Integral Means Inequalities, Convolution, and Univalent...

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Research Article Integral Means Inequalities, Convolution, and Univalent Functions Daniel Girela and Cristóbal González An´ alisis Matem´ atico, Universidad de M´ alaga, Campus de Teatinos, 29071 M´ alaga, Spain Correspondence should be addressed to Daniel Girela; [email protected] Received 19 December 2018; Accepted 5 February 2019; Published 3 March 2019 Academic Editor: Konstantin M. Dyakonov Copyright © 2019 Daniel Girela and Crist´ obal Gonz´ alez. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We use the Baernstein star-function to investigate several questions about the integral means of the convolution of two analytic functions in the unit disc. e theory of univalent functions plays a basic role in our work. 1. Introduction Let D = { ∈ C : || < 1} and T = { ∈ C : || = 1} denote the open unit disc and the unit circle in the complex plane C. We let also H(D) be the space of all analytic functions in D endowed with the topology of uniform convergence in compact subsets. If 0≤<1 and ∈ H(D), we set (, ) = (∫ − ( ) 2 ) 1/ , if 0 < < ∞, ∞ (, ) = sup ||= () . (1) For 0<≤∞, the Hardy space consists of those ∈ H(D) such that def = sup 0≤<1 (, ) < ∞. (2) We refer to [1] for the theory of -spaces. If , ∈ H(D), () = ∞ ∑ =0 , () = ∞ ∑ =0 ( ∈ D), (3) the (Hadamard) convolution ( ⋆ ) of and is defined by ( ⋆ ) () = ∞ ∑ =0 , ∈ D. (4) We have the following integral representation: ( ⋆ ) () = 1 2 ∫ ||= ( ) () , || < < 1, (5) (see [2, p. 11]). e convolution operation ⋆ makes H(D) into a commutative complex algebra with an identity () = 1 1− = ∞ ∑ =0 , ∈ D. (6) Hindawi Journal of Function Spaces Volume 2019, Article ID 7817353, 5 pages https://doi.org/10.1155/2019/7817353

Transcript of Integral Means Inequalities, Convolution, and Univalent...

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Research ArticleIntegral Means Inequalities Convolutionand Univalent Functions

Daniel Girela and Cristoacutebal Gonzaacutelez

Analisis Matematico Universidad de Malaga Campus de Teatinos 29071 Malaga Spain

Correspondence should be addressed to Daniel Girela girelaumaes

Received 19 December 2018 Accepted 5 February 2019 Published 3 March 2019

Academic Editor Konstantin M Dyakonov

Copyright copy 2019 Daniel Girela and Cristobal Gonzalez This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We use the Baernstein star-function to investigate several questions about the integral means of the convolution of two analyticfunctions in the unit disc The theory of univalent functions plays a basic role in our work

1 Introduction

Let D = 119911 isin C |119911| lt 1 and T = 119911 isin C |119911| = 1 denotethe open unit disc and the unit circle in the complex planeC We let also H119900119897(D) be the space of all analytic functionsin D endowed with the topology of uniform convergence incompact subsets

If 0 le 119903 lt 1 and 119891 isin H119900119897(D) we set

119872119901 (119903 119891) = (int120587minus120587

10038161003816100381610038161003816119891 (119903119890119894119905)10038161003816100381610038161003816119901 1198891199052120587)1119901

if 0 lt 119901 lt infin119872infin (119903 119891) = sup

|119911|=119903

1003816100381610038161003816119891 (119911)1003816100381610038161003816 (1)

For 0 lt 119901 le infin the Hardy space 119867119901 consists of those119891 isin H119900119897(D) such that

10038171003817100381710038171198911003817100381710038171003817119867119901 def= sup0le119903lt1

119872119901 (119903 119891) lt infin (2)

We refer to [1] for the theory of119867119901-spaces

If 119891 119892 isin H119900119897(D)119891 (119911) = infinsum

119899=0

119886119899119911119899

119892 (119911) = infinsum119899=0

119887119899119911119899

(119911 isin D)

(3)

the (Hadamard) convolution (119891 ⋆ 119892) of 119891 and 119892 is defined by

(119891 ⋆ 119892) (119911) = infinsum119899=0

119886119899119887119899119911119899 119911 isin D (4)

We have the following integral representation

(119891 ⋆ 119892) (119911) = 12120587119894 int|120585|=119903 119891(119911

120585)119892 (120585) 119889120585120585 |119911| lt 119903 lt 1 (5)

(see [2 p 11]) The convolution operation ⋆ makes H119900119897(D)into a commutative complex algebra with an identity

119868 (119911) = 11 minus 119911 = infinsum

119899=0

119911119899 119911 isin D (6)

HindawiJournal of Function SpacesVolume 2019 Article ID 7817353 5 pageshttpsdoiorg10115520197817353

2 Journal of Function Spaces

We refer to [2] for the theory of the convolution of analyticfunctions and its connections with geometric function the-ory

Following [3] we shall say that a function 119865 isin H119900119897(D) isbound preserving if for every 119891 isin 119867infin we have that 119891 ⋆ 119865 isin119867infin and

1003817100381710038171003817119891 ⋆ 1198651003817100381710038171003817119867infin le 10038171003817100381710038171198911003817100381710038171003817119867infin (7)

Sheil-Small [3 Theorem 13] (see also [2 p 123]) provedthat a function 119865 isin H119900119897(D) is bound preserving if and onlyif there exists a complex Borel measure 120583 on T with 120583 le 1such that

119865 (119911) = intT

119889120583 (120585)1 minus 119911120585 119911 isin D (8)

The measure 120583 is a probability measure if and only if 119865 isconvexity preserving that is for any 119891 isin H119900119897(D) the rangeof 119891⋆119865 is contained in the closed convex hull of the range of119891 [2 pp 123 124]

It turns out that if 119865 is bound preserving and 1 le 119901 le infinthen for every 119891 isin 119867119901 we have that 119891 ⋆ 119865 isin 119867119901 and

1003817100381710038171003817119891 ⋆ 1198651003817100381710038171003817119867119901 le 10038171003817100381710038171198911003817100381710038171003817119867119901 (9)

Actually the following stronger result holds

Theorem 1 Suppose that 119891 119865 isin H119900119897(D) with 119865 being boundpreserving Then

119872119901 (119903 119891 ⋆ 119865) le 119872119901 (119903 119891) 0 lt 119903 lt 1 (10)

whenever 1 le 119901 le infin

Proof Since 119865 is bound preserving there exists a complexBorel measure 120583 on T with 120583 le 1 such that

119865 (119911) = intT

119889120583 (120585)1 minus 119911120585 = infinsum

119899=0

(intT

120585119899119889120583 (120585)) 119911119899 119911 isin D (11)

If 119891(119911) = suminfin119899=0 119886119899119911119899 (119911 isin D) we have

(119891 ⋆ 119865) (119911) = infinsum119899=0

119886119899 (intT

120585119899119889120583 (120585)) 119911119899

= intT

(infinsum119899=0

119886119899120585119899119911119899)119889120583 (120585)

= intT

119891 (120585119911) 119889120583 (120585) 119911 isin D

(12)

This immediately yields (10) for 119901 = infin Now if 1 le 119901 lt infinusing Minkowskirsquos integral inequality we obtain

119872119901 (119903 119891 ⋆ 119865)= [ 1

2120587 int120587minus120587

1003816100381610038161003816100381610038161003816intT 119891 (119903120585119890119894120579) 119889120583 (120585)1003816100381610038161003816100381610038161003816119901 119889120579]1119901

le [ 12120587 int120587minus120587

(intT

10038161003816100381610038161003816119891 (119903120585119890119894120579)10038161003816100381610038161003816 119889 10038161003816100381610038161205831003816100381610038161003816 (120585))119901 119889120579]1119901

le intT

( 12120587 int120587minus120587

10038161003816100381610038161003816119891 (119903120585119890119894120579)10038161003816100381610038161003816119901 119889120579)1119901 119889 10038161003816100381610038161205831003816100381610038161003816 (120585)

= intT

119872119901 (119903 119891) 119889 10038161003816100381610038161205831003816100381610038161003816 (120585) le 119872119901 (119903 119891) (13)

2 Star-Type Inequalities

Themain purpose of this article is studying the possibility ofextending Theorem 1 to cover other integral means at leastfor some special classes of functions In order to do so weshall use the method of the star-function introduced by ABaernstein [4 5]

If 119906 is a subharmonic function in D 0 the function 119906lowastis defined by

119906lowast (119903119890119894120579) = sup|119864|=2120579

int119864119906 (119903119890119894119905) 119889119905

0 lt 119903 lt 1 0 le 120579 le 120587(14)

where |119864| denotes the Lebesgue measure of the set 119864 Thebasic properties of the star-function which make it useful tosolve extremal problems are the following [5]

(i) If 119906 is a subharmonic function in D 0 then thefunction 119906lowast is subharmonic in D+ = 119911 = 119903119890119894120579 0 lt119903 lt 1 0 lt 120579 lt 120587 and continuous in 119911 = 119903119890119894120579 0 lt119903 lt 1 0 le 120579 le 120587

(ii) If V is harmonic in D 0 and it is a symmetricdecreasing function on each of the circles |119911| = 119903(0 lt 119903 lt 1) then Vlowast is harmonic in D+ and in factVlowast(119903119890119894120579) = int120579

minus120579V(119903119890119894119905)119889119905

The relevance of the star-function to obtain integralmeans estimates comes from the following result

Proposition A (see [5]) Let 119906 and V be two subharmonicfunctions in D Then the following two conditions are equiv-alent

(i) 119906lowast le Vlowast in D+(ii) For every convex and increasing functionΦ R 997888rarr R

we have that

int120587minus120587

Φ(119906 (119903119890119894120579)) 119889120579 le int120587minus120587

Φ(V (119903119890119894120579)) 1198891205790 lt 119903 lt 1

(15)

Proposition A yields the following result about analyticfunctions

Proposition B Let 119891 and 119892 be two nonidentically zeroanalytic functions inDThen the following conditions are equi-valent

(i) (log |119891|)lowast le (log |119892|)lowast in D+

Journal of Function Spaces 3

(ii) For every convex and increasing functionΦ R 997888rarr Rwe have that

int120587minus120587

Φ(log 10038161003816100381610038161003816119891 (119903119890119894120579)10038161003816100381610038161003816) 119889120579 le int120587minus120587

Φ(log 10038161003816100381610038161003816119892 (119903119890119894120579)10038161003816100381610038161003816) 1198891205790 lt 119903 lt 1

(16)

Since for any 119901 gt 0 the function Φ defined by Φ(119909) =exp(119901119909) (119909 isin R) is convex and increasing we deduce that if119891 and 119892 are as in Proposition B and (log |119891|)lowast le (log |119892|)lowast inD+ then

119872119901 (119903 119891) le 119872119901 (119903 119892) 0 lt 119903 lt 1 (17)

for all 119901 gt 0The main achievement in the use of the star-function by

A Baernstein in [5] was the proof that the Koebe function119896(119911) = 119911(1 minus 119911)2 (119911 isin D) is extremal for the integral meansof functions in the class 119878 of univalent functions (see [1 6]for the notation and results regarding univalent functions)Namely Baernstein proved that if 119891 isin 119878 then

(plusmn log 10038161003816100381610038161198911003816100381610038161003816)lowast le (plusmn log |119896|)lowast (18)

and hence

int120587minus120587

10038161003816100381610038161003816119891 (119903119890119894120579)10038161003816100381610038161003816119901 119889120579 le int120587minus120587

10038161003816100381610038161003816119896 (119903119890119894120579)10038161003816100381610038161003816119901 119889120579 0 lt 119903 lt 1 (19)

for all 119901 isin R In particular we have that if 119891 isin 119878 and 0 lt 119901 leinfin then

119872119901 (119903 119891) le 119872119901 (119903 119896) 0 lt 119903 lt 1 (20)

Subsequently the star-function has been used in a goodnumber of papers to obtain bounds on the integral means ofdistinct classes of analytic functions (see eg [7ndash12])

Coming back to convolution the following questionsarise in a natural way

Question 1 Let 119891 119892 119865 119866 be analytic functions in D with |119865|and |119866| being symmetric decreasing on each of the circles|119911| = 119903 and suppose that

(log 10038161003816100381610038161198911003816100381610038161003816)lowast le (log |119865|)lowastand (log 10038161003816100381610038161198921003816100381610038161003816)lowast le (log |119866|)lowast (21)

Does it follow that (log |119891 ⋆ 119892|)lowast le (log |119865 ⋆ 119866|)lowastQuestion 2 Let 119865 and 119891 be two analytic functions in Dand suppose that 119865 is bound preserving Can we assert that(log |119891 ⋆ 119865|)lowast le (log |119891|)lowast

We shall show that the answer to these two questions isnegative Regarding Question 1 we have the following result

Theorem 3 There exist two functions 1198651 1198652 isin H119900119897(D) with(log 1003816100381610038161003816100381611986511989510038161003816100381610038161003816)lowast le (log |119868|)lowast 119891119900119903 119895 = 1 2 (22)

and such that

119905ℎ119890 119894119899119890119902119906119886119897119894119905119910 (log 10038161003816100381610038161198651 ⋆ 11986521003816100381610038161003816)lowastle (log |119868 ⋆ 119868|)lowast 119889119900119890119904 119899119900119905 ℎ119900119897119889 (23)

Here 119868 is the identity element of the convolution defined in(6) that is 119868(119911) = 1(1 minus 119911) (119911 isin D) Hence 119868 ⋆ 119868 = 119868Proof Let ℎ be an odd function in the class 119878 with Taylorexpansion

ℎ (119911) = 119911 + 11988631199113 + 11988651199115 + (24)

with |1198865| gt 1 The existence of such ℎ was proved by Feketeand Szego (see [13 p 104]) Set also

ℎ1 (119911) = ℎ (119911)119911 = 1 + 11988631199112 + 11988651199114 + 119911 isin D (25)

It is well known that there exists a function 119867 isin 119878 such thatℎ(119911) = radic119867(1199112) (see [13 p 64]) Set 1198962(119911) = radic119896(1199112) = 119911(1 minus1199112) and 119869(119911) = 1198962(119911)119911 = 1(1 minus 1199112) (119911 isin D) By Baernsteinrsquostheorem we have (log |119867|)lowast le (log |119896|)lowast a fact which easilyimplies that (log |ℎ1|)lowast le (log |119869|)lowast Now it is clear that 119869 issubordinate to 119868 and then using [8 Lemma 2] we see that(log |119869|)lowast le (log |119868|)lowastThus it follows that

(log 1003816100381610038161003816ℎ11003816100381610038161003816)lowast le (log |119868|)lowast (26)

For 119899 = 1 2 3 we define 119891119899 inductively as follows1198911 = ℎ1

and 119891119899 = 119891119899minus1 ⋆ 1198911 for 119899 ge 2 (27)

In other words 119891119899 =(119899)⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ℎ1 ⋆ sdot sdot sdot ⋆ ℎ1 Clearly (25) yields

119891119899 (119911) = 1 + 11988611989931199112 + 11988611989951199114 + (28)

Since |1198865| gt 1 it follows that |1198861198995 | 997888rarr infin as 119899 997888rarr infin This isequivalent to saying that

10038161003816100381610038161003816119891(4)119899 (0)10038161003816100381610038161003816 997888rarr infin as 119899 997888rarr infin (29)

Then it follows that the family 119891(4)119899 119899 = 1 2 3 is not alocally bounded family of holomorphic functions inD Using[14 Theorem 16 p 225] we see that the same is true for thefamily 119891119899 119899 = 1 2 3 Take 119901 isin (0 1) then 119868 isin 119867119901Since a bounded subset of 119867119901 is a locally bounded family [1p 36] it follows that

sup119899ge1

10038171003817100381710038171198911198991003817100381710038171003817119867119901 = infin (30)

Now (30) implies that 119891119899119867119901 gt 119868119867119901 for some 119899 UsingProposition B we see that this implies that

the inequality (log 10038161003816100381610038161198911198991003816100381610038161003816)lowastle (log |119868|)lowast is not true for some 119899 (31)

Let119873 be the smallest of all such 119899 Using (26) and the fact that1198911 = ℎ1 it follows that 119873 gt 1

4 Journal of Function Spaces

Then it is clear that (23) holds with 1198651 = 1198911 1198652 = 119891119873minus1We have the following result regarding Question 2

Theorem 4 There exist 119891 119865 analytic and univalent in D suchthat 119865 is convexity preserving and with the property that theinequality (log |119891 ⋆ 119865|)lowast le (log |119891|)lowast does not hold

The following lemma will be used in the proof ofTheorem 4

Lemma 5 Let 119891 119865 isin H119900119897((119863) and suppose that 119865(0) = 1 119865is convexity preserving and 119891 and 119891⋆119865 are zero-free inD andsatisfy the inequality (log |119891 ⋆ 119865|)lowast le (log |119891|)lowast Then we alsohave that

(log 100381610038161003816100381610038161003816100381610038161

119891 ⋆ 11986510038161003816100381610038161003816100381610038161003816)lowast le (log 10038161003816100381610038161003816100381610038161003816

111989110038161003816100381610038161003816100381610038161003816)lowast (32)

Proof Set 119906 = log |119891 ⋆ 119865| V = log |119891| Then 119906 and V areharmonic in D 119906(0) = V(0) and 119906lowast le Vlowast Then it followsthat for 0 lt 119903 lt 1 and 0 le 120579 le 120587

(minus119906)lowast (119903119890119894120579) = sup|119864|=2120579

int119864minus119906 (119903119890119894119905) 119889119905

= sup|119864|=2120579

(minusint120587minus120587

119906 (119903119890119894119905) 119889119905 + int[minus120587120587]119864

119906 (119903119890119894119905) 119889119905)= minus2120587119906 (0) + 119906lowast (119903119890119894(120587minus120579))= minus2120587V (0) + 119906lowast (119903119890119894(120587minus120579))le minus2120587V (0) + Vlowast (119903119890119894(120587minus120579)) = (minusV)lowast (119903119890119894120579)

(33)

Hence we have proved that (minus119906)lowast le (minusV)lowast which isequivalent to (32)

Proof of Theorem 4 Set

119891 (119911) = 1(1 minus 119911)2 =

infinsum119899=0

(119899 + 1) 119911119899

119865 (119911) = 1 minus 12119911

119911 isin D

(34)

Clearly 119891 and 119865 are analytic univalent and zero-free in DAlso

(119891 ⋆ 119865) (119911) = 1 minus 119911 119911 isin D (35)

Hence119891⋆119865 is also zero-free inD Notice that 1(119891⋆119865) notin 119867infinand 1119891 isin 119867infin Then it follows that

the inequality (log 100381610038161003816100381610038161003816100381610038161

119891 ⋆ 11986510038161003816100381610038161003816100381610038161003816)lowast

le (log 10038161003816100381610038161003816100381610038161003816111989110038161003816100381610038161003816100381610038161003816)lowast

does not hold(36)

Now it is a simple exercise to check that

119865 (119911) = 12120587 int120587minus120587

1 minus cos 1205791 minus 119890119894120579119911 119889120579 (37)

and then it follows that 119865 is convexity preservingThen using(36) and Lemma 5 it follows that the inequality (log |119891 ⋆119865|)lowast le (log |119891|)lowast does not hold as desired

We close the paper with a positive result determining aclass of univalent functions Z such that (10) is true for all119901 gt 0 whenever 119891 isin Z and 119865 is convexity preserving

A domain 119863 in C is said to be Steiner symmetric if itsintersection with each vertical line is either empty or is thewhole line or is a segment placed symmetrically with respectto the real axis We letZ be the class of all functions 119891 whichare analytic and univalent in D with 119891(0) = 0 1198911015840(0) gt 0 andwhose image is a Steiner symmetric domain The elementsof Z will be called Steiner symmetric functions Usingarguments similar to those used by Jenkins [15] for circularlysymmetric functions we see that a univalent function 119891with119891(0) = 0 and 1198911015840(0) gt 0 is Steiner symmetric if and only ifit satisfies the following two conditions (i) 119891 is typically realand (ii) Re119891 is a symmetric decreasing function on each of thecircles |119911| = 119903 (0 lt 119903 lt 1)Then it follows that if119891 isin Z thenfor every 119903 isin (0 1) the domain 119891(|119911| lt 119903) is a Steiner sym-metric domain and hence the function 119891119903 defined by119891119903(119911) =119891(119903119911) (119911 isin D) belongs toZ and it extends to an analytic func-tion in the closed unit discD Nowwe can state our last result

Theorem 6 Suppose that 119891 isin Z and let 119865 be an analyticfunction inDwhich is convexity preserving We have for every119901 gt 0

119872119901 (119903 119891 ⋆ 119865) le 119872119901 (119903 119891) 0 lt 119903 lt 1 (38)

Proof In view of Theorem 1 we only need to prove (38) for0 lt 119901 lt 1 Let 120583 be the probability measure on T such that119865(119911) = intT(119889120583(120585)(1 minus 119911120585)) (119911 isin D) Then we have

(119891 ⋆ 119865) (119911) = intT

119891 (120585119911) 119889120583 (120585) (39)

Since 119865 is convexity preserving for 0 lt 119903 lt 1 we have that(119891119903 ⋆ 119865) (D) is contained in the closed convex hull of 119891119903(D)This easily yields

min119911isinD

Re119891119903 (119911) le min119911isinD

Re (119891119903 ⋆ 119865) (119911) max119911isinD

Re (119891119903 ⋆ 119865) (119911) le max119911isinD

Re119891119903 (119911) (40)

By the remarks in the previous paragraph we find that for all119903 isin (0 1)119891119903 belongs toZ and extends to an analytic functionin the closed unit disc D Finally we claim that

(Re (119891119903 ⋆ 119865))lowast le (Re119891119903)lowast 0 lt 119903 lt 1 (41)

Once this is proved using Proposition 6 of [10] we deducethat119872119901 (119903 119891 ⋆ 119865) = 1003817100381710038171003817119891119903 ⋆ 1198651003817100381710038171003817119867119901 le 10038171003817100381710038171198911199031003817100381710038171003817119867119901 = 119872119901 (119903 119891)

0 lt 119901 le 2 (42)

finishing our proof

Journal of Function Spaces 5

So we proceed to prove (41) Fix 119903 isin (0 1) and set 119906 =Re(119891119903 ⋆ 119865) V = Re119891119903 Using (39) we have for 0 lt 119877 lt 1 and0 lt 120579 lt 120587

119906lowast (119877119890119894120579) = sup|119864|=2120579

int119864119906 (119877119890119894119905) 119889119905

= sup|119864|=2120579

int119864intT

V (119877119890119894119905120585) 119889120583 (120585) 119889119905

= sup|119864|=2120579

intT

int119864V (119877119890119894119905120585) 119889119905 119889120583 (120585)

le intT

Vlowast (119877119890119894120579) 119889120583 (120585) = Vlowast (119877119890119894120579)

(43)

Data Availability

No data were used to support this study

Conflicts of Interest

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

Acknowledgments

This research is supported by a grant from ldquoEl Ministeriode Economıa y Competitividadrdquo Spain (MTM2014-52865-P)and by a grant from la Junta de Andalucıa (FQM-210) Wewish also to dedicate this article to Fernando Perez Gonzalezon the occasion of his retirement

References

[1] P L Duren Theory of H119901 Spaces Academic Press New York-London 1970 Reprint Dover Mineola-New York 2000

[2] S Ruscheweyh Convolutions in Geometric Function TheorySeminaire de Mathematiques Superieures [Seminar on HigherMathematics] Fundamental Theories of Physics Presses delrsquoUniversite de Montreal Montreal Canada 1982

[3] T Sheil-Small ldquoOn the convolution of analytic functionsrdquoJournal fur die Reine und Angewandte Mathematik vol 258 pp137ndash152 1973

[4] A Baernstein ldquoProof of Edreirsquos spread conjecturerdquo Proceedingsof the London Mathematical Society vol 26 no 3 pp 418ndash4341973

[5] A Baernstein ldquoIntegralmeans univalent functions and circularsymmetrizationrdquo Acta Mathematica vol 133 pp 139ndash169 1974

[6] C Pommerenke Univalent Functions Vandenhoeck amp Ru-precht Gottingen Germany 1975

[7] A Baernstein ldquoSome sharp inequalities for conjugate func-tionsrdquo Indiana UniversityMathematics Journal vol 27 no 5 pp833ndash852 1978

[8] Y J Leung ldquoIntegral means of the derivatives of some univalentfunctionsrdquo Bulletin of the London Mathematical Society vol 11no 3 pp 289ndash294 1979

[9] J E Brown ldquoDerivatives of close-to-convex functions inte-gral means and bounded mean oscillationrdquo MathematischeZeitschrift vol 178 no 3 pp 353ndash358 1981

[10] D Girela ldquoIntegral means and BMOA-norms of logarithmsof univalent functionsrdquo Journal of The London MathematicalSociety-Second Series vol 33 no 1 pp 117ndash132 1986

[11] D Girela ldquoIntegral means bounded mean oscillation andGelrsquofer functionsrdquo Proceedings of the American MathematicalSociety vol 113 no 2 pp 365ndash370 1991

[12] M Nowak ldquoSome inequalities for BMOA functionsrdquo ComplexVariablesTheory andApplication vol 16 no 2-3 pp 81ndash86 1991

[13] P L Duren Univalent Functions Springer-Verlag New YorkNY USA 1983

[14] L V Ahlfors Complex Analysis An Introduction to the Theoryof Analytic Functions of One Complex Variable InternationalSeries in Pure and Applied Mathematics McGraw-Hill BookCo New York NY USA 3rd edition 1978

[15] J A Jenkins ldquoOn circularly symmetric functionsrdquo Proceedingsof the American Mathematical Society vol 6 pp 620ndash624 1955

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Page 2: Integral Means Inequalities, Convolution, and Univalent ...downloads.hindawi.com/journals/jfs/2019/7817353.pdf · JournalofFunctionSpaces Wereferto[] forthetheoryoftheconvolutionofanalytic

2 Journal of Function Spaces

We refer to [2] for the theory of the convolution of analyticfunctions and its connections with geometric function the-ory

Following [3] we shall say that a function 119865 isin H119900119897(D) isbound preserving if for every 119891 isin 119867infin we have that 119891 ⋆ 119865 isin119867infin and

1003817100381710038171003817119891 ⋆ 1198651003817100381710038171003817119867infin le 10038171003817100381710038171198911003817100381710038171003817119867infin (7)

Sheil-Small [3 Theorem 13] (see also [2 p 123]) provedthat a function 119865 isin H119900119897(D) is bound preserving if and onlyif there exists a complex Borel measure 120583 on T with 120583 le 1such that

119865 (119911) = intT

119889120583 (120585)1 minus 119911120585 119911 isin D (8)

The measure 120583 is a probability measure if and only if 119865 isconvexity preserving that is for any 119891 isin H119900119897(D) the rangeof 119891⋆119865 is contained in the closed convex hull of the range of119891 [2 pp 123 124]

It turns out that if 119865 is bound preserving and 1 le 119901 le infinthen for every 119891 isin 119867119901 we have that 119891 ⋆ 119865 isin 119867119901 and

1003817100381710038171003817119891 ⋆ 1198651003817100381710038171003817119867119901 le 10038171003817100381710038171198911003817100381710038171003817119867119901 (9)

Actually the following stronger result holds

Theorem 1 Suppose that 119891 119865 isin H119900119897(D) with 119865 being boundpreserving Then

119872119901 (119903 119891 ⋆ 119865) le 119872119901 (119903 119891) 0 lt 119903 lt 1 (10)

whenever 1 le 119901 le infin

Proof Since 119865 is bound preserving there exists a complexBorel measure 120583 on T with 120583 le 1 such that

119865 (119911) = intT

119889120583 (120585)1 minus 119911120585 = infinsum

119899=0

(intT

120585119899119889120583 (120585)) 119911119899 119911 isin D (11)

If 119891(119911) = suminfin119899=0 119886119899119911119899 (119911 isin D) we have

(119891 ⋆ 119865) (119911) = infinsum119899=0

119886119899 (intT

120585119899119889120583 (120585)) 119911119899

= intT

(infinsum119899=0

119886119899120585119899119911119899)119889120583 (120585)

= intT

119891 (120585119911) 119889120583 (120585) 119911 isin D

(12)

This immediately yields (10) for 119901 = infin Now if 1 le 119901 lt infinusing Minkowskirsquos integral inequality we obtain

119872119901 (119903 119891 ⋆ 119865)= [ 1

2120587 int120587minus120587

1003816100381610038161003816100381610038161003816intT 119891 (119903120585119890119894120579) 119889120583 (120585)1003816100381610038161003816100381610038161003816119901 119889120579]1119901

le [ 12120587 int120587minus120587

(intT

10038161003816100381610038161003816119891 (119903120585119890119894120579)10038161003816100381610038161003816 119889 10038161003816100381610038161205831003816100381610038161003816 (120585))119901 119889120579]1119901

le intT

( 12120587 int120587minus120587

10038161003816100381610038161003816119891 (119903120585119890119894120579)10038161003816100381610038161003816119901 119889120579)1119901 119889 10038161003816100381610038161205831003816100381610038161003816 (120585)

= intT

119872119901 (119903 119891) 119889 10038161003816100381610038161205831003816100381610038161003816 (120585) le 119872119901 (119903 119891) (13)

2 Star-Type Inequalities

Themain purpose of this article is studying the possibility ofextending Theorem 1 to cover other integral means at leastfor some special classes of functions In order to do so weshall use the method of the star-function introduced by ABaernstein [4 5]

If 119906 is a subharmonic function in D 0 the function 119906lowastis defined by

119906lowast (119903119890119894120579) = sup|119864|=2120579

int119864119906 (119903119890119894119905) 119889119905

0 lt 119903 lt 1 0 le 120579 le 120587(14)

where |119864| denotes the Lebesgue measure of the set 119864 Thebasic properties of the star-function which make it useful tosolve extremal problems are the following [5]

(i) If 119906 is a subharmonic function in D 0 then thefunction 119906lowast is subharmonic in D+ = 119911 = 119903119890119894120579 0 lt119903 lt 1 0 lt 120579 lt 120587 and continuous in 119911 = 119903119890119894120579 0 lt119903 lt 1 0 le 120579 le 120587

(ii) If V is harmonic in D 0 and it is a symmetricdecreasing function on each of the circles |119911| = 119903(0 lt 119903 lt 1) then Vlowast is harmonic in D+ and in factVlowast(119903119890119894120579) = int120579

minus120579V(119903119890119894119905)119889119905

The relevance of the star-function to obtain integralmeans estimates comes from the following result

Proposition A (see [5]) Let 119906 and V be two subharmonicfunctions in D Then the following two conditions are equiv-alent

(i) 119906lowast le Vlowast in D+(ii) For every convex and increasing functionΦ R 997888rarr R

we have that

int120587minus120587

Φ(119906 (119903119890119894120579)) 119889120579 le int120587minus120587

Φ(V (119903119890119894120579)) 1198891205790 lt 119903 lt 1

(15)

Proposition A yields the following result about analyticfunctions

Proposition B Let 119891 and 119892 be two nonidentically zeroanalytic functions inDThen the following conditions are equi-valent

(i) (log |119891|)lowast le (log |119892|)lowast in D+

Journal of Function Spaces 3

(ii) For every convex and increasing functionΦ R 997888rarr Rwe have that

int120587minus120587

Φ(log 10038161003816100381610038161003816119891 (119903119890119894120579)10038161003816100381610038161003816) 119889120579 le int120587minus120587

Φ(log 10038161003816100381610038161003816119892 (119903119890119894120579)10038161003816100381610038161003816) 1198891205790 lt 119903 lt 1

(16)

Since for any 119901 gt 0 the function Φ defined by Φ(119909) =exp(119901119909) (119909 isin R) is convex and increasing we deduce that if119891 and 119892 are as in Proposition B and (log |119891|)lowast le (log |119892|)lowast inD+ then

119872119901 (119903 119891) le 119872119901 (119903 119892) 0 lt 119903 lt 1 (17)

for all 119901 gt 0The main achievement in the use of the star-function by

A Baernstein in [5] was the proof that the Koebe function119896(119911) = 119911(1 minus 119911)2 (119911 isin D) is extremal for the integral meansof functions in the class 119878 of univalent functions (see [1 6]for the notation and results regarding univalent functions)Namely Baernstein proved that if 119891 isin 119878 then

(plusmn log 10038161003816100381610038161198911003816100381610038161003816)lowast le (plusmn log |119896|)lowast (18)

and hence

int120587minus120587

10038161003816100381610038161003816119891 (119903119890119894120579)10038161003816100381610038161003816119901 119889120579 le int120587minus120587

10038161003816100381610038161003816119896 (119903119890119894120579)10038161003816100381610038161003816119901 119889120579 0 lt 119903 lt 1 (19)

for all 119901 isin R In particular we have that if 119891 isin 119878 and 0 lt 119901 leinfin then

119872119901 (119903 119891) le 119872119901 (119903 119896) 0 lt 119903 lt 1 (20)

Subsequently the star-function has been used in a goodnumber of papers to obtain bounds on the integral means ofdistinct classes of analytic functions (see eg [7ndash12])

Coming back to convolution the following questionsarise in a natural way

Question 1 Let 119891 119892 119865 119866 be analytic functions in D with |119865|and |119866| being symmetric decreasing on each of the circles|119911| = 119903 and suppose that

(log 10038161003816100381610038161198911003816100381610038161003816)lowast le (log |119865|)lowastand (log 10038161003816100381610038161198921003816100381610038161003816)lowast le (log |119866|)lowast (21)

Does it follow that (log |119891 ⋆ 119892|)lowast le (log |119865 ⋆ 119866|)lowastQuestion 2 Let 119865 and 119891 be two analytic functions in Dand suppose that 119865 is bound preserving Can we assert that(log |119891 ⋆ 119865|)lowast le (log |119891|)lowast

We shall show that the answer to these two questions isnegative Regarding Question 1 we have the following result

Theorem 3 There exist two functions 1198651 1198652 isin H119900119897(D) with(log 1003816100381610038161003816100381611986511989510038161003816100381610038161003816)lowast le (log |119868|)lowast 119891119900119903 119895 = 1 2 (22)

and such that

119905ℎ119890 119894119899119890119902119906119886119897119894119905119910 (log 10038161003816100381610038161198651 ⋆ 11986521003816100381610038161003816)lowastle (log |119868 ⋆ 119868|)lowast 119889119900119890119904 119899119900119905 ℎ119900119897119889 (23)

Here 119868 is the identity element of the convolution defined in(6) that is 119868(119911) = 1(1 minus 119911) (119911 isin D) Hence 119868 ⋆ 119868 = 119868Proof Let ℎ be an odd function in the class 119878 with Taylorexpansion

ℎ (119911) = 119911 + 11988631199113 + 11988651199115 + (24)

with |1198865| gt 1 The existence of such ℎ was proved by Feketeand Szego (see [13 p 104]) Set also

ℎ1 (119911) = ℎ (119911)119911 = 1 + 11988631199112 + 11988651199114 + 119911 isin D (25)

It is well known that there exists a function 119867 isin 119878 such thatℎ(119911) = radic119867(1199112) (see [13 p 64]) Set 1198962(119911) = radic119896(1199112) = 119911(1 minus1199112) and 119869(119911) = 1198962(119911)119911 = 1(1 minus 1199112) (119911 isin D) By Baernsteinrsquostheorem we have (log |119867|)lowast le (log |119896|)lowast a fact which easilyimplies that (log |ℎ1|)lowast le (log |119869|)lowast Now it is clear that 119869 issubordinate to 119868 and then using [8 Lemma 2] we see that(log |119869|)lowast le (log |119868|)lowastThus it follows that

(log 1003816100381610038161003816ℎ11003816100381610038161003816)lowast le (log |119868|)lowast (26)

For 119899 = 1 2 3 we define 119891119899 inductively as follows1198911 = ℎ1

and 119891119899 = 119891119899minus1 ⋆ 1198911 for 119899 ge 2 (27)

In other words 119891119899 =(119899)⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ℎ1 ⋆ sdot sdot sdot ⋆ ℎ1 Clearly (25) yields

119891119899 (119911) = 1 + 11988611989931199112 + 11988611989951199114 + (28)

Since |1198865| gt 1 it follows that |1198861198995 | 997888rarr infin as 119899 997888rarr infin This isequivalent to saying that

10038161003816100381610038161003816119891(4)119899 (0)10038161003816100381610038161003816 997888rarr infin as 119899 997888rarr infin (29)

Then it follows that the family 119891(4)119899 119899 = 1 2 3 is not alocally bounded family of holomorphic functions inD Using[14 Theorem 16 p 225] we see that the same is true for thefamily 119891119899 119899 = 1 2 3 Take 119901 isin (0 1) then 119868 isin 119867119901Since a bounded subset of 119867119901 is a locally bounded family [1p 36] it follows that

sup119899ge1

10038171003817100381710038171198911198991003817100381710038171003817119867119901 = infin (30)

Now (30) implies that 119891119899119867119901 gt 119868119867119901 for some 119899 UsingProposition B we see that this implies that

the inequality (log 10038161003816100381610038161198911198991003816100381610038161003816)lowastle (log |119868|)lowast is not true for some 119899 (31)

Let119873 be the smallest of all such 119899 Using (26) and the fact that1198911 = ℎ1 it follows that 119873 gt 1

4 Journal of Function Spaces

Then it is clear that (23) holds with 1198651 = 1198911 1198652 = 119891119873minus1We have the following result regarding Question 2

Theorem 4 There exist 119891 119865 analytic and univalent in D suchthat 119865 is convexity preserving and with the property that theinequality (log |119891 ⋆ 119865|)lowast le (log |119891|)lowast does not hold

The following lemma will be used in the proof ofTheorem 4

Lemma 5 Let 119891 119865 isin H119900119897((119863) and suppose that 119865(0) = 1 119865is convexity preserving and 119891 and 119891⋆119865 are zero-free inD andsatisfy the inequality (log |119891 ⋆ 119865|)lowast le (log |119891|)lowast Then we alsohave that

(log 100381610038161003816100381610038161003816100381610038161

119891 ⋆ 11986510038161003816100381610038161003816100381610038161003816)lowast le (log 10038161003816100381610038161003816100381610038161003816

111989110038161003816100381610038161003816100381610038161003816)lowast (32)

Proof Set 119906 = log |119891 ⋆ 119865| V = log |119891| Then 119906 and V areharmonic in D 119906(0) = V(0) and 119906lowast le Vlowast Then it followsthat for 0 lt 119903 lt 1 and 0 le 120579 le 120587

(minus119906)lowast (119903119890119894120579) = sup|119864|=2120579

int119864minus119906 (119903119890119894119905) 119889119905

= sup|119864|=2120579

(minusint120587minus120587

119906 (119903119890119894119905) 119889119905 + int[minus120587120587]119864

119906 (119903119890119894119905) 119889119905)= minus2120587119906 (0) + 119906lowast (119903119890119894(120587minus120579))= minus2120587V (0) + 119906lowast (119903119890119894(120587minus120579))le minus2120587V (0) + Vlowast (119903119890119894(120587minus120579)) = (minusV)lowast (119903119890119894120579)

(33)

Hence we have proved that (minus119906)lowast le (minusV)lowast which isequivalent to (32)

Proof of Theorem 4 Set

119891 (119911) = 1(1 minus 119911)2 =

infinsum119899=0

(119899 + 1) 119911119899

119865 (119911) = 1 minus 12119911

119911 isin D

(34)

Clearly 119891 and 119865 are analytic univalent and zero-free in DAlso

(119891 ⋆ 119865) (119911) = 1 minus 119911 119911 isin D (35)

Hence119891⋆119865 is also zero-free inD Notice that 1(119891⋆119865) notin 119867infinand 1119891 isin 119867infin Then it follows that

the inequality (log 100381610038161003816100381610038161003816100381610038161

119891 ⋆ 11986510038161003816100381610038161003816100381610038161003816)lowast

le (log 10038161003816100381610038161003816100381610038161003816111989110038161003816100381610038161003816100381610038161003816)lowast

does not hold(36)

Now it is a simple exercise to check that

119865 (119911) = 12120587 int120587minus120587

1 minus cos 1205791 minus 119890119894120579119911 119889120579 (37)

and then it follows that 119865 is convexity preservingThen using(36) and Lemma 5 it follows that the inequality (log |119891 ⋆119865|)lowast le (log |119891|)lowast does not hold as desired

We close the paper with a positive result determining aclass of univalent functions Z such that (10) is true for all119901 gt 0 whenever 119891 isin Z and 119865 is convexity preserving

A domain 119863 in C is said to be Steiner symmetric if itsintersection with each vertical line is either empty or is thewhole line or is a segment placed symmetrically with respectto the real axis We letZ be the class of all functions 119891 whichare analytic and univalent in D with 119891(0) = 0 1198911015840(0) gt 0 andwhose image is a Steiner symmetric domain The elementsof Z will be called Steiner symmetric functions Usingarguments similar to those used by Jenkins [15] for circularlysymmetric functions we see that a univalent function 119891with119891(0) = 0 and 1198911015840(0) gt 0 is Steiner symmetric if and only ifit satisfies the following two conditions (i) 119891 is typically realand (ii) Re119891 is a symmetric decreasing function on each of thecircles |119911| = 119903 (0 lt 119903 lt 1)Then it follows that if119891 isin Z thenfor every 119903 isin (0 1) the domain 119891(|119911| lt 119903) is a Steiner sym-metric domain and hence the function 119891119903 defined by119891119903(119911) =119891(119903119911) (119911 isin D) belongs toZ and it extends to an analytic func-tion in the closed unit discD Nowwe can state our last result

Theorem 6 Suppose that 119891 isin Z and let 119865 be an analyticfunction inDwhich is convexity preserving We have for every119901 gt 0

119872119901 (119903 119891 ⋆ 119865) le 119872119901 (119903 119891) 0 lt 119903 lt 1 (38)

Proof In view of Theorem 1 we only need to prove (38) for0 lt 119901 lt 1 Let 120583 be the probability measure on T such that119865(119911) = intT(119889120583(120585)(1 minus 119911120585)) (119911 isin D) Then we have

(119891 ⋆ 119865) (119911) = intT

119891 (120585119911) 119889120583 (120585) (39)

Since 119865 is convexity preserving for 0 lt 119903 lt 1 we have that(119891119903 ⋆ 119865) (D) is contained in the closed convex hull of 119891119903(D)This easily yields

min119911isinD

Re119891119903 (119911) le min119911isinD

Re (119891119903 ⋆ 119865) (119911) max119911isinD

Re (119891119903 ⋆ 119865) (119911) le max119911isinD

Re119891119903 (119911) (40)

By the remarks in the previous paragraph we find that for all119903 isin (0 1)119891119903 belongs toZ and extends to an analytic functionin the closed unit disc D Finally we claim that

(Re (119891119903 ⋆ 119865))lowast le (Re119891119903)lowast 0 lt 119903 lt 1 (41)

Once this is proved using Proposition 6 of [10] we deducethat119872119901 (119903 119891 ⋆ 119865) = 1003817100381710038171003817119891119903 ⋆ 1198651003817100381710038171003817119867119901 le 10038171003817100381710038171198911199031003817100381710038171003817119867119901 = 119872119901 (119903 119891)

0 lt 119901 le 2 (42)

finishing our proof

Journal of Function Spaces 5

So we proceed to prove (41) Fix 119903 isin (0 1) and set 119906 =Re(119891119903 ⋆ 119865) V = Re119891119903 Using (39) we have for 0 lt 119877 lt 1 and0 lt 120579 lt 120587

119906lowast (119877119890119894120579) = sup|119864|=2120579

int119864119906 (119877119890119894119905) 119889119905

= sup|119864|=2120579

int119864intT

V (119877119890119894119905120585) 119889120583 (120585) 119889119905

= sup|119864|=2120579

intT

int119864V (119877119890119894119905120585) 119889119905 119889120583 (120585)

le intT

Vlowast (119877119890119894120579) 119889120583 (120585) = Vlowast (119877119890119894120579)

(43)

Data Availability

No data were used to support this study

Conflicts of Interest

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

Acknowledgments

This research is supported by a grant from ldquoEl Ministeriode Economıa y Competitividadrdquo Spain (MTM2014-52865-P)and by a grant from la Junta de Andalucıa (FQM-210) Wewish also to dedicate this article to Fernando Perez Gonzalezon the occasion of his retirement

References

[1] P L Duren Theory of H119901 Spaces Academic Press New York-London 1970 Reprint Dover Mineola-New York 2000

[2] S Ruscheweyh Convolutions in Geometric Function TheorySeminaire de Mathematiques Superieures [Seminar on HigherMathematics] Fundamental Theories of Physics Presses delrsquoUniversite de Montreal Montreal Canada 1982

[3] T Sheil-Small ldquoOn the convolution of analytic functionsrdquoJournal fur die Reine und Angewandte Mathematik vol 258 pp137ndash152 1973

[4] A Baernstein ldquoProof of Edreirsquos spread conjecturerdquo Proceedingsof the London Mathematical Society vol 26 no 3 pp 418ndash4341973

[5] A Baernstein ldquoIntegralmeans univalent functions and circularsymmetrizationrdquo Acta Mathematica vol 133 pp 139ndash169 1974

[6] C Pommerenke Univalent Functions Vandenhoeck amp Ru-precht Gottingen Germany 1975

[7] A Baernstein ldquoSome sharp inequalities for conjugate func-tionsrdquo Indiana UniversityMathematics Journal vol 27 no 5 pp833ndash852 1978

[8] Y J Leung ldquoIntegral means of the derivatives of some univalentfunctionsrdquo Bulletin of the London Mathematical Society vol 11no 3 pp 289ndash294 1979

[9] J E Brown ldquoDerivatives of close-to-convex functions inte-gral means and bounded mean oscillationrdquo MathematischeZeitschrift vol 178 no 3 pp 353ndash358 1981

[10] D Girela ldquoIntegral means and BMOA-norms of logarithmsof univalent functionsrdquo Journal of The London MathematicalSociety-Second Series vol 33 no 1 pp 117ndash132 1986

[11] D Girela ldquoIntegral means bounded mean oscillation andGelrsquofer functionsrdquo Proceedings of the American MathematicalSociety vol 113 no 2 pp 365ndash370 1991

[12] M Nowak ldquoSome inequalities for BMOA functionsrdquo ComplexVariablesTheory andApplication vol 16 no 2-3 pp 81ndash86 1991

[13] P L Duren Univalent Functions Springer-Verlag New YorkNY USA 1983

[14] L V Ahlfors Complex Analysis An Introduction to the Theoryof Analytic Functions of One Complex Variable InternationalSeries in Pure and Applied Mathematics McGraw-Hill BookCo New York NY USA 3rd edition 1978

[15] J A Jenkins ldquoOn circularly symmetric functionsrdquo Proceedingsof the American Mathematical Society vol 6 pp 620ndash624 1955

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 3: Integral Means Inequalities, Convolution, and Univalent ...downloads.hindawi.com/journals/jfs/2019/7817353.pdf · JournalofFunctionSpaces Wereferto[] forthetheoryoftheconvolutionofanalytic

Journal of Function Spaces 3

(ii) For every convex and increasing functionΦ R 997888rarr Rwe have that

int120587minus120587

Φ(log 10038161003816100381610038161003816119891 (119903119890119894120579)10038161003816100381610038161003816) 119889120579 le int120587minus120587

Φ(log 10038161003816100381610038161003816119892 (119903119890119894120579)10038161003816100381610038161003816) 1198891205790 lt 119903 lt 1

(16)

Since for any 119901 gt 0 the function Φ defined by Φ(119909) =exp(119901119909) (119909 isin R) is convex and increasing we deduce that if119891 and 119892 are as in Proposition B and (log |119891|)lowast le (log |119892|)lowast inD+ then

119872119901 (119903 119891) le 119872119901 (119903 119892) 0 lt 119903 lt 1 (17)

for all 119901 gt 0The main achievement in the use of the star-function by

A Baernstein in [5] was the proof that the Koebe function119896(119911) = 119911(1 minus 119911)2 (119911 isin D) is extremal for the integral meansof functions in the class 119878 of univalent functions (see [1 6]for the notation and results regarding univalent functions)Namely Baernstein proved that if 119891 isin 119878 then

(plusmn log 10038161003816100381610038161198911003816100381610038161003816)lowast le (plusmn log |119896|)lowast (18)

and hence

int120587minus120587

10038161003816100381610038161003816119891 (119903119890119894120579)10038161003816100381610038161003816119901 119889120579 le int120587minus120587

10038161003816100381610038161003816119896 (119903119890119894120579)10038161003816100381610038161003816119901 119889120579 0 lt 119903 lt 1 (19)

for all 119901 isin R In particular we have that if 119891 isin 119878 and 0 lt 119901 leinfin then

119872119901 (119903 119891) le 119872119901 (119903 119896) 0 lt 119903 lt 1 (20)

Subsequently the star-function has been used in a goodnumber of papers to obtain bounds on the integral means ofdistinct classes of analytic functions (see eg [7ndash12])

Coming back to convolution the following questionsarise in a natural way

Question 1 Let 119891 119892 119865 119866 be analytic functions in D with |119865|and |119866| being symmetric decreasing on each of the circles|119911| = 119903 and suppose that

(log 10038161003816100381610038161198911003816100381610038161003816)lowast le (log |119865|)lowastand (log 10038161003816100381610038161198921003816100381610038161003816)lowast le (log |119866|)lowast (21)

Does it follow that (log |119891 ⋆ 119892|)lowast le (log |119865 ⋆ 119866|)lowastQuestion 2 Let 119865 and 119891 be two analytic functions in Dand suppose that 119865 is bound preserving Can we assert that(log |119891 ⋆ 119865|)lowast le (log |119891|)lowast

We shall show that the answer to these two questions isnegative Regarding Question 1 we have the following result

Theorem 3 There exist two functions 1198651 1198652 isin H119900119897(D) with(log 1003816100381610038161003816100381611986511989510038161003816100381610038161003816)lowast le (log |119868|)lowast 119891119900119903 119895 = 1 2 (22)

and such that

119905ℎ119890 119894119899119890119902119906119886119897119894119905119910 (log 10038161003816100381610038161198651 ⋆ 11986521003816100381610038161003816)lowastle (log |119868 ⋆ 119868|)lowast 119889119900119890119904 119899119900119905 ℎ119900119897119889 (23)

Here 119868 is the identity element of the convolution defined in(6) that is 119868(119911) = 1(1 minus 119911) (119911 isin D) Hence 119868 ⋆ 119868 = 119868Proof Let ℎ be an odd function in the class 119878 with Taylorexpansion

ℎ (119911) = 119911 + 11988631199113 + 11988651199115 + (24)

with |1198865| gt 1 The existence of such ℎ was proved by Feketeand Szego (see [13 p 104]) Set also

ℎ1 (119911) = ℎ (119911)119911 = 1 + 11988631199112 + 11988651199114 + 119911 isin D (25)

It is well known that there exists a function 119867 isin 119878 such thatℎ(119911) = radic119867(1199112) (see [13 p 64]) Set 1198962(119911) = radic119896(1199112) = 119911(1 minus1199112) and 119869(119911) = 1198962(119911)119911 = 1(1 minus 1199112) (119911 isin D) By Baernsteinrsquostheorem we have (log |119867|)lowast le (log |119896|)lowast a fact which easilyimplies that (log |ℎ1|)lowast le (log |119869|)lowast Now it is clear that 119869 issubordinate to 119868 and then using [8 Lemma 2] we see that(log |119869|)lowast le (log |119868|)lowastThus it follows that

(log 1003816100381610038161003816ℎ11003816100381610038161003816)lowast le (log |119868|)lowast (26)

For 119899 = 1 2 3 we define 119891119899 inductively as follows1198911 = ℎ1

and 119891119899 = 119891119899minus1 ⋆ 1198911 for 119899 ge 2 (27)

In other words 119891119899 =(119899)⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ℎ1 ⋆ sdot sdot sdot ⋆ ℎ1 Clearly (25) yields

119891119899 (119911) = 1 + 11988611989931199112 + 11988611989951199114 + (28)

Since |1198865| gt 1 it follows that |1198861198995 | 997888rarr infin as 119899 997888rarr infin This isequivalent to saying that

10038161003816100381610038161003816119891(4)119899 (0)10038161003816100381610038161003816 997888rarr infin as 119899 997888rarr infin (29)

Then it follows that the family 119891(4)119899 119899 = 1 2 3 is not alocally bounded family of holomorphic functions inD Using[14 Theorem 16 p 225] we see that the same is true for thefamily 119891119899 119899 = 1 2 3 Take 119901 isin (0 1) then 119868 isin 119867119901Since a bounded subset of 119867119901 is a locally bounded family [1p 36] it follows that

sup119899ge1

10038171003817100381710038171198911198991003817100381710038171003817119867119901 = infin (30)

Now (30) implies that 119891119899119867119901 gt 119868119867119901 for some 119899 UsingProposition B we see that this implies that

the inequality (log 10038161003816100381610038161198911198991003816100381610038161003816)lowastle (log |119868|)lowast is not true for some 119899 (31)

Let119873 be the smallest of all such 119899 Using (26) and the fact that1198911 = ℎ1 it follows that 119873 gt 1

4 Journal of Function Spaces

Then it is clear that (23) holds with 1198651 = 1198911 1198652 = 119891119873minus1We have the following result regarding Question 2

Theorem 4 There exist 119891 119865 analytic and univalent in D suchthat 119865 is convexity preserving and with the property that theinequality (log |119891 ⋆ 119865|)lowast le (log |119891|)lowast does not hold

The following lemma will be used in the proof ofTheorem 4

Lemma 5 Let 119891 119865 isin H119900119897((119863) and suppose that 119865(0) = 1 119865is convexity preserving and 119891 and 119891⋆119865 are zero-free inD andsatisfy the inequality (log |119891 ⋆ 119865|)lowast le (log |119891|)lowast Then we alsohave that

(log 100381610038161003816100381610038161003816100381610038161

119891 ⋆ 11986510038161003816100381610038161003816100381610038161003816)lowast le (log 10038161003816100381610038161003816100381610038161003816

111989110038161003816100381610038161003816100381610038161003816)lowast (32)

Proof Set 119906 = log |119891 ⋆ 119865| V = log |119891| Then 119906 and V areharmonic in D 119906(0) = V(0) and 119906lowast le Vlowast Then it followsthat for 0 lt 119903 lt 1 and 0 le 120579 le 120587

(minus119906)lowast (119903119890119894120579) = sup|119864|=2120579

int119864minus119906 (119903119890119894119905) 119889119905

= sup|119864|=2120579

(minusint120587minus120587

119906 (119903119890119894119905) 119889119905 + int[minus120587120587]119864

119906 (119903119890119894119905) 119889119905)= minus2120587119906 (0) + 119906lowast (119903119890119894(120587minus120579))= minus2120587V (0) + 119906lowast (119903119890119894(120587minus120579))le minus2120587V (0) + Vlowast (119903119890119894(120587minus120579)) = (minusV)lowast (119903119890119894120579)

(33)

Hence we have proved that (minus119906)lowast le (minusV)lowast which isequivalent to (32)

Proof of Theorem 4 Set

119891 (119911) = 1(1 minus 119911)2 =

infinsum119899=0

(119899 + 1) 119911119899

119865 (119911) = 1 minus 12119911

119911 isin D

(34)

Clearly 119891 and 119865 are analytic univalent and zero-free in DAlso

(119891 ⋆ 119865) (119911) = 1 minus 119911 119911 isin D (35)

Hence119891⋆119865 is also zero-free inD Notice that 1(119891⋆119865) notin 119867infinand 1119891 isin 119867infin Then it follows that

the inequality (log 100381610038161003816100381610038161003816100381610038161

119891 ⋆ 11986510038161003816100381610038161003816100381610038161003816)lowast

le (log 10038161003816100381610038161003816100381610038161003816111989110038161003816100381610038161003816100381610038161003816)lowast

does not hold(36)

Now it is a simple exercise to check that

119865 (119911) = 12120587 int120587minus120587

1 minus cos 1205791 minus 119890119894120579119911 119889120579 (37)

and then it follows that 119865 is convexity preservingThen using(36) and Lemma 5 it follows that the inequality (log |119891 ⋆119865|)lowast le (log |119891|)lowast does not hold as desired

We close the paper with a positive result determining aclass of univalent functions Z such that (10) is true for all119901 gt 0 whenever 119891 isin Z and 119865 is convexity preserving

A domain 119863 in C is said to be Steiner symmetric if itsintersection with each vertical line is either empty or is thewhole line or is a segment placed symmetrically with respectto the real axis We letZ be the class of all functions 119891 whichare analytic and univalent in D with 119891(0) = 0 1198911015840(0) gt 0 andwhose image is a Steiner symmetric domain The elementsof Z will be called Steiner symmetric functions Usingarguments similar to those used by Jenkins [15] for circularlysymmetric functions we see that a univalent function 119891with119891(0) = 0 and 1198911015840(0) gt 0 is Steiner symmetric if and only ifit satisfies the following two conditions (i) 119891 is typically realand (ii) Re119891 is a symmetric decreasing function on each of thecircles |119911| = 119903 (0 lt 119903 lt 1)Then it follows that if119891 isin Z thenfor every 119903 isin (0 1) the domain 119891(|119911| lt 119903) is a Steiner sym-metric domain and hence the function 119891119903 defined by119891119903(119911) =119891(119903119911) (119911 isin D) belongs toZ and it extends to an analytic func-tion in the closed unit discD Nowwe can state our last result

Theorem 6 Suppose that 119891 isin Z and let 119865 be an analyticfunction inDwhich is convexity preserving We have for every119901 gt 0

119872119901 (119903 119891 ⋆ 119865) le 119872119901 (119903 119891) 0 lt 119903 lt 1 (38)

Proof In view of Theorem 1 we only need to prove (38) for0 lt 119901 lt 1 Let 120583 be the probability measure on T such that119865(119911) = intT(119889120583(120585)(1 minus 119911120585)) (119911 isin D) Then we have

(119891 ⋆ 119865) (119911) = intT

119891 (120585119911) 119889120583 (120585) (39)

Since 119865 is convexity preserving for 0 lt 119903 lt 1 we have that(119891119903 ⋆ 119865) (D) is contained in the closed convex hull of 119891119903(D)This easily yields

min119911isinD

Re119891119903 (119911) le min119911isinD

Re (119891119903 ⋆ 119865) (119911) max119911isinD

Re (119891119903 ⋆ 119865) (119911) le max119911isinD

Re119891119903 (119911) (40)

By the remarks in the previous paragraph we find that for all119903 isin (0 1)119891119903 belongs toZ and extends to an analytic functionin the closed unit disc D Finally we claim that

(Re (119891119903 ⋆ 119865))lowast le (Re119891119903)lowast 0 lt 119903 lt 1 (41)

Once this is proved using Proposition 6 of [10] we deducethat119872119901 (119903 119891 ⋆ 119865) = 1003817100381710038171003817119891119903 ⋆ 1198651003817100381710038171003817119867119901 le 10038171003817100381710038171198911199031003817100381710038171003817119867119901 = 119872119901 (119903 119891)

0 lt 119901 le 2 (42)

finishing our proof

Journal of Function Spaces 5

So we proceed to prove (41) Fix 119903 isin (0 1) and set 119906 =Re(119891119903 ⋆ 119865) V = Re119891119903 Using (39) we have for 0 lt 119877 lt 1 and0 lt 120579 lt 120587

119906lowast (119877119890119894120579) = sup|119864|=2120579

int119864119906 (119877119890119894119905) 119889119905

= sup|119864|=2120579

int119864intT

V (119877119890119894119905120585) 119889120583 (120585) 119889119905

= sup|119864|=2120579

intT

int119864V (119877119890119894119905120585) 119889119905 119889120583 (120585)

le intT

Vlowast (119877119890119894120579) 119889120583 (120585) = Vlowast (119877119890119894120579)

(43)

Data Availability

No data were used to support this study

Conflicts of Interest

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

Acknowledgments

This research is supported by a grant from ldquoEl Ministeriode Economıa y Competitividadrdquo Spain (MTM2014-52865-P)and by a grant from la Junta de Andalucıa (FQM-210) Wewish also to dedicate this article to Fernando Perez Gonzalezon the occasion of his retirement

References

[1] P L Duren Theory of H119901 Spaces Academic Press New York-London 1970 Reprint Dover Mineola-New York 2000

[2] S Ruscheweyh Convolutions in Geometric Function TheorySeminaire de Mathematiques Superieures [Seminar on HigherMathematics] Fundamental Theories of Physics Presses delrsquoUniversite de Montreal Montreal Canada 1982

[3] T Sheil-Small ldquoOn the convolution of analytic functionsrdquoJournal fur die Reine und Angewandte Mathematik vol 258 pp137ndash152 1973

[4] A Baernstein ldquoProof of Edreirsquos spread conjecturerdquo Proceedingsof the London Mathematical Society vol 26 no 3 pp 418ndash4341973

[5] A Baernstein ldquoIntegralmeans univalent functions and circularsymmetrizationrdquo Acta Mathematica vol 133 pp 139ndash169 1974

[6] C Pommerenke Univalent Functions Vandenhoeck amp Ru-precht Gottingen Germany 1975

[7] A Baernstein ldquoSome sharp inequalities for conjugate func-tionsrdquo Indiana UniversityMathematics Journal vol 27 no 5 pp833ndash852 1978

[8] Y J Leung ldquoIntegral means of the derivatives of some univalentfunctionsrdquo Bulletin of the London Mathematical Society vol 11no 3 pp 289ndash294 1979

[9] J E Brown ldquoDerivatives of close-to-convex functions inte-gral means and bounded mean oscillationrdquo MathematischeZeitschrift vol 178 no 3 pp 353ndash358 1981

[10] D Girela ldquoIntegral means and BMOA-norms of logarithmsof univalent functionsrdquo Journal of The London MathematicalSociety-Second Series vol 33 no 1 pp 117ndash132 1986

[11] D Girela ldquoIntegral means bounded mean oscillation andGelrsquofer functionsrdquo Proceedings of the American MathematicalSociety vol 113 no 2 pp 365ndash370 1991

[12] M Nowak ldquoSome inequalities for BMOA functionsrdquo ComplexVariablesTheory andApplication vol 16 no 2-3 pp 81ndash86 1991

[13] P L Duren Univalent Functions Springer-Verlag New YorkNY USA 1983

[14] L V Ahlfors Complex Analysis An Introduction to the Theoryof Analytic Functions of One Complex Variable InternationalSeries in Pure and Applied Mathematics McGraw-Hill BookCo New York NY USA 3rd edition 1978

[15] J A Jenkins ldquoOn circularly symmetric functionsrdquo Proceedingsof the American Mathematical Society vol 6 pp 620ndash624 1955

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 4: Integral Means Inequalities, Convolution, and Univalent ...downloads.hindawi.com/journals/jfs/2019/7817353.pdf · JournalofFunctionSpaces Wereferto[] forthetheoryoftheconvolutionofanalytic

4 Journal of Function Spaces

Then it is clear that (23) holds with 1198651 = 1198911 1198652 = 119891119873minus1We have the following result regarding Question 2

Theorem 4 There exist 119891 119865 analytic and univalent in D suchthat 119865 is convexity preserving and with the property that theinequality (log |119891 ⋆ 119865|)lowast le (log |119891|)lowast does not hold

The following lemma will be used in the proof ofTheorem 4

Lemma 5 Let 119891 119865 isin H119900119897((119863) and suppose that 119865(0) = 1 119865is convexity preserving and 119891 and 119891⋆119865 are zero-free inD andsatisfy the inequality (log |119891 ⋆ 119865|)lowast le (log |119891|)lowast Then we alsohave that

(log 100381610038161003816100381610038161003816100381610038161

119891 ⋆ 11986510038161003816100381610038161003816100381610038161003816)lowast le (log 10038161003816100381610038161003816100381610038161003816

111989110038161003816100381610038161003816100381610038161003816)lowast (32)

Proof Set 119906 = log |119891 ⋆ 119865| V = log |119891| Then 119906 and V areharmonic in D 119906(0) = V(0) and 119906lowast le Vlowast Then it followsthat for 0 lt 119903 lt 1 and 0 le 120579 le 120587

(minus119906)lowast (119903119890119894120579) = sup|119864|=2120579

int119864minus119906 (119903119890119894119905) 119889119905

= sup|119864|=2120579

(minusint120587minus120587

119906 (119903119890119894119905) 119889119905 + int[minus120587120587]119864

119906 (119903119890119894119905) 119889119905)= minus2120587119906 (0) + 119906lowast (119903119890119894(120587minus120579))= minus2120587V (0) + 119906lowast (119903119890119894(120587minus120579))le minus2120587V (0) + Vlowast (119903119890119894(120587minus120579)) = (minusV)lowast (119903119890119894120579)

(33)

Hence we have proved that (minus119906)lowast le (minusV)lowast which isequivalent to (32)

Proof of Theorem 4 Set

119891 (119911) = 1(1 minus 119911)2 =

infinsum119899=0

(119899 + 1) 119911119899

119865 (119911) = 1 minus 12119911

119911 isin D

(34)

Clearly 119891 and 119865 are analytic univalent and zero-free in DAlso

(119891 ⋆ 119865) (119911) = 1 minus 119911 119911 isin D (35)

Hence119891⋆119865 is also zero-free inD Notice that 1(119891⋆119865) notin 119867infinand 1119891 isin 119867infin Then it follows that

the inequality (log 100381610038161003816100381610038161003816100381610038161

119891 ⋆ 11986510038161003816100381610038161003816100381610038161003816)lowast

le (log 10038161003816100381610038161003816100381610038161003816111989110038161003816100381610038161003816100381610038161003816)lowast

does not hold(36)

Now it is a simple exercise to check that

119865 (119911) = 12120587 int120587minus120587

1 minus cos 1205791 minus 119890119894120579119911 119889120579 (37)

and then it follows that 119865 is convexity preservingThen using(36) and Lemma 5 it follows that the inequality (log |119891 ⋆119865|)lowast le (log |119891|)lowast does not hold as desired

We close the paper with a positive result determining aclass of univalent functions Z such that (10) is true for all119901 gt 0 whenever 119891 isin Z and 119865 is convexity preserving

A domain 119863 in C is said to be Steiner symmetric if itsintersection with each vertical line is either empty or is thewhole line or is a segment placed symmetrically with respectto the real axis We letZ be the class of all functions 119891 whichare analytic and univalent in D with 119891(0) = 0 1198911015840(0) gt 0 andwhose image is a Steiner symmetric domain The elementsof Z will be called Steiner symmetric functions Usingarguments similar to those used by Jenkins [15] for circularlysymmetric functions we see that a univalent function 119891with119891(0) = 0 and 1198911015840(0) gt 0 is Steiner symmetric if and only ifit satisfies the following two conditions (i) 119891 is typically realand (ii) Re119891 is a symmetric decreasing function on each of thecircles |119911| = 119903 (0 lt 119903 lt 1)Then it follows that if119891 isin Z thenfor every 119903 isin (0 1) the domain 119891(|119911| lt 119903) is a Steiner sym-metric domain and hence the function 119891119903 defined by119891119903(119911) =119891(119903119911) (119911 isin D) belongs toZ and it extends to an analytic func-tion in the closed unit discD Nowwe can state our last result

Theorem 6 Suppose that 119891 isin Z and let 119865 be an analyticfunction inDwhich is convexity preserving We have for every119901 gt 0

119872119901 (119903 119891 ⋆ 119865) le 119872119901 (119903 119891) 0 lt 119903 lt 1 (38)

Proof In view of Theorem 1 we only need to prove (38) for0 lt 119901 lt 1 Let 120583 be the probability measure on T such that119865(119911) = intT(119889120583(120585)(1 minus 119911120585)) (119911 isin D) Then we have

(119891 ⋆ 119865) (119911) = intT

119891 (120585119911) 119889120583 (120585) (39)

Since 119865 is convexity preserving for 0 lt 119903 lt 1 we have that(119891119903 ⋆ 119865) (D) is contained in the closed convex hull of 119891119903(D)This easily yields

min119911isinD

Re119891119903 (119911) le min119911isinD

Re (119891119903 ⋆ 119865) (119911) max119911isinD

Re (119891119903 ⋆ 119865) (119911) le max119911isinD

Re119891119903 (119911) (40)

By the remarks in the previous paragraph we find that for all119903 isin (0 1)119891119903 belongs toZ and extends to an analytic functionin the closed unit disc D Finally we claim that

(Re (119891119903 ⋆ 119865))lowast le (Re119891119903)lowast 0 lt 119903 lt 1 (41)

Once this is proved using Proposition 6 of [10] we deducethat119872119901 (119903 119891 ⋆ 119865) = 1003817100381710038171003817119891119903 ⋆ 1198651003817100381710038171003817119867119901 le 10038171003817100381710038171198911199031003817100381710038171003817119867119901 = 119872119901 (119903 119891)

0 lt 119901 le 2 (42)

finishing our proof

Journal of Function Spaces 5

So we proceed to prove (41) Fix 119903 isin (0 1) and set 119906 =Re(119891119903 ⋆ 119865) V = Re119891119903 Using (39) we have for 0 lt 119877 lt 1 and0 lt 120579 lt 120587

119906lowast (119877119890119894120579) = sup|119864|=2120579

int119864119906 (119877119890119894119905) 119889119905

= sup|119864|=2120579

int119864intT

V (119877119890119894119905120585) 119889120583 (120585) 119889119905

= sup|119864|=2120579

intT

int119864V (119877119890119894119905120585) 119889119905 119889120583 (120585)

le intT

Vlowast (119877119890119894120579) 119889120583 (120585) = Vlowast (119877119890119894120579)

(43)

Data Availability

No data were used to support this study

Conflicts of Interest

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

Acknowledgments

This research is supported by a grant from ldquoEl Ministeriode Economıa y Competitividadrdquo Spain (MTM2014-52865-P)and by a grant from la Junta de Andalucıa (FQM-210) Wewish also to dedicate this article to Fernando Perez Gonzalezon the occasion of his retirement

References

[1] P L Duren Theory of H119901 Spaces Academic Press New York-London 1970 Reprint Dover Mineola-New York 2000

[2] S Ruscheweyh Convolutions in Geometric Function TheorySeminaire de Mathematiques Superieures [Seminar on HigherMathematics] Fundamental Theories of Physics Presses delrsquoUniversite de Montreal Montreal Canada 1982

[3] T Sheil-Small ldquoOn the convolution of analytic functionsrdquoJournal fur die Reine und Angewandte Mathematik vol 258 pp137ndash152 1973

[4] A Baernstein ldquoProof of Edreirsquos spread conjecturerdquo Proceedingsof the London Mathematical Society vol 26 no 3 pp 418ndash4341973

[5] A Baernstein ldquoIntegralmeans univalent functions and circularsymmetrizationrdquo Acta Mathematica vol 133 pp 139ndash169 1974

[6] C Pommerenke Univalent Functions Vandenhoeck amp Ru-precht Gottingen Germany 1975

[7] A Baernstein ldquoSome sharp inequalities for conjugate func-tionsrdquo Indiana UniversityMathematics Journal vol 27 no 5 pp833ndash852 1978

[8] Y J Leung ldquoIntegral means of the derivatives of some univalentfunctionsrdquo Bulletin of the London Mathematical Society vol 11no 3 pp 289ndash294 1979

[9] J E Brown ldquoDerivatives of close-to-convex functions inte-gral means and bounded mean oscillationrdquo MathematischeZeitschrift vol 178 no 3 pp 353ndash358 1981

[10] D Girela ldquoIntegral means and BMOA-norms of logarithmsof univalent functionsrdquo Journal of The London MathematicalSociety-Second Series vol 33 no 1 pp 117ndash132 1986

[11] D Girela ldquoIntegral means bounded mean oscillation andGelrsquofer functionsrdquo Proceedings of the American MathematicalSociety vol 113 no 2 pp 365ndash370 1991

[12] M Nowak ldquoSome inequalities for BMOA functionsrdquo ComplexVariablesTheory andApplication vol 16 no 2-3 pp 81ndash86 1991

[13] P L Duren Univalent Functions Springer-Verlag New YorkNY USA 1983

[14] L V Ahlfors Complex Analysis An Introduction to the Theoryof Analytic Functions of One Complex Variable InternationalSeries in Pure and Applied Mathematics McGraw-Hill BookCo New York NY USA 3rd edition 1978

[15] J A Jenkins ldquoOn circularly symmetric functionsrdquo Proceedingsof the American Mathematical Society vol 6 pp 620ndash624 1955

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 5: Integral Means Inequalities, Convolution, and Univalent ...downloads.hindawi.com/journals/jfs/2019/7817353.pdf · JournalofFunctionSpaces Wereferto[] forthetheoryoftheconvolutionofanalytic

Journal of Function Spaces 5

So we proceed to prove (41) Fix 119903 isin (0 1) and set 119906 =Re(119891119903 ⋆ 119865) V = Re119891119903 Using (39) we have for 0 lt 119877 lt 1 and0 lt 120579 lt 120587

119906lowast (119877119890119894120579) = sup|119864|=2120579

int119864119906 (119877119890119894119905) 119889119905

= sup|119864|=2120579

int119864intT

V (119877119890119894119905120585) 119889120583 (120585) 119889119905

= sup|119864|=2120579

intT

int119864V (119877119890119894119905120585) 119889119905 119889120583 (120585)

le intT

Vlowast (119877119890119894120579) 119889120583 (120585) = Vlowast (119877119890119894120579)

(43)

Data Availability

No data were used to support this study

Conflicts of Interest

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

Acknowledgments

This research is supported by a grant from ldquoEl Ministeriode Economıa y Competitividadrdquo Spain (MTM2014-52865-P)and by a grant from la Junta de Andalucıa (FQM-210) Wewish also to dedicate this article to Fernando Perez Gonzalezon the occasion of his retirement

References

[1] P L Duren Theory of H119901 Spaces Academic Press New York-London 1970 Reprint Dover Mineola-New York 2000

[2] S Ruscheweyh Convolutions in Geometric Function TheorySeminaire de Mathematiques Superieures [Seminar on HigherMathematics] Fundamental Theories of Physics Presses delrsquoUniversite de Montreal Montreal Canada 1982

[3] T Sheil-Small ldquoOn the convolution of analytic functionsrdquoJournal fur die Reine und Angewandte Mathematik vol 258 pp137ndash152 1973

[4] A Baernstein ldquoProof of Edreirsquos spread conjecturerdquo Proceedingsof the London Mathematical Society vol 26 no 3 pp 418ndash4341973

[5] A Baernstein ldquoIntegralmeans univalent functions and circularsymmetrizationrdquo Acta Mathematica vol 133 pp 139ndash169 1974

[6] C Pommerenke Univalent Functions Vandenhoeck amp Ru-precht Gottingen Germany 1975

[7] A Baernstein ldquoSome sharp inequalities for conjugate func-tionsrdquo Indiana UniversityMathematics Journal vol 27 no 5 pp833ndash852 1978

[8] Y J Leung ldquoIntegral means of the derivatives of some univalentfunctionsrdquo Bulletin of the London Mathematical Society vol 11no 3 pp 289ndash294 1979

[9] J E Brown ldquoDerivatives of close-to-convex functions inte-gral means and bounded mean oscillationrdquo MathematischeZeitschrift vol 178 no 3 pp 353ndash358 1981

[10] D Girela ldquoIntegral means and BMOA-norms of logarithmsof univalent functionsrdquo Journal of The London MathematicalSociety-Second Series vol 33 no 1 pp 117ndash132 1986

[11] D Girela ldquoIntegral means bounded mean oscillation andGelrsquofer functionsrdquo Proceedings of the American MathematicalSociety vol 113 no 2 pp 365ndash370 1991

[12] M Nowak ldquoSome inequalities for BMOA functionsrdquo ComplexVariablesTheory andApplication vol 16 no 2-3 pp 81ndash86 1991

[13] P L Duren Univalent Functions Springer-Verlag New YorkNY USA 1983

[14] L V Ahlfors Complex Analysis An Introduction to the Theoryof Analytic Functions of One Complex Variable InternationalSeries in Pure and Applied Mathematics McGraw-Hill BookCo New York NY USA 3rd edition 1978

[15] J A Jenkins ldquoOn circularly symmetric functionsrdquo Proceedingsof the American Mathematical Society vol 6 pp 620ndash624 1955

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 6: Integral Means Inequalities, Convolution, and Univalent ...downloads.hindawi.com/journals/jfs/2019/7817353.pdf · JournalofFunctionSpaces Wereferto[] forthetheoryoftheconvolutionofanalytic

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


CHAPTER 5 HARMONIC UNIVALENT AND MULTIVALENT FUNCTIONSshodhganga.inflibnet.ac.in/bitstream/10603/3418/13/13_chapter 5.pdf · Harmonic Univalent and Multivalent Functions 5.1 Introduction

CHAPTER 5 HARMONIC UNIVALENT AND MULTIVALENT FUNCTIONSshodhganga.inflibnet.ac.in/bitstream/10603/3418/13/13_chapter 5.pdf · Harmonic Univalent and Multivalent Functions 5.1 Introduction

Parametric representation of univalent self-maps with ...gumenyuk/slides/TalkBilbao2014_flat.pdf · Parametric representation of univalent self-maps with given boundary fixed points

Parametric representation of univalent self-maps with ...gumenyuk/slides/TalkBilbao2014_flat.pdf · Parametric representation of univalent self-maps with given boundary fixed points

Criteria for univalent, starlike and ... - Project Euclid

Criteria for univalent, starlike and ... - Project Euclid

JFS Newsletter - January 2011

JFS Newsletter - January 2011

Axiomatizing Cubical Sets Models of Univalent Foundations · HoTT/UF from the outside in Why study models of univalent type theory? (instead of just developing univalent foundations)

Axiomatizing Cubical Sets Models of Univalent Foundations · HoTT/UF from the outside in Why study models of univalent type theory? (instead of just developing univalent foundations)

ResearchArticle Integration in Orlicz-Bochner Spacesdownloads.hindawi.com/journals/jfs/2018/9380350.pdf · JournalofFunctionSpaces Let T [T ,T0](briey T ) denote the natural mixed

ResearchArticle Integration in Orlicz-Bochner Spacesdownloads.hindawi.com/journals/jfs/2018/9380350.pdf · JournalofFunctionSpaces Let T [T ,T0](briey T ) denote the natural mixed

Homotopy Type Theory - heidelberg-laureate-forum.org · 03.02.2011 · Homotopy Type Theory Univalent Foundations of Mathematics The Univalent Foundations Program Institute for Advanced

Homotopy Type Theory - heidelberg-laureate-forum.org · 03.02.2011 · Homotopy Type Theory Univalent Foundations of Mathematics The Univalent Foundations Program Institute for Advanced

Jfs 351900193

Jfs 351900193

i Jfs 2281328905800

i Jfs 2281328905800

An introduction to univalent foundations for mathematicians

An introduction to univalent foundations for mathematicians

JFS Tuning

JFS Tuning

JFS Social Media Presentation

JFS Social Media Presentation

Duren Peter - Univalent Functions

Duren Peter - Univalent Functions

Jfs Final Presentation

Jfs Final Presentation

JFS Family Matters 2014

JFS Family Matters 2014

JFS New Orleans

JFS New Orleans

Zippers and univalent functions

Zippers and univalent functions

JFS Annual Report 2011

JFS Annual Report 2011

JFS Solutions

JFS Solutions

CV JFS Int.

CV JFS Int.