Integral means inequalities, convolution, and univalent functions

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.


Introduction
Let D = {z ∈ C : |z| < 1} and T = {z ∈ C : |z| = 1} denote the open unit disc and the unit circle in the complex plane C. We let also Hol(D) be the space of all analytic functions in D endowed with the topology of uniform convergence in compact subsets.
If 0 ≤ r < 1 and f ∈ Hol(D), we set For 0 < p ≤ ∞, the Hardy space H p consists of those f ∈ Hol(D) such that We refer to [6] for the theory of H p -spaces. If f, g ∈ Hol(D), f (z) = ∞ n=0 a n z n , g(z) = ∞ n=0 b n z n (z ∈ D), the (Hadamard) convolution (f ⋆ g) of f and g is defined by (f ⋆ g)(z) = ∞ n=0 a n b n z n , z ∈ D.
Following [15], we shall say that a function F ∈ Hol(D) is bound preserving if for every Sheil-Small [15,Theorem 1. 3] (see also [14, p. 123]) proved that a function F ∈ Hol(D) is bound preserving if and only if there exists a complex Borel measure µ on T with µ ≤ 1 such that The measure µ is a probability measure if and only if F is convexity preserving, that is, for any f ∈ Hol(D) the range of f ⋆ F is contained in the closed convex hull of the range of f [14, pp. 123, 124]. It turns out that if F is bound preserving and 1 ≤ p ≤ ∞, then for every f ∈ H p we have that f ⋆ F ∈ H p and Actually, the following stronger result holds. Theorem 1. Suppose that f, F ∈ Hol(D) with F being bound preserving. Then Proof. Since F is bound preserving, there exists a complex Borel measure µ on T with µ ≤ 1 such that If f (z) = ∞ n=0 a n z n (z ∈ D), we have This immediately yields (1.2) for p = ∞. Now, if 1 ≤ p < ∞, using Minkowski's integral inequality we obtain

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 [2,3].
If u is a subharmonic function in D \ {0}, the function u * is defined by where |E| denotes the Lebesgue measure of the set E. The basic properties of the starfunction which make it useful to solve extremal problems are the following [3]: • If u is a subharmonic function in D \ {0}, then the function u * is subharmonic in , and it is a symmetric decreasing function on each of the The relevance of the star-function to obtain integral means estimates comes from the following result.
). Let u and v be two subharmonic functions in D. Then the following two conditions are equivalent: Proposition A yields the following result about analytic functions.
Proposition B. Let f and g be two non-identically zero analytic functions in D. Then the following conditions are equivalent: (ii) For every convex and increasing function Φ : Since for any p > 0 the function Φ defined by Φ(x) = exp(px) (x ∈ R) is convex and increasing we deduce that if f and g are as in Proposition B and (log |f |) The main achievement in the use of the star-function by A. Baernstein in [3], was the proof that the Koebe function is extremal for the integral means of functions in the class S of univalent functions (see [6] and [13] for the notation and results regarding univalent functions). Namely, Baernstein proved that if f ∈ S then for all p ∈ R. In particular, we have that if f ∈ S and 0 < p ≤ ∞, 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., [4,11,5,8,9,12]).
Coming back to convolution, the following questions arise in a natural way. Does it follow that log |f ⋆ g| * ≤ log |F ⋆ G| * ?
Question 2. Let F and f be two analytic functions in D and suppose that F is bound preserving. Can we assert that (log |f ⋆ F |) * ≤ (log |f |) * ?
We shall show that the answer to these two questions is negative. Regarding Question 1 we have the following result.
Then it follows that the family {f Now, (2.4) implies that f n H p > I H p for some n. Using Proposition B, we see that this implies that the inequality (log |f n |) * ≤ (log |I|) * is not true for some n.
Let N be the smallest of all such n. Using (2.3) and the fact that f 1 = h 1 , it follows that that N > 1.
Then it is clear that (2.1) holds with We have the following result regarding Question 2.
Theorem 3. There exist f, F analytic and univalent in D such that F is convexity preserving and with the property that the inequality (log |f ⋆ F |) * ≤ (log |f |) * does not hold.
The following lemma will be used in the proof of Theorem 3.
Proof of Theorem 3. Set Clearly, f and F are analytic, univalent, and zero-free in D. Also Hence f ⋆ F is also zero-free in D.
Notice that 1 f ⋆F ∈ H ∞ and 1 f ∈ H ∞ . Then it follows that Now, it is a simple exercise to check that and then it follows that F is convexity preserving. Then, using (2.6) and Lemma 1, it follows that the inequality (log |f ⋆ F |) * ≤ (log |f |) * does not hold, as desired.
We close the paper with a positive result, determining a class of univalent functions Z such that (1.2) is true for all p > 0, whenever f ∈ Z and F is convexity preserving.
A domain D in C 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 Z be the class of all functions f which are analytic and univalent in D with f (0) = 0, f ′ (0) > 0, and whose image is a Steiner symmetric domain. The elements of Z will be called Steiner symmetric functions. Using arguments similar to those used by Jenkins [10] for circularly symmetric functions, we see that a univalent function f with f (0) = 0 and f ′ (0) > 0 is Steiner symmetric if and only if it satisfies the following two conditions: (i) f is typically real and (ii) Re f is a symmetric decreasing function on each of the circles {|z| = r} (0 < r < 1). Then it follows that if f ∈ Z then for every r ∈ (0, 1), the domain f ({|z| < r}) is a Steiner symmetric domain and, hence, the function f r defined by f r (z) = f (rz) (z ∈ D) belongs to Z and it extends to an analytic function in the closed unit disc D. Now we can state our last result.
Theorem 4. Suppose that f ∈ Z and let F be an analytic function in D which is convexity preserving. We have, for every p > 0, Proof. In view of Theorem 1 we only need to prove (2.7) for 0 < p < 1. Let µ be the probability measure on T such that F (z) = T dµ(ξ) 1−zξ (z ∈ D). Then we have Since F is convexity preserving, for 0 < r < 1, we have that (f r ⋆ F ) (D) is contained in the closed convex hull of f r (D). This easily yields min z∈D Re f r (z) ≤ min z∈D Re (f r ⋆ F )(z), max z∈D Re (f r ⋆ F )(z) ≤ max z∈D Re f r (z).
By the remarks in the previous paragraph, we find that, for all r ∈ (0, 1), f r belongs to Z and extends to an analytic function in the closed unit disc D. Finally, we claim that (2.9) Re (f r ⋆ F ) * ≤ Re f r * , 0 < r < 1.