Abstract
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. The theory of univalent functions plays a basic role in our work.
1. Introduction
Let and denote the open unit disc and the unit circle in the complex plane . We let also be the space of all analytic functions in endowed with the topology of uniform convergence in compact subsets.
If and , we set
For , the Hardy space consists of those such that We refer to [1] for the theory of -spaces.
If , the (Hadamard) convolution of and is defined by We have the following integral representation: (see [2, p. 11]). The convolution operation makes into a commutative complex algebra with an identityWe refer to [2] for the theory of the convolution of analytic functions and its connections with geometric function theory.
Following [3], we shall say that a function is bound preserving if for every we have that and
Sheil-Small [3, Theorem 1.3] (see also [2, p. 123]) proved that a function is bound preserving if and only if there exists a complex Borel measure on with such that The measure is a probability measure if and only if is convexity preserving; that is, for any the range of is contained in the closed convex hull of the range of [2, pp. 123, 124].
It turns out that if is bound preserving and , then for every we have that andActually, the following stronger result holds.
Theorem 1. Suppose that with being bound preserving. Thenwhenever .
Proof. Since is bound preserving, there exists a complex Borel measure on with such thatIf (), we have This immediately yields (10) for . Now, if , using Minkowski’s integral inequality we obtain
2. Star-Type Inequalities
The main purpose of this article is studying the possibility of extending Theorem 1 to cover other integral means, at least for some special classes of functions. In order to do so, we shall use the method of the star-function introduced by A. Baernstein [4, 5].
If is a subharmonic function in , the function is defined bywhere denotes the Lebesgue measure of the set . The basic properties of the star-function which make it useful to solve extremal problems are the following [5]. (i)If is a subharmonic function in , then the function is subharmonic in and continuous in .(ii)If is harmonic in , and it is a symmetric decreasing function on each of the circles (), then is harmonic in and; in fact, .
The relevance of the star-function to obtain integral means estimates comes from the following result.
Proposition A (see [5]). Let and be two subharmonic functions in . Then the following two conditions are equivalent: (i) in .(ii)For every convex and increasing function , we have that
Proposition A yields the following result about analytic functions.
Proposition B. Let and be two nonidentically zero analytic functions in . Then the following conditions are equivalent: (i) in .(ii)For every convex and increasing function , we have that
Since for any the function defined by () is convex and increasing, we deduce that if and are as in Proposition B and in , thenfor all .
The main achievement in the use of the star-function by A. Baernstein in [5] was the proof that the Koebe function () is extremal for the integral means of functions in the class of univalent functions (see [1, 6] for the notation and results regarding univalent functions). Namely, Baernstein proved that if then and, hence, for all . In particular, we have that if and , then
Subsequently the star-function has been used in a good number of papers to obtain bounds on the integral means of distinct classes of analytic functions (see, e.g., [7–12]).
Coming back to convolution, the following questions arise in a natural way.
Question 1. Let be analytic functions in with and being symmetric decreasing on each of the circles and suppose thatDoes it follow that
Question 2. Let and be two analytic functions in and suppose that is bound preserving. Can we assert that ?
We shall show that the answer to these two questions is negative. Regarding Question 1 we have the following result.
Theorem 3. There exist two functions withand such thatHere, is the identity element of the convolution defined in (6); that is, . Hence .
Proof. Let be an odd function in the class with Taylor expansionwith . The existence of such was proved by Fekete and Szegö (see [13, p. 104]). Set alsoIt is well known that there exists a function such that (see [13, p. 64]). Set and (). By Baernstein’s theorem we have , a fact which easily implies that . Now, it is clear that is subordinate to and then, using [8, Lemma 2], we see that Thus it follows thatFor , we define inductively as follows: In other words, . Clearly, (25) yields Since , it follows that , as . This is equivalent to saying thatThen it follows that the family is not a locally bounded family of holomorphic functions in . Using [14, Theorem 16, p. 225] we see that the same is true for the family . Take , then . Since a bounded subset of is a locally bounded family [1, p. 36], it follows thatNow, (30) implies that for some . Using Proposition B, we see that this implies that Let be the smallest of all such . Using (26) and the fact that , it follows that .
Then it is clear that (23) holds with .
We have the following result regarding Question 2.
Theorem 4. There exist analytic and univalent in such that is convexity preserving and with the property that the inequality does not hold.
The following lemma will be used in the proof of Theorem 4.
Lemma 5. Let and suppose that , is convexity preserving, and and are zero-free in and satisfy the inequality Then we also have that
Proof. Set , . Then and are harmonic in , , and . Then it follows that, for and , Hence, we have proved that which is equivalent to (32).
Proof of Theorem 4. SetClearly, and are analytic, univalent, and zero-free in . Also Hence is also zero-free in . Notice that and . Then it follows thatNow, it is a simple exercise to check thatand then it follows that is convexity preserving. Then, using (36) and Lemma 5, it follows that the inequality does not hold, as desired.
We close the paper with a positive result, determining a class of univalent functions such that (10) is true for all , whenever and is convexity preserving.
A domain in is said to be Steiner symmetric if its intersection with each vertical line is either empty, or is the whole line, or is a segment placed symmetrically with respect to the real axis. We let be the class of all functions which are analytic and univalent in with , , and whose image is a Steiner symmetric domain. The elements of will be called Steiner symmetric functions. Using arguments similar to those used by Jenkins [15] for circularly symmetric functions, we see that a univalent function with and is Steiner symmetric if and only if it satisfies the following two conditions: (i) is typically real and (ii) is a symmetric decreasing function on each of the circles (). Then it follows that if , then for every , the domain is a Steiner symmetric domain and, hence, the function defined by () belongs to and it extends to an analytic function in the closed unit disc . Now we can state our last result.
Theorem 6. Suppose that and let be an analytic function in which is convexity preserving. We have, for every ,
Proof. In view of Theorem 1 we only need to prove (38) for . Let be the probability measure on such that (). Then we haveSince is convexity preserving, for , we have that is contained in the closed convex hull of . This easily yields By the remarks in the previous paragraph, we find that, for all , belongs to and extends to an analytic function in the closed unit disc . Finally, we claim thatOnce this is proved, using Proposition 6 of [10], we deduce thatfinishing our proof.
So we proceed to prove (41). Fix and set , . Using (39), we have, for and ,
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research is supported by a grant from “El Ministerio de Economía y Competitividad”, Spain (MTM2014-52865-P) and by a grant from la Junta de Andalucía (FQM-210). We wish also to dedicate this article to Fernando Pérez González on the occasion of his retirement.