Integral means inequalities, convolution, and univalent functions

TL;DR

This paper explores inequalities related to integral means of convolutions of analytic functions within the unit disc, utilizing the Baernstein star-function and univalent function theory.

Contribution

It introduces new methods for analyzing integral means inequalities using the Baernstein star-function in the context of univalent functions.

Findings

Established new inequalities for integral means of convolutions.

Applied the Baernstein star-function to problems in univalent function theory.

Enhanced understanding of convolution behavior in analytic functions.

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.

Equations

$$M_p(r,f)=$$

$$M_{\infty }(r,f)=$$

$$\Vert f{\Vert }_{H^p}\stackrel{\text{def}}{=}\underset{0\le r<1}{\sup }{M}_p(r,f)<\infty .$$

$$f(z)=\sum _{n=0}^{\infty }{a}_n{z}^n,g(z)=\sum _{n=0}^{\infty }{b}_n{z}^n(z\in \mathbb{D}),$$

$$(f\star g)(z)=\sum _{n=0}^{\infty }{a}_n{b}_n{z}^n,z\in \mathbb{D}.$$

$$(f\star g)(z)={\frac{1}{2\pi i}\int_{|\xi|=r}f\Bigl{(}\frac{z}{\xi}\Bigr{)}g(\xi)\frac{d\xi}{\xi}},\quad|z|<r<1,$$

$$I(z)=\frac{1}{1-z}=\sum _{n=0}^{\infty }{z}^n,z\in \mathbb{D}.$$

$$\Vert f\star F{\Vert }_{H^{\infty }}\le \Vert f{\Vert }_{H^{\infty }}.$$

$$F(z)={\int }_{\mathbb{T}}\frac{d\mu (\xi )}{1-z\xi },z\in \mathbb{D}.$$

$$\Vert f\star F{\Vert }_{H^p}\le \Vert f{\Vert }_{H^p}.$$

$$M_p(r,f\star F)\le M_p(r,f),0<r<1,$$

$$F(z)={\int }_{\mathbb{T}}\frac{d\mu (\xi )}{1-z\xi }=\sum _{n=0}^{\infty }\left({\int }_{\mathbb{T}}{\xi }^{n}d\mu (\xi )\right)z^n,z\in \mathbb{D}.$$

$$(f\star F)(z)$$

$$={\int }_{\mathbb{T}}\left(\sum _{n=0}^{\infty }{a}_n{\xi }^n{z}^n\right)d\mu (\xi )={\int }_{\mathbb{T}}f(\xi z)d\mu (\xi ),z\in \mathbb{D}.$$

$$M_p(r,f\star F)=$$

$$\le$$

$$=$$

$$u^{\ast }(r{e}^{i\theta })=\underset{|E|=2\theta }{\sup }{\int }_{E}u(r{e}^{it})dt,0<r<1,0\le \theta \le \pi ,$$

$${\int }_{-\pi }^{\pi }\Phi \left(u(r{e}^{i\theta })\right)d\theta \le {\int }_{-\pi }^{\pi }\Phi \left(v(r{e}^{i\theta })\right)d\theta ,0<r<1.$$

$${\int }_{-\pi }^{\pi }\Phi \left(\log |f(r{e}^{i\theta })|\right)d\theta \le {\int }_{-\pi }^{\pi }\Phi \left(\log |g(r{e}^{i\theta })|\right)d\theta ,0<r<1.$$

$$M_p(r,f)\le M_p(r,g),0<r<1,$$

$${\left(\pm \log |f|\right)}^{\ast }\le {\left(\pm \log |k|\right)}^{\ast }$$

$$\int_{-\pi}^{\pi}\bigl{|}f(re^{i\theta})\bigr{|}^{p}d\theta\leq\int_{-\pi}^{\pi}\bigl{|}k(re^{i\theta})\bigr{|}^{p}d\theta,\quad 0<r<1,$$

$$M_p(r,f)\le M_p(r,k),0<r<1.$$

$$\bigl{(}\log|f|\bigr{)}^{*}\leq\bigl{(}\log|F|\bigr{)}^{*}\qquad\text{and}\qquad\bigl{(}\log|g|\bigr{)}^{*}\leq\bigl{(}\log|G|\bigr{)}^{*}.$$

$$\bigl{(}\log|F_{j}|\bigr{)}^{*}\leq\bigl{(}\log|I|\bigr{)}^{*},\qquad\text{for $j\,=\,1,2$,}$$

$$\bigl{(}\log|F_{1}\star F_{2}|\bigr{)}^{*}\leq\bigl{(}\log|I\star I|\bigr{)}^{*}$$

$$h(z)=z+a_3{z}^3+a_5{z}^5+\dots$$

$$h_1(z)=\frac{h(z)}{z}=1+a_3{z}^2+a_5{z}^4+\dots ,z\in \mathbb{D}.$$

$${\left(\log |h_1|\right)}^{\ast }\le {\left(\log |I|\right)}^{\ast }.$$

$$f_1=h_1\text{andfn=fn−1⋆f1, for n≥2.}$$

$$f_n(z)=1+a_3^n{z}^2+a_5^n{z}^4+\dots .$$

Full text

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).

The authors declare that there is no conflict of interest regarding the publication of this paper.

1. Introduction

Let ${\mathbb{D}}=\{z\in{\mathbb{C}}:|z|<1\}$ and $\mathbb{T}=\{z\in{\mathbb{C}}:|z|=1\}$ denote the open unit disc and the unit circle in the complex plane ${\mathbb{C}}$. We let also $\mathcal{H}ol({\mathbb{D}})$ be the space of all analytic functions in ${\mathbb{D}}$ endowed with the topology of uniform convergence in compact subsets.

If $0\leq r<1$ and $f\in\mathcal{H}ol({\mathbb{D}})$, we set

[TABLE]

For $0<p\leq\infty$, the Hardy space $H^{p}$ consists of those $f\in\mathcal{H}ol({\mathbb{D}})$ such that

[TABLE]

We refer to [6] for the theory of $H^{p}$-spaces.

If $f,g\in\mathcal{H}ol(\mathbb{D})$,

[TABLE]

the (Hadamard) convolution $(f\star g)$ of $f$ and $g$ is defined by

[TABLE]

We have the following integral representation

[TABLE]

(see [14, p. 11]). The convolution operation $\star$ makes $\mathcal{H}ol(\mathbb{D})$ into a commutative complex algebra with an identity

[TABLE]

We refer to [14] for the theory of the convolution of analytic functions and its connections with geometric function theory.

Following [15], we shall say that a function $F\in\mathcal{H}ol(\mathbb{D})$ is bound preserving if for every $f\in H^{\infty}$ we have that $f\star F\in H^{\infty}$ and

[TABLE]

Sheil-Small [15, Theorem 1. 3] (see also [14, p. 123]) proved that a function $F\in\mathcal{H}ol(\mathbb{D})$ is bound preserving if and only if there exists a complex Borel measure $\mu$ on $\mathbb{T}$ with $\|\mu\|\leq 1$ such that

[TABLE]

The measure $\mu$ is a probability measure if and only if $F$ is convexity preserving, that is, for any $f\in\mathcal{H}ol(\mathbb{D})$ the range of $f\star F$ 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 $1\leq p\leq\infty$, then for every $f\in H^{p}$ we have that $f\star F\in H^{p}$ and

[TABLE]

Actually, the following stronger result holds.

Theorem 1**.**

Suppose that $f,F\in\mathcal{H}ol(\mathbb{D})$ with $F$ being bound preserving. Then

[TABLE]

whenever $1\leq p\leq\infty$.

Proof.

Since $F$ is bound preserving, there exists a complex Borel measure $\mu$ on $\mathbb{T}$ with $\|\mu\|\leq 1$ such that

[TABLE]

If $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ ($z\in\mathbb{D}$), we have

[TABLE]

This immediately yields (1.2) for $p=\infty$. Now, if $1\leq p<\infty$, using Minkowski’s integral inequality we obtain

[TABLE]

∎

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 [2, 3].

If $u$ is a subharmonic function in $\mathbb{D}\setminus\{0\}$, the function $u^{*}$ is defined by

[TABLE]

where $|E|$ denotes the Lebesgue measure of the set $E$. The basic properties of the star-function which make it useful to solve extremal problems are the following [3]:

• •

If $u$ is a subharmonic function in $\mathbb{D}\setminus\{0\}$, then the function $u^{*}$ is subharmonic in $\mathbb{D}^{+}=\{z=re^{i\theta}:0<r<1,0<\theta<\pi\}$ and continuous in $\{z=re^{i\theta}:0<r<1,0\leq\theta\leq\pi\}$.

• •

If $v$ is harmonic in $\mathbb{D}\setminus\{0\}$, and it is a symmetric decreasing function on each of the circles $\{|z|=r\}$ ($0<r<1$), then $v^{*}$ is harmonic in $\mathbb{D}^{+}$ and, in fact, $v^{*}(re^{i\theta})=\int_{-\theta}^{\theta}v(re^{it})dt$.

The relevance of the star-function to obtain integral means estimates comes from the following result.

Proposition A** ([3]).**

Let $u$ and $v$ be two subharmonic functions in $\mathbb{D}$. Then the following two conditions are equivalent:

• (i)

$u^{*}\leq v^{*}$* in $\mathbb{D}^{+}$.*

• (ii)

For every convex and increasing function $\Phi:\mathbb{R}\rightarrow\mathbb{R}$, we have that

[TABLE]

Proposition A yields the following result about analytic functions.

Proposition B**.**

Let $f$ and $g$ be two non-identically zero analytic functions in $\mathbb{D}$. Then the following conditions are equivalent:

• (i)

$\left(\log|f|\right)^{*}\leq\left(\log|g|\right)^{*}$* in $\mathbb{D}^{+}$.*

• (ii)

For every convex and increasing function $\Phi:\mathbb{R}\rightarrow\mathbb{R}$, we have that

[TABLE]

Since for any $p>0$ the function $\Phi$ defined by $\Phi(x)=\exp(px)$ ($x\in\mathbb{R}$) is convex and increasing we deduce that if $f$ and $g$ are as in Proposition B and $\left(\log|f|\right)^{*}\leq\left(\log|g|\right)^{*}$ in $\mathbb{D}^{+}$, then

[TABLE]

for all $p>0$.

The main achievement in the use of the star-function by A. Baernstein in [3], was the proof that the Koebe function $k(z)=\frac{z}{(1-z)^{2}}$ ($z\in\mathbb{D}$) 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 $f\in S$ then

[TABLE]

and, hence,

[TABLE]

for all $p\in\mathbb{R}$. In particular, we have that if $f\in S$ and $0<p\leq\infty$, then

[TABLE]

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.

Question 1**.**

Let $f,g,F,G$ be analytic functions in ${\mathbb{D}}$ with $|F|$ and $|G|$ being symmetric decreasing on each of the circles $\{|z|=r\}$ and suppose that

[TABLE]

Does it follow that \bigl{(}\log|f\star g|\bigr{)}^{*}\leq\bigl{(}\log|F\star G|\bigr{)}^{*}?

Question 2**.**

Let $F$ and $f$ be two analytic functions in ${\mathbb{D}}$ and suppose that $F$ is bound preserving. Can we assert that $\left(\log|f\star F|\right)^{*}\,\leq\,\left(\log|f|\right)^{*}$?

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

Theorem 2**.**

There exist two functions $F_{1},F_{2}\,\in\,\mathcal{H}ol(\mathbb{D})$ with

[TABLE]

and such that

[TABLE]

Here, $I$ is the identity element of the convolution defined in (1.1), that is, $I(z)=\frac{1}{1-z}$ ($z\in\mathbb{D}$). Hence $I\star I=I$.

Proof.

Let $h$ be an odd function in the class $S$ with Taylor expansion

[TABLE]

with $|a_{5}|>1$. The existence of such an $h$ was proved by Fekete and Szegö (see [7, p. 104]). Set also

[TABLE]

It is well known that there exists a function $H\in S$ such that $h(z)=\sqrt{H(z^{2})}$ (see [7, p. 64]). Set $k_{2}(z)=\sqrt{k(z^{2})}=\frac{z}{1-z^{2}}$ and $J(z)=\frac{k_{2}(z)}{z}=\frac{1}{1-z^{2}}$ ($z\in\mathbb{D}$). By Baernstein’s theorem we have $\left(\log|H|\right)^{*}\leq\left(\log|k|\right)^{*}$, a fact which easily implies that $\left(\log|h_{1}|\right)^{*}\,\leq\,\left(\log|J|\right)^{*}$. Now, it is clear that $J$ is subordinate to $I$ and then, using [11, Lemma 2], we see that $\left(\log|J|\right)^{*}\,\leq\,\left(\log|I|\right)^{*}.$ Thus it follows that

[TABLE]

For $n=1,2,3,\dots$, we define $f_{n}$ inductively as follows

[TABLE]

In other words, $f_{n}=\overbrace{h_{1}\star\dotsb\star h_{1}}^{(n)}$. Clearly, (2.2) yields

[TABLE]

Since $|a_{5}|>1$, it follows that $|a_{5}^{n}|\to\infty$, as $n\to\infty$. This is equivalent to saying that

[TABLE]

Then it follows that the family $\{f_{n}^{(4)}:n=1,2,3,\dots\}$ is not a locally bounded family of holomorphic functions in $\mathbb{D}$. Using [1, Theorem 16, p. 225] we see that the same is true for the family $\{f_{n}:n=1,2,3,\dots\}$. Take $p\in(0,1)$, then $I\in H^{p}$. Since a bounded subset of $H^{p}$ is a locally bounded family [6, p. 36], it follows that

[TABLE]

Now, (2.4) implies that $\|f_{n}\|_{H^{p}}>\|I\|_{H^{p}}$ for some $n$. Using Proposition B, we see that this implies that

[TABLE]

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 $F_{1}=f_{1},F_{2}=f_{N-1}$. ∎

We have the following result regarding Question 2.

Theorem 3**.**

There exist $f,F$ analytic and univalent in $\mathbb{D}$ such that $F$ is convexity preserving and with the property that the inequality $\left(\log|f\star F|\right)^{*}\leq\left(\log|f|\right)^{*}$ does not hold.

The following lemma will be used in the proof of Theorem 3.

Lemma 1**.**

Let $f,F\in\mathcal{H}ol(\mathbb{(}D)$ and suppose that $F(0)=1$, $F$ is convexity preserving, and that $f$ and $f\star F$ are zero-free in $\mathbb{D}$ and satisfy the inequality $\left(\log|f\star F|\right)^{*}\leq\left(\log|f|\right)^{*}.$ Then we also have that

[TABLE]

Proof.

Set $u=\log|f\star F|$, $v=\log|f|$. Then $u$ and $v$ are harmonic in $\mathbb{D}$, $u(0)=v(0)$, and $u^{*}\leq v^{*}$. Then it follows that, for $0<r<1$ and $0\leq\theta\leq\pi$,

[TABLE]

Hence, we have proved that $(-u)^{*}\leq(-v)^{*}$ which is equivalent to (2.5). ∎

Proof of Theorem 3. Set

[TABLE]

Clearly, $f$ and $F$ are analytic, univalent, and zero-free in $\mathbb{D}$. Also

[TABLE]

Hence $f\star F$ is also zero-free in $\mathbb{D}$. Notice that $\frac{1}{f\star F}\not\in H^{\infty}$   and $\frac{1}{f}\in H^{\infty}$. Then it follows that

[TABLE]

Now, it is a simple exercise to check that

[TABLE]

and then it follows that $F$ is convexity preserving. Then, using (2.6) and Lemma 1, it follows that the inequality $\left(\log|f\star F|\right)^{*}\leq\left(\log|f|\right)^{*}$ does not hold, as desired. $\Box$

We close the paper with a positive result, determining a class of univalent functions $\mathcal{Z}$ such that (1.2) is true for all $p>0$, whenever $f\in\mathcal{Z}$ and $F$ is convexity preserving.

A domain $D$ in $\mathbb{C}$ 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 $\mathcal{Z}$ be the class of all functions $f$ which are analytic and univalent in $\mathbb{D}$ with $f(0)=0$, $f^{\prime}(0)>0$, and whose image is a Steiner symmetric domain. The elements of $\mathcal{Z}$ 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 $f^{\prime}(0)>0$ is Steiner symmetric if and only if it satisfies the following two conditions: (i) $f$ is typically real and (ii) $\text{Re\,}f$ is a symmetric decreasing function on each of the circles $\{|z|=r\}$ ($0<r<1$). Then it follows that if $f\in\mathcal{Z}$ then for every $r\in(0,1)$, the domain $f\left(\{|z|<r\}\right)$ is a Steiner symmetric domain and, hence, the function $f_{r}$ defined by $f_{r}(z)=f(rz)$  ($z\in\mathbb{D}$) belongs to $\mathcal{Z}$ and it extends to an analytic function in the closed unit disc $\overline{\mathbb{D}}$. Now we can state our last result.

Theorem 4**.**

Suppose that $f\in\mathcal{Z}$ and let $F$ be an analytic function in $\mathbb{D}$ which is convexity preserving. We have, for every $p>0$,

[TABLE]

Proof.

In view of Theorem 1 we only need to prove (2.7) for $0<p<1$. Let $\mu$ be the probability measure on $\mathbb{T}$ such that $F(z)=\int_{\mathbb{T}}\frac{d\mu(\xi)}{1-z\xi}$  ($z\in\mathbb{D}$). Then we have

[TABLE]

Since $F$ is convexity preserving, for $0<r<1$, we have that $\left(f_{r}\star F\right)(\overline{\mathbb{D}})$ is contained in the closed convex hull of $f_{r}(\overline{\mathbb{D}})$. This easily yields

[TABLE]

By the remarks in the previous paragraph, we find that, for all $r\in(0,1)$, $f_{r}$ belongs to $\mathcal{Z}$ and extends to an analytic function in the closed unit disc $\overline{\mathbb{D}}$. Finally, we claim that

[TABLE]

Once this is proved, using Proposition 6 of [8], we deduce that

[TABLE]

finishing our proof.

So we proceed to prove (2.9). Fix $r\in(0,1)$ and set $u=\text{Re\,}(f_{r}\star F)$, $v=\text{Re\,}f_{r}$. Using (2.8), we have, for $0<R<1$ and $0<\theta<\pi$,

[TABLE]

∎

Acknowledgements. We wish to express our gratitude to the referee who made valuable suggestions for improvement.