Univalency of convolutions of harmonic mappings

Abstract

We consider the convolution or Hadamard product of planar harmonic mappings that are the vertical shears of the canonical half-plane mapping $\phi (z)=z/(1-z)$ with respective dilatations $-\mathit{xz}$ and $-\mathit{yz}$, where $|x|=|y|=1$. We prove that any such convolution is univalent. Furthermore, in the case that $x=y=-1$, we show the resulting convolution is convex.

Introduction

Let $\mathbb{D}$ be the unit disk. We consider the family of complex-valued harmonic functions $f=u+\mathit{iv}$ defined in $\mathbb{D}$, where u and v are real harmonic in $\mathbb{D}$. Such functions can be expressed as $f=h+\bar{g}$, where

$$h(z)={\displaystyle \sum _{n=0}^{\infty }}{a}_n{z}^n\text{and}g(z)={\displaystyle \sum _{n=1}^{\infty }}{b}_n{z}^n\text{,}$$

are analytic in $\mathbb{D}$. By the Lewy theorem, a harmonic function $f=h+\bar{g}$ is locally univalent and sense-preserving if and only if its dilatation $\omega =g^{\prime }/h^{\prime }$ satisfies $|\omega (z)|<1$ for $z\in \mathbb{D}$, see e.g. [5].

For functions $F(z)={\sum }_{n=0}^{\infty }{A}_n{z}^n$ and $G(z)={\sum }_{n=0}^{\infty }{B}_n{z}^n$, analytic in $\mathbb{D}$, their convolution (or Hadamard product) is defined as

$$F\ast G(z)={\displaystyle \sum _{n=0}^{\infty }}{A}_n{B}_n{z}^n\text{.}$$

If we take two harmonic functions $f_1\text{,}{f}_2$, we can define $f_1\ast f_2$ in the natural way as $h_1\ast h_2+\overline{g_1\ast g_2}$ where $f_j=h_j+\overline{g_j}\text{,}j=1\text{,}2$. Convolutions of two harmonic functions were studied in [2], [3], [4], [6], [11].

Let ${\mathbf{S}}_H^O$ be the class of complex-valued harmonic sense-preserving univalent functions f in $\mathbb{D}$, normalized so that $f(0)=0$, $f_z(0)=1$, and $f_{\bar{z}}(0)=0$.

It is known that if $f=h+\bar{g}\in {\mathbf{S}}_H^O$ maps the unit disk onto the right half-plane $R=\{w:\mathrm{Re}w>-1/2\}$, then it must satisfy the following condition

$$h(z)+g(z)=\frac{z}{1-z}\text{.}$$

Let ${\mathbf{S}}_H^O(R)$ denote the class of harmonic mappings $f=h+\bar{g}\in {\mathbf{S}}_H^O$ that satisfy (1). They are the so called vertical shears of the conformal half-plane mapping $\phi (z)=z/(1-z)$. It was proved in [3] that if $f_j=h_j+\overline{g_j}\in {\mathbf{S}}_H^O(R)\text{,}j=1\text{,}2$, and $f_1\ast f_2$ is locally univalent and sense-preserving, then $f_1\ast f_2\in {\mathbf{S}}_H^O$ and is convex in the direction of the real axis. As observed in [4] the assumption of the local univalency of the convolution function in this statement cannot be omitted. However, in some cases these harmonic convolutions are locally univalent. It was shown in [4] that if $f_0=h_0+\overline{g_0}\in {\mathbf{S}}_H^O(R)$ with dilatation ${\omega }_0(z)=-z$ and $f=h+\bar{g}\in {\mathbf{S}}_H^O(R)$ with dilatation $\omega (z)=e^{i\theta }{z}^n\text{,}\theta \in \mathbb{R}$, then the convolution $f_0\ast f$ is locally univalent for all $\theta$ if and only if $n=1\text{,}2$.

In this paper we prove the following.

Theorem

If $f_k=h_k+{\bar{g}}_k\in {\mathbf{S}}_H^O(R)\text{,}k=1\text{,}2$, and ${\omega }_1(z)=g_1^{\prime }(z)/h_1^{\prime }(z)=-\mathit{xz}\text{,}{\omega }_2(z)=g_2^{\prime }(z)/h_2^{\prime }(z)=-\mathit{yz}$ with $|x|=|y|=1$, then the function $\stackrel{\sim }{f}=f_1\ast f_2$ is convex in the direction of the real axis. In particular, if $x=y=-1$, then $\stackrel{\sim }{f}$ is convex.

We will use the following characterization of the class of analytic functions mapping $\mathbb{D}$ conformally onto a domain convex in one direction due to Royster and Ziegler [10].

Theorem A

A nonconstant and analytic function F maps $\mathbb{D}$ univalently onto a domain $\mathrm{\Omega }$ convex in the direction of the imaginary axis if and only if there are numbers $\mu \in [0\text{,}2\pi )$ and $\nu \in [0\text{,}\pi ]$, such that

$$\mathrm{Re}\left\{-{\mathit{ie}}^{i\mu }\left(1-2\cos \nu e^{-i\mu }z+e^{-2i\mu }{z}^2\right)F^{\prime }(z)\right\}⩾0\text{,}z\in \mathbb{D}\text{.}$$

In the proof of our theorem we will also take advantage of the theory of harmonic Hardy spaces. A function $f(z)$ harmonic in the unit disk is said to be of class $h^p(0<p<\infty )$ if the integral means

$$\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{\left|f({\mathit{re}}^{i\theta })\right|}^{p}d\theta \text{,}$$

are bounded for $0⩽r<1$; $h^{\infty }$ is simply the collection of bounded harmonic functions on $\mathbb{D}$. The usual Hardy spaces $H^p$ consist of the functions in $h^p$ that are analytic in $\mathbb{D}$. It is clear that an analytic function belongs to $H^p$ if and only if its real and imaginary parts are both in $h^p$. It is well known that every function $f\in H^p$ has a non-tangential limit $f(e^{i\theta })$ for almost every $\theta \in (-\pi \text{,}\pi ]$. We will apply the following theorem concerning harmonic Hardy spaces, see, e.g. [9, p. 15], [8, p. 38].

Theorem B

Let $1<p⩽\infty$, and assume that $f\in h^p$. Then for almost all $\theta \text{,}f(z)$ tends to a finite limit $f(e^{i\theta })$, as $z\to e^{i\theta }$ nontangentially, $f(e^{i\theta })\in L^p(-\pi \text{,}\pi )$, and, for $0⩽r<1$,

$$f({\mathit{re}}^{i\theta })=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\frac{1-r^2}{1+r^2-2r\cos (\theta -t)}f(e^{i\theta })\mathit{dt}\text{.}$$

Let us emphasize that Theorem B does not hold for $p=1$. In the Remark we give an example of a function $f\in h^1$ which cannot be recovered from its boundary behavior.

We also note that this theorem actually implies an extended version of the maximum (minimum) principle for real harmonic functions in $\mathbb{D}$.