is devoted to a generalization of an inequality introduced by D. S.

Mitrinovic and J. Pecaric in 1988. Mean value theorems of Cauchy type and the

positive semi-definiteness of the matrices generated by the difference of the inequality

which implies the exponential convexity and logarithmic convexity is proved also, new

means of Cauchy type and their monotonicity is proved. Furthermore, n-exponential

convexity of the linear functionals obtained by taking the positive difference of Hardy-

type inequalities is discussed and some related examples are given. The results given

in this chapter can be seen in [37] and [47].

5.1 Generalization of an inequality for integral trans-

forms with kernel and related results

5.1.1 Introduction

Let k(x, t) be a non-negative kernel. Consider a function u : [a, b] → R where u ∈U(v, k) and the representation of u is

u(x) =

b∫a

k(x, t)v(t)dt

94

95

for any continuous function v on [a, b]. Throughout the chapter, it is assumed that

all integrals under consideration exist and that they are finite.

The following theorem is given in [56] (see also [65, p. 235]).

Theorem 5.1.1. Let ui ∈ U(v, k) (i = 1, 2) and r(t) ≥ 0 for all t ∈ [a, b]. Also let

Φ : R+ → R be a function such that Φ be convex and increasing for x > 0. Then

b∫a

r(x)Φ

(∣∣∣∣u1(x)

u2(x)

∣∣∣∣) dx ≤b∫

a

s(x)Φ

(∣∣∣∣v1(x)

v2(x)

∣∣∣∣) dx,where

s(x) = v2(x)

b∫a

r(t)k(t, x)

u2(t)dt, u2(t) 6= 0.

5.1.2 Main results

Our main result is given in the following theorem.

Theorem 5.1.2. Let ui ∈ U(v, k) (i = 1, 2) and r(x) ≥ 0 for all x ∈ [a, b]. Also let

I ⊆ R be an interval, Φ : I → R be convex and u1(x)/u2(x), v1(x)/v2(x) ∈ I. Then

b∫a

r(x)Φ

(u1(x)

u2(x)

)dx ≤

b∫a

q(x)Φ

(v1(x)

v2(x)

)dx, (5.1.1)

where

q(x) = v2(x)

b∫a

r(t)k(t, x)

u2(t)dt, u2(t) 6= 0. (5.1.2)

Proof. Since u1 =b∫a

k(x, t)v1(t)dt and v2(t) > 0, we have

b∫a

r(x)Φ

(u1(x)

u2(x)

)dx =

b∫a

r(x)Φ

1

u2(x)

b∫a

k(x, t)v1(t)dt

dx

=

b∫a

r(x)Φ

b∫a

k(x, t)v2(t)

u2(x)

v1(t)

v2(t)dt

dx.

96

By Jensen’s inequality and Fubini’s theorem, we get

b∫a

r(x)Φ

(u1(x)

u2(x)

)dx ≤

b∫a

r(x)

b∫a

k(x, t)v2(t)

u2(x)Φ

(v1(t)

v2(t)

)dt

dx

=

b∫a

Φ

(v1(t)

v2(t)

)v2(t)

b∫a

r(x)k(x, t)

u2(x)dx

dt

=

b∫a

q(t)Φ

(v1(t)

v2(t)

)dt.


Remark 5.1.1. If Φ is strictly convex on I and v1(x)v2(x)

is non-constant, then the inequality

in (5.1.1) is strict.

Remark 5.1.2. Let us note that the Theorem 5.1.1 follows from Theorem 5.1.2. Indeed,

let the conditions of Theorem 5.1.1 be satisfied, and let ui ∈ U(|v|, k), i.e.

u1(x) =

b∫a

k(x, t)|v1(t)| dt.

So, by Theorem 5.1.2, we have

b∫a

q(x)Φ

(∣∣∣∣v1(x)

v2(x)

∣∣∣∣) dx =

b∫a

q(x)Φ

(|v1(x)|v2(x)

)dx,

≥b∫

a

r(x) Φ

(u1(x)

u2(x)

)dx. (5.1.3)

On the other hand, Φ is increasing function, we have

Φ

(u1(x)

u2(x)

)= Φ

1

u2(x)

b∫a

k(x, t) |v1(t)| dt

≥ Φ

1

u2(x)

∣∣∣∣∣∣b∫

a

k(x, t)v1(t)dt

∣∣∣∣∣∣

97

= Φ

(|u1(x)|u2(x)

)= Φ

(∣∣∣∣u1(x)

u2(x)

∣∣∣∣) . (5.1.4)

From (5.1.3) and (5.1.4), we get (5.1.1).

Here, we will give the corollaries by taking the kernels of Riemann-Liouville frac-

tional integral, L∞ fractional derivative and Caputo fractional derivative respectively.

Corollary 5.1.3. Let ui ∈ C[a, b] (i = 1, 2) and r(x) ≥ 0 for all x ∈ [a, b]. Also let

I ⊆ R be an interval, Φ : I → R be convex, u1(x)/u2(x), Iαa+u1(x)/Iαa+u2(x) ∈ I and

u1(x), u2(x) have Riemann-Liouville fractional integral of order α > 0. Then

b∫a

r(x)Φ

(Iαa+u1(x)

Iαa+u2(x)

)dx ≤

b∫a

Φ

(u1(t)

u2(t)

)QI(t)dt,

where

QI(t) =u2(t)

Γ(α)

b∫t

r(x)(x− t)α−1

Iαa+u2(x)dx, Iαa+u2(x) 6= 0. (5.1.5)

Corollary 5.1.4. Let ui ∈ ACn[a, b] (i = 1, 2) and r(x) ≥ 0 for all x ∈ [a, b]. Also

let I ⊆ R be an interval, Φ : I → R be convex, u(n)1 (t)/u

(n)2 (t), Dα

∗au1(x)/Dα∗au2(x) ∈ I

and u1(x), u2(x) have Caputo fractional derivative of order α > 0. Then

b∫a

r(x)Φ

(Dα∗au1(x)

Dα∗au2(x)

)dx ≤

b∫a

Φ

(u

(n)1 (t)

u(n)2 (t)

)QD(t)dt,

where

QD(t) =u

(n)2 (t)

Γ(n− α)

b∫t

r(x)(x− t)n−α−1

Dα∗au2(x)

dx, Dα∗au2(x) 6= 0. (5.1.6)

Corollary 5.1.5. Let β > α ≥ 0, ui ∈ L1(a, b) (i = 1, 2) has an L∞ fractional

derivative Dβaui in [a, b] and r(x) ≥ 0 for all x ∈ [a, b]. Also let Dβ−k

a ui(a) =

0 for k = 1, . . . , [β] + 1 (i = 1, 2), Φ : I → R be convex and Dαau1(x)/Dα

au2(x),

Dβau1(x)/Dβ

au2(x) ∈ I. Then

b∫a

r(x)Φ

(Dαau1(x)

Dαau2(x)

)dx ≤

b∫a

Φ

(Dβau1(t)

Dβau2(t)

)QL(t)dt,

where

QL(t) =Dβau2(t)

Γ(β − α)

b∫t

r(x)(x− t)β−α−1

Dαau2(x)

dx, Dαau2(x) 6= 0. (5.1.7)

98

Lemma 5.1.6. Let f ∈ C2(I), I be a compact interval, such that

m ≤ f′′(x) ≤M, for all x ∈ I.

Consider two functions Φ1 and Φ2 defined as:

Φ1(x) =Mx2

2− f(x),

and

Φ2(x) = f(x)− mx2

2.

Then Φ1 and Φ2 are convex on I.

Theorem 5.1.7. Let f ∈ C2(I), I be a compact interval, ui ∈ U(v, k) (i = 1, 2),

and r(x) ≥ 0 for all x ∈ [a, b]. Also let u1(x)/u2(x), v1(x)/v2(x) ∈ I, v1(x)/v2(x) be

non-constant and q(x) be given in (5.1.2). Then there exists ξ ∈ I such that

b∫a

(q(x)f

(v1(x)

v2(x)

)− r(x)f

(u1(x)

u2(x)

))dx

=f′′(ξ)

2

b∫a

(q(x)

(v1(x)

v2(x)

)2

− r(x)

(u1(x)

u2(x)

)2)dx.

Proof. Since f ∈ C2(I) and I is a compact interval therefore, suppose m = min f′′,

M = max f′′. Using the Theorem 5.1.2 for the function Φ1 defined in Lemma 5.1.6,

we have

b∫a

r(x)

(M

2

(u1(x)

u2(x)

)2

− f(u1(x)

u2(x)

))dx

≤b∫

a

q(x)

(M

2

(v1(x)

v2(x)

)2

− f(v1(x)

v2(x)

))dx. (5.1.8)

Using Remark 5.1.1, (5.1.8) can be written as

2b∫a

(q(x)f

(v1(x)v2(x)

)− r(x)f

(u1(x)u2(x)

))dx

b∫a

(q(x)

(v1(x)v2(x)

)2

− r(x)(u1(x)u2(x)

)2)dx

≤M. (5.1.9)

99

We have the similar result for the function Φ2 defined in Lemma 5.1.6 as follows.

2b∫a

(q(x)f

(v1(x)v2(x)

)− r(x)f

(u1(x)u2(x)

))dx

b∫a

(q(x)

(v1(x)v2(x)

)2

− r(x)(u1(x)u2(x)

)2)dx

≥ m. (5.1.10)

Combining (5.1.9) and (5.1.10), we have

m ≤2

b∫a

(q(x)f

(v1(x)v2(x)

)− r(x)f

(u1(x)u2(x)

))dx

b∫a

(q(x)

(v1(x)v2(x)

)2

− r(x)(u1(x)u2(x)

)2)dx

≤M.

By Lemma 5.1.6, there exists ξ ∈ I such that

b∫a

(q(x)f

(v1(x)v2(x)

)− r(x)f

(u1(x)u2(x)

))dx

b∫a

(q(x)

(v1(x)v2(x)

)2

− r(x)(u1(x)u2(x)

)2)dx

=f′′(ξ)

2.

This is the claim of theorem.

Let us note that a generalized mean value theorem for fractional derivative was

given in [70]. Here we will give some related results as consequences of Theorem 5.1.7.

Corollary 5.1.8. Let f ∈ C2(I), I be a compact interval, ui ∈ C[a, b] (i = 1, 2) and

r(x) ≥ 0 for all x ∈ [a, b]. Also let u1(x)/u2(x), Iαa+u1(x)/Iαa+u2(x) ∈ I, u1(x)/u2(x)

be non-constant, QI(t) be given in (5.1.5) and u1(x), u2(x) has Riemman-Liouville

fractional integral of order α > 0. Then there exists ξ ∈ I such that

b∫a

(QI(x)f

(u1(x)

u2(x)

)− r(x)f

(Iαa+u1(x)

Iαa+u2(x)

))dx

=f′′(ξ)

2

b∫a

(QI(x)

(u1(x)

u2(x)

)2

− r(x)

(Iαa+u1(x)

Iαa+u2(x)

)2)dx.

Corollary 5.1.9. Let f ∈ C2(I), I be a compact interval, ui ∈ ACn[a, b] (i = 1, 2)

and r(x) ≥ 0 for all x ∈ [a, b]. Also let u(n)1 (t)/u

(n)2 (t) Dα

∗au1(x)/Dα∗au2(x) ∈ I,

100

u(n)1 (x)/u

(n)2 (x) be non-constant, QD(t) be given in (5.1.6) and u1(x), u2(x) have Ca-

puto derivative of order α > 0. Then there exists ξ ∈ I such that

b∫a

(QD(x)f

(u

(n)1 (x)

u(n)2 (x)

)− r(x)f

(Dα∗au1(x)

Dα∗au2(x)

))dx

=f′′(ξ)

2

b∫a

QD(x)

(u

(n)1 (x)

u(n)2 (x)

)2

− r(x)

(Dα∗au1(x)

Dα∗au2(x)

)2 dx.

Corollary 5.1.10. Let β > α ≥ 0, f ∈ C2(I), I be a compact interval, ui ∈ L1(a, b)

(i = 1, 2) has an L∞ fractional derivative and r(x) ≥ 0 for all x ∈ [a, b]. Let

Dβ−ka ui(a) = 0 for k = 1, . . . , [β] + 1 (i = 1, 2), Dα

au1(x)/Dαau2(x), Dβ

au1(x)/Dβau2(x)

∈ I, Dβau1(x)/Dβ

au2(x) be non-constant and QL(t) be given in (5.1.7). Then there

exists ξ ∈ I such that

b∫a

(QL(x)f

(Dβau1(x)

Dβau2(x)

)− r(x)f

(Dαau1(x)

Dαau2(x)

))dx

=f′′(ξ)

2

b∫a

(QL(x)

(Dβau1(x)

Dβau2(x)

)2

− r(x)

(Dαau1(x)

Dαau2(x)

)2)dx.

Theorem 5.1.11. Let f, g ∈ C2(I), I be a compact interval, ui ∈ U(v, k) (i = 1, 2)

and r(x) ≥ 0 for all x ∈ [a, b]. Also let u1(x)/u2(x), v1(x)/v2(x) ∈ I, v1(x)/v2(x) be

non-constant and q(x) be given in (5.1.2). Then there exists ξ ∈ I such that

b∫a

q(x)f(v1(x)v2(x)

)dx−

b∫a

r(x)f(u1(x)u2(x)

)dx

b∫a

q(x)g(v1(x)v2(x)

)dx−

b∫a

r(x)g(u1(x)u2(x)

)dx

=f′′(ξ)

g′′(ξ).

It is provided that denominators are not equal to zero.

Proof. Let us take a function h ∈ C2(I) defined as

h(x) = c1f(x)− c2g(x),

where

c1 =

b∫a

q(x)g

(v1(x)

v2(x)

)dx−

b∫a

r(x)g

(u1(x)

u2(x)

)dx,

101

and

c2 =

b∫a

q(x)f

(v1(x)

v2(x)

)dx−

b∫a

r(x)f

(u1(x)

u2(x)

)dx.

By Theorem 5.1.7 with f = h, we have

0 =(c1

2f′′(ξ)− c2

2g′′(ξ)) b∫

a

q(x)

(v1(x)

v2(x)

)2

dx−b∫

a

r(x)

(u1(x)

u2(x)

)2

dx

.

Using Remark 5.1.1, we getc2

c1

=f′′(ξ)

g′′(ξ).

This is the claim of the theorem.

Let us note that a generalized Cauchy mean valued theorem for fractional deriva-

tive was given in [67]. Here we will give some related results as consequences of

Theorem 5.1.11.

Corollary 5.1.12. Let f, g ∈ C2(I), I be a compact interval, ui ∈ C[a, b] (i =

1, 2) and r(x) ≥ 0 for all x ∈ [a, b]. Also let u1(x)/u2(x), Iαa+u1(x)/Iαa+u2(x) ∈ I,

u1(x)/u2(x) be non-constant, QI(t) be given in (5.1.5) and u1(x), u2(x) have Riemann-

Liouville fractional derivative of order α > 0. Then there exists ξ ∈ I such that

b∫a

QI(x)f(u1(x)u2(x)

)dx−

b∫a

r(x)f(Iαa+u1(x)

Iαa+u2(x)

)dx

b∫a

QI(x)g(u1(x)u2(x)

)dx−

b∫a

r(x)g(Iαa+u1(x)

Iαa+u2(x)

)dx

=f′′(ξ)

g′′(ξ).


Corollary 5.1.13. Let f, g ∈ C2(I), I be a compact interval, ui ∈ ACn[a, b] (i =

1, 2) and r(x) ≥ 0 for all x ∈ [a, b]. Also let u(n)1 (t)/u

(n)2 (t), Dα

∗au1(x)/Dα∗au2(x) ∈

I, u(n)1 (x)/u

(n)2 (x) be non-constant, QD(t) be given in (5.1.6) and u1(x), u2(x) have

Caputo fractional derivative of order α > 0. Then there exists ξ ∈ I such that

b∫a

QD(x)f

(u

(n)1 (x)

u(n)2 (x)

)dx−

b∫a

r(x)f(Dα∗au1(x)Dα∗au2(x)

)dx

b∫a

QD(x)g

(u

(n)1 (x)

u(n)2 (x)

)dx−

b∫a

r(x)g(Dα∗au1(x)Dα∗au2(x)

)dx

=f′′(ξ)

g′′(ξ).


102

Corollary 5.1.14. Let β > α ≥ 0, f, g ∈ C2(I), I be a compact interval, ui ∈L1(a, b) (i = 1, 2) has an L∞ fractional derivative Dβ

aui in [a, b] and r(x) ≥ 0 for all

x ∈ [a, b]. Also let Dβ−ka ui(a) = 0 for k = 1, . . . , [β] + 1 (i = 1, 2), Dα

au1(x)/Dαau2(x),

Dβau1(x)/Dβ

au2(x) ∈ I, Dβau1(x)/Dβ

au2(x) be non-constant and QL(t) be given in

(5.1.7). Then there exists ξ ∈ I such that

b∫a

QL(x)f(Dβau1(x)

Dβau2(x)

)dx−

b∫a

r(x)f(Dαa u1(x)Dαa u2(x)

)dx

b∫a

QL(x)g(Dβau1(x)

Dβau2(x)

)dx−

b∫a

r(x)g(Dαa u1(x)Dαa u2(x)

)dx

=f′′(ξ)

g′′(ξ).


Corollary 5.1.15. Let I ⊆ R+, I be a compact interval, ui ∈ U(v, k) (i = 1, 2)

and r(x) ≥ 0 for all x ∈ [a, b]. Let u1(x)/u2(x), v1(x)/v2(x) ∈ I, v1(x)/v2(x) be non-

constant and q(x) be given in (5.1.2). Then for s, t ∈ R\0, 1 and s 6= t, there exists

ξ ∈ I such that

ξ =

s(s− 1)

t(t− 1)

b∫a

q(x)(v1(x)v2(x)

)tdx−

b∫a

r(x)(u1(x)u2(x)

)tdx

b∫a

q(x)(v1(x)v2(x)

)sdx−

b∫a

r(x)(u1(x)u2(x)

)sdx

1t−s

. (5.1.11)

Proof. We set f(x) = xt and g(x) = xs, t 6= s and s, t 6= 0, 1. By Theorem 5.1.11,

we haveb∫a

q(x)(v1(x)v2(x)

)tdx−

b∫a

r(x)(u1(x)u2(x)

)tdx

b∫a

q(x)(v1(x)v2(x)

)sdx−

b∫a

r(x)(u1(x)u2(x)

)sdx

=t(t− 1)ξt−2

s(s− 1)ξs−2.

This implies

ξt−s =s(s− 1)

t(t− 1)

b∫a

q(x)(v1(x)v2(x)

)tdx−

b∫a

r(x)(u1(x)u2(x)

)tdx

b∫a

q(x)(v1(x)v2(x)

)sdx−

b∫a

r(x)(u1(x)u2(x)

)sdx

.


103

Remark 5.1.3. Since the function ξ → ξt−s is invertible and from (5.1.11), we have

m ≤

s(s− 1)

t(t− 1)

b∫a

q(x)(v1(x)v2(x)

)tdx−

b∫a

r(x)(u1(x)u2(x)

)tdx

b∫a

q(x)(v1(x)v2(x)

)sdx−

b∫a

r(x)(u1(x)u2(x)

)sdx

1t−s

≤M. (5.1.12)

Now we can suppose that f′′/g′′

is invertible function, then from (5.1.11) we have

ξ =

(f′′

g′′

)−1

b∫a

q(x)f(v1(x)v2(x)

)dx−

b∫a

r(x)f(u1(x)

u2(x)

)dx

b∫a

q(x)g(v1(x)v2(x)

)dx−

b∫a

r(x)g(u1(x)u2(x)

)dx

. (5.1.13)

We see that on the right hand side of the (5.1.13) is mean, then for distinct s, t ∈ Rit can be written as

Ms,t =

(∧t∧s

) 1t−s

(5.1.14)

as mean in broader sense. Moreover we can extend these means, so in limiting cases

for s, t 6= 0, 1,

limt→s

Ms,t = Ms,s

= exp

b∫a

q(x)(v1(x)v2(x)

)slog(v1(x)v2(x)

)dx−

b∫a

r(x)(u1(x)u2(x)

)slog(u1(x)u2(x)

)dx

b∫a

q(x)(v1(x)v2(x)

)sdx−

b∫a

r(x)(u1(x)u2(x)

)sdx

− 2s− 1

s(s− 1)

,

lims→0

Ms,s = M0,0 = exp

b∫a

q(x) log2(v1(x)v2(x)

)dx−

b∫a

r(x) log2(u1(x)u2(x)

)dx

2

[b∫a

q(x) log(v1(x)v2(x)

)dx−

b∫a

r(x) log(u1(x)u2(x)

)dx

] + 1

,

lims→1

Ms,s = M1,1

= exp

b∫a

q(x)(v1(x)v2(x)

)log2

(v1(x)v2(x)

)dx−

b∫a

r(x)(u1(x)u2(x)

)log2

(u1(x)u2(x)

)dx

2

[b∫a

q(x)(v1(x)v2(x)

)log(v1(x)v2(x)

)dx−

b∫a

r(x)(u1(x)u2(x)

)log(u1(x)u2(x)

)dx

] − 1

,

104

limt→0

Ms,t = Ms,0 =

b∫a

q(x)(v1(x)v2(x)

)sdx−

b∫a

r(x)(u1(x)u2(x)

)sdx[

b∫a

q(x) log(v1(x)v2(x)

)dx−

b∫a

r(x) log(u1(x)u2(x)

)dx

]s(s− 1)

1s

,

limt→1

Ms,t = Ms,1

=

[b∫a

q(x)(v1(x)v2(x)

)log(v1(x)v2(x)

)dx−

b∫a

r(x)(u1(x)u2(x)

)log(u1(x)u2(x)

)dx

]s(s− 1)

b∫a

q(x)(v1(x)v2(x)

)sdx−

b∫a

r(x)(u1(x)u2(x)

)sdx

1

1−s

.

Remark 5.1.4. In the case of Riemann-Liouville fractional integral of order α > 0,

we will use the notation Ms,t instead of Ms,t and we replace vi(x) to ui(x), ui(x) to

Iαa+ui(x) and q(x) to QI(x).

Remark 5.1.5. In the case of Caputo fractional derivative of order α > 0, we will use

the notation Ms,t instead of Ms,t and we replace vi(x) to u(n)i (x), ui(x) to Dα

∗aui(x)

and q(x) to QD(x).

Remark 5.1.6. In the case of L∞ fractional derivative, we will use the notation Ms,t

instead of Ms,t and we replace vi(x) to Dβaui(x), ui(x) to Dα

aui(x) and q(x) to QL(x).

5.1.3 Exponential Convexity

Theorem 5.1.16. Let ui ∈ U(v, k) (i = 1, 2), ui(x), vi(x) > 0, (i = 1, 2), r(x) ≥ 0

for all x ∈ [a, b], q(x) be given in (5.1.2) and

∧t =

b∫a

q(x)ϕt

(v1(x)

v2(x)

)dx−

b∫a

r(x)ϕt

(u1(x)

u2(x)

)dx. (5.1.15)

Then the following statements are valid:

(a) For n ∈ N and si ∈ R, i = 1, . . . , n, the matrix[∧ si+sj

2

]ni,j=1

is positive semi-

definite matrix. Particulary

det[∧ si+sj

2

]ki,j=1≥ 0 for k = 1, ...n.

(b) The function s 7→ ∧s is exponentially convex on R.

105

(c) The function s 7→ ∧s is log-convex on R and the following inequality holds for

−∞ < r < s < t <∞ :

∧t−rs ≤ ∧t−sr ∧s−rt . (5.1.16)

Proof. (a) Here we define a new function µ,

µ(x) =k∑

i,j=1

aiajϕsij(x)

for k = 1, . . . , n, ai ∈ R, sij ∈ R, where sij =si+sj

2.

µ′′(x) =

n∑i,j=1

aiajxsij−2 =

(n∑i=1

aixsi2−1

)2

≥ 0.

This shows that µ(x) is convex for x ≥ 0. Using Theorem 5.1.2, we have

k∑i,j=1

aiaj∧sij ≥ 0.

From the above result, it shows that the matrix[∧ si+sj

2

]ni,j=1

is positive semi-definite

matrix. Specially, we get

det[∧ si+sj

2

]ki,j=1≥ 0 for all k = 1, . . . , n.

(b) Since

lims→1∧s = ∧1 and lim

s→0∧s = ∧0,

it follows that ∧s is continuous for s ∈ R. Then by using Definition 1.1.4, we get the

exponential convexity of the function s 7→ ∧s.

(c) Since ∧s is continuous for s ∈ R and using Definition 1.1.3, we get ∧s is log-convex.

Now by Remark 1.1.1 with f(t) = log∧t and r, s, t ∈ R such that r < s < t, we get

log∧t−rs ≤ log∧t−sr + log∧s−rt ,

which is equivalent to (5.1.16).

106

Corollary 5.1.17. Let ui ∈ C[a, b] (i = 1, 2) and r(x) ≥ 0 for all x ∈ [a, b]. Also let

u1(x)/u2(x), Iαa+u1(x)/Iαa+u2(x) ∈ R+, u1(x), u2(x) has Riemann-Liouville fractional

integral of order α > 0, let QI(t) be given in (5.1.5) and

∧t =

b∫a

QI(x)ϕt

(u1(x)

u2(x)

)dx−

b∫a

r(x)ϕt

(Iαa+u1(x)

Iαa+u2(x)

)dx.

Then the statement of Theorem 5.1.16 with ∧t instead of ∧t is valid.

Corollary 5.1.18. Let ui ∈ ACn[a, b] (i = 1, 2) and r(x) ≥ 0 for all x ∈ [a, b].

Also let u(n)1 (t)/u

(n)2 (t), Dα

∗au1(x)/Dα∗au2(x) ∈ R+, u1(x), u2(x) has Caputo fractional

derivative of order α > 0, let QD(t) be given in (5.1.6) and

∧t =

b∫a

QD(x)ϕt

(u

(n)1 (x)

u(n)2 (x)

)dx−

b∫a

r(x)ϕt

(Dα∗au1(x)

Dα∗au2(x)

)dx.


Corollary 5.1.19. Let β > α ≥ 0, ui ∈ L1 (a, b) (i = 1, 2) has L∞ fractional

derivative and r(x) ≥ 0 for all x ∈ [a, b]. Also let Dβ−ka ui(a) = 0 for k = 1, . . . , [β] +

1 (i = 1, 2), Dαau1(x)/Dα

au2(x), Dβau1(x)/Dβ

au2(x) ∈ R+, QL(t) be given in (5.1.7) and

∧t =

b∫a

QL(x)ϕt

(Dβau1(x)

Dβau2(x)

)dx−

b∫a

r(x)ϕt

(Dαau1(x)

Dαau2(x)

)dx.


In the following theorem we prove the monotonicity of Ms,t defined in (5.1.14).

Theorem 5.1.20. Let the assumption of Theorem 5.1.16 be satisfied, also let ∧t be

defined in (5.1.15) and t, s,m, n ∈ R such that s ≤ m, t ≤ n. Then following inequality

is true:

Ms,t ≤Mm,n. (5.1.17)

Proof. For a convex function ϕ, using the Remark 1.1.1, we get the following inequality

ϕ(x2)− ϕ(x1)

x2 − x1

≤ ϕ(y2)− ϕ(y1)

y2 − y1

107

with x1 ≤ y1, x2 ≤ y2, x1 6= x2, y1 6= y2. Since by Theorem 5.1.16, we get that ∧t is

log-convex. We set ϕ(t) = log∧t, x1 = s, x2 = t, y1 = m, y2 = n, s 6= t, m 6= n.

Therefore, we getlog∧t − log∧s

t− s≤ log∧n − log∧m

n−m,

log

(∧t∧s

) 1t−s

≤ log

(∧n∧m

) 1n−m

, (5.1.18)

which is equivalent to (5.1.17) for s 6= t, m 6= n.

For s = t, m = n, we get that the required result by taking limit in (5.1.18).

Corollary 5.1.21. Let ui ∈ C[a, b] (i = 1, 2) and let the assumption of Corollary

5.1.17 be satisfied, also let ∧t be defined by (5.1.17). For t, s,m, n ∈ R such that

s ≤ m, t ≤ n, then following inequality holds:

Ms,t ≤ Mm,n.

Corollary 5.1.22. Let ui ∈ ACn[a, b] (i = 1, 2) and let the assumption of Corollary

5.1.18 be satisfied, also let ∧t defined by (5.1.18). For t, s,m, n ∈ R such that s ≤ m,

t ≤ n, then following inequality holds:

Ms,t ≤ Mm,n.

Corollary 5.1.23. Let β > α ≥ 0, ui ∈ L1(a, b) (i = 1, 2) and let the assumption of

Corollary 5.1.19 be satisfied, also let ∧t be defined by (5.1.19). For t, s,m, n ∈ R such

that s ≤ m, t ≤ n, then following inequality holds:

Ms,t ≤ Mm,n.

5.2 n-Exponential convexity of Hardy-type func-

tionals

In this part, we discuss and produce the exponential convexity and n-exponential

convexity of the linear functionals obtained by taking the positive difference of Hardy-

type inequalities.

Under assumptions of the Theorem 2.3.1 and Theorem 5.1.2, we define linear

functionals by taking the positive differences of left hand-side and right hand-side of

the inequalities stated in (2.3.2) and (5.1.1) respectively:

∆1(Φ) =

∫Ω2

v(y)Φ (f(y)) dµ2(y)−∫Ω1

u(x)Φ (Akf(x)) dµ1(x). (5.2.1)

108

∆2(Φ) =

b∫a

q(x)Φ

(v1(x)

v2(x)

)dx−

b∫a

r(x)Φ

(u1(x)

u2(x)

)dx. (5.2.2)

The discrete results about Hardy-type inequalities are given in [24, Theorem 2.1].

Here, we consider a special case of [24, Theorem 2.1], that is for convex functions this

result holds.

Theorem 5.2.1. Let M,N ∈ N and let non-negative real numbers um, vn, kmn, where

m ∈ NM , n ∈ NN , be such that

Km =N∑n=1

kmn > 0, m ∈ NM ,

and

vn =M∑m=1

umkmnKm

, n ∈ NN .

If Φ is a convex function on the interval I ⊆ R, then the inequality

N∑n=1

vnΦ(an)−M∑m=1

umΦ(Am) ≥ 0 (5.2.3)

holds for all real numbers an ∈ I, for n ∈ NN , where

Am =1

Km

N∑n=1

kmnan.

We define linear functional from (5.2.3) as:

∆3(Φ) =N∑n=1

vnΦ(an)−M∑m=1

umΦ(Am). (5.2.4)

The main purpose of this part is to discuss the n-exponential convexity of the three

non-negative Hardy-type linear functionals obtained by taking the positive difference

of Hardy-type inequalities stated in [52] (see also [28]), [37], [24] and defined by (5.2.1),

(5.2.2) and (5.2.4).

109

5.2.1 Main results

First we give some necessary details about the divided differences.

Let I ⊆ R be an interval and f : I → R be a function. Then for distinct points

zi ∈ I, (i = 0, 1, 2), the divided differences of first and second order are defined by:

[zi, zi+1; f ] =f(zi+1)− f(zi)

zi+1 − zi, (i = 0, 1),

[z0, z1, z2; f ] =[z1, z2; f ]− [z0, z1; f ]

z2 − z0

. (5.2.5)

One can observe that if for all z0, z1 ∈ I, [z0, z1, f ] ≥ 0, then f is increasing on I

and if for all z0, z1, z2 ∈ I, [z0, z1, z2; f ] ≥ 0, then f is convex on I.

Now we will produce n-exponentially convex and exponentially convex functions

applying functionals ∆i, (i = 1, 2, 3) on a given family with the same property. In

the sequel J and I will be intervals in R. The proofs of our results are similar to the

proofs in [66] but for completeness of results and for the reader’s convenience we will

also give them.

Theorem 5.2.2. Let Γ = Φp : p ∈ J be a family of functions defined on I such

that the function p 7→ [z0, z1, z2; Φp] is n-exponentially convex in the Jensen sense on

J for every three distinct points z0, z1, z2 ∈ I. Let ∆i (i = 1, 2, 3) be linear functionals

defined by (5.2.1), (5.2.2) and (5.2.4). Then the functions p 7→ ∆i(Φp) (i = 1, 2, 3)

are n-exponentially convex in the Jensen sense on J. If the functions p 7→ ∆i(Φp) are

continuous on J , then it is n-exponentially convex on J .

Proof. For ai ∈ R, i = 1, . . . , n and pi ∈ J, i = 1, . . . , n, we define the function

Υ(z) =n∑

i,j=1

aiajΦ pi+pj2

(z).

Using the assumption that the function p 7→ [z0, z1, z2; Φp] is n-exponentially convex

in the Jensen sense, we have

[z0, z1, z2; Υ] =n∑

i,j=1

aiaj[z0, z1, z2; Φ pi+pj2

] ≥ 0,

which shows that Υ is convex on I and therefore we have ∆i(Υ) ≥ 0 for (i = 1, 2, 3).

Hencen∑

i,j=1

aiaj∆i(Φ pi+pj2

) ≥ 0.

110

We conclude that the function p 7→ ∆i(Φp) for (i = 1, 2, 3) are n-exponentially convex

in Jensen sense on J .

If the functions p 7→ ∆i(Φp) for (i = 1, 2, 3) are also continuous on J , then

p 7→ ∆i(Φp) is n-exponentially convex by definition.

As a direct consequence of the above theorem, we can give the following corollary.

Corollary 5.2.3. Let Γ = Φp : I → R, p ∈ J ⊆ R be a family of functions such

that the function p 7→ [z0, z1, z2; Φp] is exponentially convex in the Jensen sense on J

for every three distinct points z0, z1, z2 ∈ I. Let ∆i (i = 1, 2, 3) be linear functionals

defined by (5.2.1), (5.2.2) and (5.2.4). Then p 7→ ∆i(Φp) are exponentially convex in

the Jensen sense on J. If the functions p 7→ ∆i(Φp) are continuous on J , then these

are exponentially convex on J .

Using analogous arguing as in the proof of [66, Corollary 3.11], we have the fol-

lowing corollary.

Corollary 5.2.4. Let Γ = Φp : I → R, p ∈ J ⊆ R be a family, such that the

function p → [z0, z1, z2; Φp] is 2-exponentially convex in the Jensen sense on J for

every three distinct points z0, z1, z2 ∈ I. Let ∆i (i = 1, 2, 3) be a linear functionals

defined in (5.2.1), (5.2.2) and (5.2.4). Then the following statements hold:

(i) If the function p 7→ ∆i(Φp) is continuous on J , then it is 2-exponentially convex

function on J , thus log-convex on J and for p, q, r ∈ I such that p < q < r, we

have

∆i(Φq)r−p ≤ ∆i(Φp)

r−q∆i(Φr)q−p, (i = 1, 2, 3).

(ii) If the function p 7→ ∆i(Φp) is strictly positive and differentiable on J , then for

every p, q,m, n ∈ J such that p ≤ m, q ≤ n, we have

Bp,q(f,∆i; Γ) ≤ Bm,n(f,∆i; Γ), (i = 1, 2, 3), (5.2.6)

where

Bp,q(f,∆i; Γ) =

(

∆i(Φp)

∆i(Φq)

) 1p−q

, p 6= q,

exp

(ddp

(∆i(Φp))

∆i(Φp)

), p = q,

(5.2.7)

for Φp,Φq ∈ Γ.

111

Proof. (i) This can be obtained as a direct consequence of Theorem 5.2.2 and Remark

1.1.3.

(ii) Since by (i) the function p 7→ ∆i(Φp) for (i = 1, 2, 3) is log-convex on J, that is

the function p 7→ log ∆i(Φp) for (i = 1, 2, 3) is convex on J. Applying Remark 1.1.1,

we obtainlog ∆i(Φp)− log ∆i(Φq)

p− q≤ log ∆i(Φm)− log ∆i(Φn)

m− n(5.2.8)

for p ≤ m, q ≤ n, p 6= q, m 6= n and we conclude that

Bp,q(f,∆i; Γ) ≤ Bm,n(f,∆i; Γ), (i = 1, 2, 3).

Cases for p = q, m = n follows from (5.2.8) as limiting case.

Remark 5.2.1. Note that the results of Theorem 5.2.2, Corollary 5.2.3 and Corollary

5.2.4 still hold when two of the points z0, z1, z2 ∈ I coincides for a family of differ-

entiable functions Φp such that p 7→ [z0, z1, z2; Φp] is n-exponentially convex in the

Jensen sense (exponentially convex in the Jensen sense), further, they still hold when

all three point coincide for a family of twice differentiable functions with the same

property. The proofs are obtained using (5.2.1) and some facts about the exponential

convexity.

5.2.2 Examples

Example 5.2.1. Consider a family of functions

Γ1 = gp : (0,∞)→ (0,∞) : p ∈ (0,∞),

defined by

gp(t) =e−t√p

p.

Since p 7→ d2gp(t)

dt2= e−t

√p is the Laplace transform of a non-negative function, it is

exponentially convex (see [49], also see [71]). Clearly gp are convex functions for each

p > 0. It is obvious that ∆i(gp) for (i = 1, 2, 3) are continuous. It is easy to prove

that the function p 7→ [z0, z1, z2; gp] is also exponentially convex for arbitrary points

z0, z1, z2 ∈ I. For this family of functions, Bp,q(f,∆i; Γ1) becomes

Bp,q(f,∆i(gp); Γ1) =

(

∆i(gp)

∆i(gq)

) 1p−q

, p 6= q;

exp(− ∆i(id·gp)

2√p∆i(gp)

− 1p

), p = q,

and from (5.2.6) it follows that the function Bp,q(f,∆i; Γ1) are monotonous in param-

eters p and q.

112

Example 5.2.2. Let

Γ2 = hp : (0,∞)→ (0,∞) : p ∈ (0,∞),

be a family of functions defined by

hp(t) =

p−t

(ln p)2 , p ∈ R+ \ 1,t2

2, p = 1.

Since p 7→ d2

dt2hp(t) = p−t is the Laplace transform of a non-negative function (see

[71]), it is exponentially convex. Obviously hp are convex functions for every p > 0.

It is easy to prove that the function p 7→ [z0, z1, z2;hp] is also exponentially convex for

arbitrary points z0, z1, z2 ∈ I. Using Corollary 5.2.3, it follows that p 7→ ∆i(hp) for

(i = 1, 2, 3) are exponentially convex (it is easy to verify that these are continuous)

and thus log-convex. From (5.2.7), we can write

Bp,q(f,∆i(hp); Γ2) =

(

∆i(hp)

∆i(hq)

) 1p−q

, p 6= q,

exp(−∆i(id·hp)

p∆i(hp)− 2

p ln p

), p = q 6= 1,

exp(−∆i(id·h1)

3∆i(h1)

), p = q = 1,

and from (5.2.6) it follows monotonicity of the function Bp,q(f,∆i(hp); Γ2) in param-

eters p and q for hp, hq ∈ Γ2.


Γ3 = ψp : R→ [0,∞) : p ∈ (0,∞),

defined with

ψp(t) =

1p2 e

tp, p ∈ R \ 0,12t2, p = 0,

Since d2

dt2(ψp(t)) = etp > 0 which shows that ψp is convex on R for every p ∈ R and

p 7→ d2

dt2(ψp(t)) is exponentially convex by definition. Using the analogous arguments

as in Theorem 5.2.2, we also have that p 7→ [z0, z1, z2;ψp] is exponentially convex

(also exponentially convex in J-sense). For the family of the function Bp,q(f,∆i; Γ3)

for (i = 1, 2, 3), then (5.2.7) becomes

Bp,q(f,∆i(ψp); Γ3) =

(

∆i(ψp)

∆i(ψq)

) 1p−q

, p 6= q,

exp(

∆i(id·ψp)

∆i(ψp)− 2

p

), p = q 6= 0,

exp(

∆i(id·ψ0)3∆i(ψ0)

), p = q = 0,

113

and using (5.2.6) we can see that these are monotonous function in parameters p and

q for ψp, ψq ∈ Γ3.


Γ4 = φp : (0,∞)→ R : p ∈ R,

defined by

φp(t) =

tp

p(p−1)p 6= 1, 0,

− ln t p = 0,

t ln t p = 1.

Since p 7→ d2

dt2(φp(t)) = tp−2 = e(p−2) ln t > 0, is the Laplace transform of a non-negative

function (see [71]), it is exponentially convex. Obviously φp are convex functions

for every t > 0. It is easy to prove that the function p 7→ [z0, z1, z2;φp] is also

exponentially convex for arbitrary points z0, z1, z2 ∈ I. Using Corollary 5.2.3 it follows

that p 7→ ∆i(φp) for (i = 1, 2, 3) are exponentially convex (it is easy to verify that

these are continuous), and thus log-convex. From (5.2.7), we see that

Bp,q(f,∆i(φp); Γ4) =

(∆i(φp)

∆i(φq)

) 1p−q

, p 6= q,

exp(

1−2pp(p−1)

− ∆i(φpφ0)

∆i(φp)

), p = q 6= 0, 1,

exp(

1− ∆i(φ20)

2∆i(φ0)

), p = q = 0,

exp(−1− ∆i(φ0φ1)

2∆i(φ1)

), p = q = 1,

(5.2.9)

for φp, φq ∈ Γ4.

Remark 5.2.2. For the case i = 1, the means given in (5.2.9) were already presented

in [28] in explicit form.

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