Harmonic mappings related to the m-fold starlike functions

Abstract

In the present paper we will give some properties of the subclass of harmonic mappings which is related to m-fold starlike functions in the open unit disc $\mathbb{D}=\{z||z|<1\}$. Throughout this paper we restrict ourselves to the study of sense-preserving harmonic mappings. We also note that an elegant and complete treatment theory of the harmonic mapping is given in Durens monograph (Duren, 1983). The main aim of us to investigate some properties of the new class of us which represented as in the following form,

$$S^{\ast }H(m)=\left\{f=h(z)+\overline{g(z)}|f\in \mathit{SH}(m)\text{,}\frac{g^{\prime }(z)}{h^{\prime }(z)}\prec b_{1}p(z)\text{,}h(z)\in S^{\ast }(m)\text{,}p(z)\in P^{(m)}\right\}\text{,}$$

where $h(z)=z+{\sum }_{n=1}^{\infty }{a}_{\mathit{mn}+1}{z}^{\mathit{mn}+1}$, $g(z)={\sum }_{n=0}^{\infty }{b}_{\mathit{mn}+1}{z}^{\mathit{mn}+1}\text{,}|b_1|<1$.

Introduction

Let $\mathrm{\Omega }$ be the family of functions $\varphi (z)$ which are analytic in $\mathbb{D}$ and satisfying the conditions $\varphi (0)=0\text{,}|\varphi (z)|<1$ for every $z\in \mathbb{D}$. Let $P^{(m)}$ denote the set of functions of the form $p(z)=1+p_m{z}^m+p_{2m}{z}^{2m}+\dots$ for which $\mathit{Rep}(z)>0$ in $\mathbb{D}$. For brevity we say that $p(z)$ has $m$-fold symmetry. A function is called $m$-fold symmetric is its power series has the form [1], [4], [5].

$$s(z)={\displaystyle \sum _{n=0}^{\infty }}{d}_{\mathit{mn}+1}{z}^{\mathit{mn}+1}$$

This condition on the power series is equivalent to the condition $s\left(e^{\frac{2\pi i}{m}}z\right)=e^{\frac{2\pi i}{m}}s(z)$ for $z\in \mathbb{D}$. The class of such functions is denoted by $S^{(m)}$. We denote by $S^{\ast }(m)$ the families $m$-fold symmetric starlike functions. That is

$$s(z)\in S^{\ast }(m)\iff \mathit{Re}\left(z\frac{s^{\prime }(z)}{s(z)}\right)>0$$

[1], [4], [5]. Let $s_1(z)=z+d_2{z}^2+\dots$ and $s_2(z)=z+e_2{z}^2+\dots$ be analytic functions in $\mathbb{D}$. If there exists a function $\varphi (z)\in \mathrm{\Omega }$ such that $s_1(z)=s_2(\varphi (z))$ for every $z\in \mathbb{D}$. Then we say that $s_1(z)$ is subordinate to $s_2(z)$ and we write $s_1(z)\prec s_2(z)$. Specially if $s_2(z)$ is univalent in $\mathbb{D}$, then $s_1(z)\prec s_2(z)$ if and only if $s_1(\mathbb{D})\subset s_2(\mathbb{D})$ and $s_1(0)=s_2(0)$ implies $s_1({\mathbb{D}}_r)\subset s_2({\mathbb{D}}_r)$ where ${\mathbb{D}}_r=\left\{z||z|<r\text{,}0<r<1\right\}$ [1], [4].

A planar harmonic mapping in the open unit disc $\mathbb{D}$ is a complex valued harmonic function f, which maps $\mathbb{D}$ onto the some planar domain $f(\mathbb{D})$. Since $\mathbb{D}$ is a simply connected domain the mapping f has a canonical decomposition $f=h(z)+\overline{g(z)}$, where $h(z)$ and $g(z)$ are analytic in $\mathbb{D}$ and have the following power series expansions,

$$h(z)=z+{\displaystyle \sum _{n=2}^{\infty }}{a}_n{z}^n\text{,}g(z)={\displaystyle \sum _{n=1}^{\infty }}{b}_n{z}^n$$

as usual, we call $h(z)$ the analytic part and $g(z)$ is the co-analytic part of f, respectively and let the class of such harmonic mappings is denoted by SH, (see [2]) proved in 1936 that the harmonic mapping f is locally univalent in $\mathbb{D}$ if and only if its Jacobian $J_f=(|h^{\prime }(z){|}^2-|g^{\prime }(z){|}^2)$ is strictly positive in $\mathbb{D}$. In view of this result, locally univalent harmonic mapping in the unit disc are either sense-reversing if $\left|g^{\prime }(z)\right|>\left|h^{\prime }(z)\right|$ or sense-preserving if $\left|h^{\prime }(z)\right|>\left|g^{\prime }(z)\right|$ in $\mathbb{D}$. Throughout this paper we restrict ourselves to the study of sense-preserving harmonic mappings. We also note that an elegant and complete treatment theory of the harmonic mapping is given Duren’s monograph [2].

The main aim of this paper is to investigate the some properties of the following class.

$$S^{\ast }H(m)=\left\{f=h(z)+\overline{g(z)}|f\in \mathit{SH}(m)\text{,}\frac{g^{\prime }(z)}{h^{\prime }(z)}\prec b_{1}p(z)\text{,}h(z)\in S^{\ast }(m)\text{,}p(z)\in P^{(m)}\right\}\text{,}$$

where

$h(z)=z+{\sum }_{n=1}^{\infty }{a}_{\mathit{mn}+1}{z}^{\mathit{mn}+1}$, $g(z)={\sum }_{n=0}^{\infty }{b}_{\mathit{mn}+1}{z}^{\mathit{mn}+1}\text{,}\left|b_1\right|<1$. and for this aim we will need the following lemmas.

Lemma 1.1 [4]

Let $m⩾1$ be a fixed integer and assume that

$$\varphi (z)=B_{2m}{z}^{2m}+B_{2m+1}{z}^{2m+1}+\dots ={\displaystyle \sum _{n=2m}^{\infty }}{B}_n{z}^n$$

satisfies the conditions of Schwarz Lemma. If $0<r<1$, then

$$\left|\varphi (z)\right|⩽r^{2m}$$

If equality occurs in (1.3) for one point $z_0={\mathit{re}}^{i\theta }$ with $0<r<1$, then $\varphi (z)=e^{i\alpha }{z}^{2m}$ and the equal sign holds in (1.3) for all $z\in \mathbb{D}$.

Lemma 1.2 [3]

Let $\varphi (z)={\alpha }_n{z}^n+{\alpha }_{n+1}{z}^{n+1}+\dots$$({\alpha }_n\ne 0\text{,}n⩾1)$ be analytic in $\mathbb{D}$. If the maximum value of $\left|\varphi (z)\right|$ on the circle $\left|z\right|=r<1$ is attained at $z=z_0$, then we have $z_0{\varphi }^{\prime }(z_0)=m\varphi (z_0)\text{,}m⩾n$ and every $z\in \mathbb{D}$.

Lemma 1.3 [4]

Let $p(z)$ be an element of $p^{(m)}$, then $P({\mathbb{D}}_r)$ is the open disc with diameter end points $w_1=\frac{1-r^k}{1+r^k}$ and $w_2=\frac{1}{w_1}$.

Lemma 1.4 [5]

Let $h(z)=z+a_{m+1}{z}^{m+1}+a_{2m+1}{z}^{2m+1}+\dots$ be a starlike function on $\mathbb{D}$. Then

$$\frac{r}{{(1+r^m)}^{\frac{2}{m}}}⩽\left|h(z)\right|⩽\frac{r}{{(1-r^m)}^{\frac{2}{m}}}$$

$$\frac{{(1-r^m)}^{\frac{1}{m}}}{{(1+r^m)}^{\frac{2}{m}}}⩽\left|h^{\prime }(z)\right|⩽\frac{{(1+r^m)}^{\frac{1}{m}}}{{(1-r^m)}^{\frac{2}{m}}}$$