An investigation on a new class of harmonic mappings

Abstract

In the present paper, we give an extension of the idea which was introduced by Sakaguchi (J. Math. Soc. Jpn. 11:72-75, 1959), and we give some applications of this extended idea for the investigation of the class of harmonic mappings.

MSC:30C45, 30C55.

1 Introduction

Let $\mathbb{D}=\left\{z\mid |z|<1\right\}$ be the open unit disc in the complex plane . A complex valued harmonic function $f:\mathbb{D}\to \mathbb{C}$ has the representation

$f=h\left(z\right)+\overline{g\left(z\right)},$
(1.1)

where $h\left(z\right)$ and $g\left(z\right)$ are analytic in and have the following power series expansions:

$h\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}{a}_{n}{z}^{n},\phantom{\rule{2em}{0ex}}g\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}{b}_{n}{z}^{n},\phantom{\rule{1em}{0ex}}z\in \mathbb{D},$

where ${a}_{n},{b}_{n}\in \mathbb{C}$, $n=0,1,2,\dots$ . Choose $g\left(0\right)=0$ (i.e., ${b}_{0}=0$), so the representation (1.1) is unique in and is called the canonical representation of f in . It is convenient to make further normalization (without loss of generality) $h\left(0\right)=0$ (i.e., ${a}_{0}=0$) and ${h}^{\prime }\left(0\right)=1$ (i.e., ${a}_{1}=1$). The family of such functions f is denoted by ${\mathcal{S}}_{\mathcal{H}}$ [1]. It is known that if f is a sense-preserving harmonic mapping of onto some other region, then by Lewy’s theorem its Jacobian is strictly positive, i.e.,

${J}_{f\left(z\right)}={|{h}^{\prime }\left(z\right)|}^{2}-{|{g}^{\prime }\left(z\right)|}^{2}>0.$

Equivalently, the inequality $|{g}^{\prime }\left(z\right)|<|{h}^{\prime }\left(z\right)|$ holds for all $z\in \mathbb{D}$. The family of all functions $f\in {\mathcal{S}}_{\mathcal{H}}$ with the additional property that ${g}^{\prime }\left(0\right)=0$ (i.e., ${b}_{1}=0$) is denoted by ${\mathcal{S}}_{\mathcal{H}}^{0}$ [1]. Observe that the classical family of univalent functions consists of all functions $f\in {\mathcal{S}}_{\mathcal{H}}^{0}$ such that $g\left(z\right)=0$ for all $z\in \mathbb{D}$. Thus, it is clear that $\mathcal{S}\subset {\mathcal{S}}_{\mathcal{H}}^{0}\subset {\mathcal{S}}_{\mathcal{H}}$.

Let Ω be the family of functions $\varphi \left(z\right)$ regular in the open unit disc and satisfy the conditions $\varphi \left(0\right)=0$, $|\varphi \left(z\right)|<1$ for all $z\in \mathbb{D}$.

Denote by the family of functions $p\left(z\right)=1+{p}_{1}z+{p}_{2}{z}^{2}+\cdots$ regular in such that $p\left(z\right)$ in if and only if

$p\left(z\right)=\frac{1+\varphi \left(z\right)}{1-\varphi \left(z\right)}$
(1.2)

for some $\varphi \left(z\right)\in \mathrm{\Omega }$ and every $z\in \mathbb{D}$.

If $s\left(z\right)=z+{c}_{2}{z}^{2}+\cdots$ is regular in the open unit disc and satisfies the condition

$Re\left({e}^{i\alpha }z\frac{{s}^{\prime }\left(z\right)}{s\left(z\right)}\right)>0,\phantom{\rule{1em}{0ex}}z\in \mathbb{D}$
(1.3)

for some real α, $|\alpha |<\pi /2$, then $s\left(z\right)$ is said to be an α-spirallike function in [2, 3]. Such functions are known to be univalent [4]. The class of such functions is denoted by ${\mathcal{S}}_{\alpha }^{\ast }$.

Let ${F}_{1}\left(z\right)=z+{\alpha }_{2}{z}^{2}+{\alpha }_{3}{z}^{3}+\cdots$ and ${F}_{2}\left(z\right)=z+{\beta }_{2}{z}^{2}+{\beta }_{3}{z}^{3}+\cdots$ be analytic functions in . If there exists a function $\varphi \left(z\right)\in \mathrm{\Omega }$ such that ${F}_{1}\left(z\right)={F}_{2}\left(\varphi \left(z\right)\right)$ for every $z\in \mathbb{D}$, then we say that ${F}_{1}\left(z\right)$ is subordinate to ${F}_{2}\left(z\right)$ and we write ${F}_{1}\prec {F}_{2}$. We also note that if ${F}_{1}\prec {F}_{2}$, then ${F}_{1}\left(\mathbb{D}\right)\subset {F}_{2}\left(\mathbb{D}\right)$ [2, 3]. We also note that $w\left(z\right)={g}^{\prime }\left(z\right)/{h}^{\prime }\left(z\right)$ is the analytic second dilatation of f and $|w\left(z\right)|<1$ for every $z\in \mathbb{D}$.

In this paper we investigate the class of harmonic mappings defined by

${\mathcal{S}}_{\mathcal{H}\left(\alpha \right)}=\left\{f=h\left(z\right)+\overline{g\left(z\right)}|w\left(z\right)\prec \frac{1+z}{1-z},h\left(z\right)\in {\mathcal{S}}_{\alpha }^{\ast }\right\}.$

For this aim we need the following theorems and lemmas.

Theorem 1.1 [5]

Let $s\left(z\right)\in {\mathcal{S}}_{\alpha }^{\ast }$, then

$rF\left(cos\alpha ,-r\right)\le |s\left(z\right)|\le rF\left(cos\alpha ,r\right)$
(1.4)

and

$\begin{array}{r}\left[\left(1-r\right)cos\alpha -\left(1+r\right)sin\alpha \right]F\left(cos\alpha ,-r\right)\\ \phantom{\rule{1em}{0ex}}\le |{s}^{\prime }\left(z\right)|\le \left[\left(1+r\right)cos\alpha +\left(1-r\right)sin\alpha \right]F\left(cos\alpha ,r\right),\end{array}$
(1.5)

where

$F\left(cos\alpha ,r\right)=\frac{1}{{\left(1+r\right)}^{cos\alpha \left(cos\alpha -1\right)}{\left(1-r\right)}^{cos\alpha \left(cos\alpha +1\right)}}$
(1.6)

for all $|z|=r<1$ and $|\alpha |<\pi /2$.

Lemma 1.2 [6]

Let $\varphi \left(z\right)$ be regular in the unit disc with $\varphi \left(0\right)=0$, then if $|\varphi \left(z\right)|$ attains its maximum value on the circle $|z|=r$ at the point ${z}_{1}$, one has ${z}_{1}{\varphi }^{\prime }\left({z}_{1}\right)=k\varphi \left({z}_{1}\right)$ for some $k\ge 1$.

Lemma 1.3 [7]

If ${s}_{1}\left(z\right)$ and ${s}_{2}\left(z\right)$ are regular in , ${s}_{1}\left(0\right)={s}_{2}\left(0\right)$, ${s}_{2}\left(z\right)$ maps onto a many-sheeted region which is starlike with respect to the origin, and ${s}_{1}^{\prime }\left(z\right)/{s}_{2}^{\prime }\left(z\right)\in \mathcal{P}$, then ${s}_{1}\left(z\right)/{s}_{2}\left(z\right)\in \mathcal{P}$.

Lemma 1.4 [8]

Let $f=h\left(z\right)+\overline{g\left(z\right)}$ be an element of ${\mathcal{S}}_{\mathcal{HPST}{\left(\alpha \right)}^{\ast }}$, then

$\frac{|{b}_{1}|-r}{1-|{b}_{1}|r}\le |\frac{{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}|\le \frac{|{b}_{1}|+r}{1+|{b}_{1}|r}$
(1.7)

for all $|z|=r<1$. This inequality is sharp because the extremal function is

${e}^{i\alpha }\frac{{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}=\frac{z+b}{1+\overline{b}z},$

where $b={e}^{i\alpha }{b}_{1}$.

2 Main results

Theorem 2.1 $s\left(z\right)\in {\mathcal{S}}_{\alpha }^{\ast }$ if and only if

$z\frac{{s}^{\prime }\left(z\right)}{s\left(z\right)}-1\prec \frac{2cos\alpha {e}^{-i\alpha }z}{1-z}$
(2.1)

for all $z\in \mathbb{D}$.

Proof Let $s\left(z\right)$ in ${\mathcal{S}}_{\alpha }^{\ast }$, then we have

${e}^{i\alpha }z\frac{{s}^{\prime }\left(z\right)}{s\left(z\right)}=cos\alpha \frac{1+\varphi \left(z\right)}{1-\varphi \left(z\right)}+isin\alpha$

or

${e}^{i\alpha }z\frac{{s}^{\prime }\left(z\right)}{s\left(z\right)}={e}^{i\alpha }\frac{1+\varphi \left(z\right)}{1-\varphi \left(z\right)}$

for some $\varphi \left(z\right)\in \mathrm{\Omega }$ and all $z\in \mathbb{D}$. Thus

$\begin{array}{rl}z\frac{{s}^{\prime }\left(z\right)}{s\left(z\right)}-1& =\frac{1+{e}^{-2i\alpha }\varphi \left(z\right)}{1-\varphi \left(z\right)}-1\\ =\frac{1+\left(cos2\alpha -isin2\alpha \right)\varphi \left(z\right)-1+\varphi \left(z\right)}{1-\varphi \left(z\right)}\\ =\frac{2cos\alpha {e}^{-i\alpha }\varphi \left(z\right)}{1-\varphi \left(z\right)}\end{array}$

for some $\varphi \left(z\right)\in \mathrm{\Omega }$ and all $z\in \mathbb{D}$. Since $\varphi \left(z\right)\in \mathrm{\Omega }$, we have that (2.1) is true. The sufficient part of the proof can be seen by following the above steps in the opposite direction by considering the subordination principle. □

Theorem 2.2 Let $f=h\left(z\right)+\overline{g\left(z\right)}$ be an element of ${\mathcal{S}}_{\mathcal{H}\left(\alpha \right)}$. If $w\left(z\right)=\frac{{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}\in \mathcal{P}$, then $\frac{g\left(z\right)}{h\left(z\right)}\in \mathcal{P}$ for all $z\in \mathbb{D}$.

Proof A version of this theorem was proved by Sakaguchi for a univalent starlike function [7, 9]. Since $f=h\left(z\right)+\overline{g\left(z\right)}\in {\mathcal{S}}_{\mathcal{H}\left(\alpha \right)}$, then $h\left(z\right)$ and $g\left(z\right)$ are regular in and $h\left(0\right)=g\left(0\right)=0$. On the other hand, we have

$w\left(z\right)=\frac{{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}\in \mathcal{P}\phantom{\rule{1em}{0ex}}\text{if and only if}\phantom{\rule{1em}{0ex}}\frac{{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}\prec \frac{1+z}{1+z}$
(2.2)

for all $z\in \mathbb{D}$. Geometrically, this means that $\frac{{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}$ maps inside the open disc centered on the real axis with diameter end points $\frac{1-r}{1+r}$ and $\frac{1+r}{1-r}$. Now we define a function $\varphi \left(z\right)$ by

$\frac{g\left(z\right)}{h\left(z\right)}=\frac{1+\varphi \left(z\right)}{1-\varphi \left(z\right)}\phantom{\rule{1em}{0ex}}\left(z\in \mathbb{D}\right).$
(2.3)

Then $\varphi \left(z\right)$ is analytic in , and $\varphi \left(0\right)=0$. On the other hand,

$w\left(z\right)=\frac{{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}=\frac{2z{\varphi }^{\prime }\left(z\right)}{1-\varphi \left(z\right)}\frac{1}{{e}^{i\alpha }\left(1+{e}^{-2i\alpha }\right)\varphi \left(z\right)}+\frac{1+\varphi \left(z\right)}{1-\varphi \left(z\right)}\phantom{\rule{1em}{0ex}}\left(z\in \mathbb{D}\right).$
(2.4)

Now, it is easy to realize that the subordination (2.2) is equivalent to $|\varphi \left(z\right)|<1$ in (2.3) for all $z\in \mathbb{D}$. Indeed, assume to the contrary that there exists ${z}_{1}\in \mathbb{D}$ such that $|\varphi \left({z}_{1}\right)|=1$. Then by Jack’s lemma (Lemma 1.2), ${z}_{1}{\varphi }^{\prime }\left({z}_{1}\right)=k\varphi \left({z}_{1}\right)$, $k\ge 1$, for such ${z}_{1}$ we have

$w\left({z}_{1}\right)=\frac{{g}^{\prime }\left({z}_{1}\right)}{{h}^{\prime }\left({z}_{1}\right)}=\frac{2k\varphi \left({z}_{1}\right)}{1-\varphi \left({z}_{1}\right)}\frac{1}{{e}^{i\alpha }\left(1+{e}^{-2i\alpha }\right)\varphi \left({z}_{1}\right)}+\frac{1+\varphi \left({z}_{1}\right)}{1-\varphi \left({z}_{1}\right)}=w\left(\varphi \left({z}_{1}\right)\right)\notin w\left(\mathbb{D}\right),$

since $|\varphi \left({z}_{1}\right)|=1$ and $k\ge 1$. But this is a contradiction to the condition $w\left(z\right)=\frac{{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}\prec \frac{1+z}{1-z}$, and so the assumption is wrong, i.e., $|\varphi \left(z\right)|<1$ for all $z\in \mathbb{D}$. □

Remark 2.3 Theorem 2.2 is an extension of Lemma 1.3 to the harmonic mappings.

Corollary 2.4 Let $f=h\left(z\right)+\overline{g\left(z\right)}$ be an element of ${\mathcal{S}}_{\mathcal{H}\left(\alpha \right)}$, then

$\frac{r\left(1-r\right)}{1+r}F\left(cos\alpha ,-r\right)\le |g\left(z\right)|\le \frac{r\left(1+r\right)}{1-r}F\left(cos\alpha ,r\right)$
(2.5)

and

$\begin{array}{r}\left[\left(1-r\right)cos\alpha -\left(1+r\right)sin\alpha \right]\frac{1-r}{1+r}F\left(cos\alpha ,-r\right)\\ \phantom{\rule{1em}{0ex}}\le |{g}^{\prime }\left(z\right)|\le \left[\left(1+r\right)cos\alpha +\left(1-r\right)sin\alpha \right]\frac{1+r}{1-r}F\left(cos\alpha ,r\right),\end{array}$
(2.6)

where $F\left(cos\alpha ,r\right)$ is given by (1.6) for all $|z|=r<1$ and $|\alpha |<\pi /2$.

Proof The proof of this theorem is a simple consequence of Theorem 1.1 and Theorem 2.2 since

$Rew\left(z\right)=Re\frac{{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}\frac{{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}\prec \frac{1+z}{1-z}\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}\frac{1-r}{1+r}\le |\frac{{g}^{\prime }\left(z\right)}{{h}^{\prime }\left(z\right)}|\le \frac{1+r}{1-r},$

then

$|{h}^{\prime }\left(z\right)|\frac{1-r}{1+r}\le |{g}^{\prime }\left(z\right)|\le |{h}^{\prime }\left(z\right)|\frac{1+r}{1-r}$

and

$Re\frac{g\left(z\right)}{h\left(z\right)}>0\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}\frac{g\left(z\right)}{h\left(z\right)}\prec \frac{1+z}{1-z}\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}\frac{1-r}{1+r}\le |\frac{g\left(z\right)}{h\left(z\right)}|\le \frac{1+r}{1-r},$

then

$|h\left(z\right)|\frac{1-r}{1+r}\le |g\left(z\right)|\le |h\left(z\right)|\frac{1+r}{1-r}.$

□

Corollary 2.5 Let $f=h\left(z\right)+\overline{g\left(z\right)}$ be an element of ${\mathcal{S}}_{\mathcal{H}\left(\alpha \right)}$, then

$\begin{array}{r}\frac{{\left[\left(1-r\right)cos\alpha -\left(1+r\right)sin\alpha \right]}^{2}{\left(F\left(cos\alpha ,-r\right)\right)}^{2}\left(1-{r}^{2}\right)\left(1-{|{b}_{1}|}^{2}\right)}{{\left(1+|{b}_{1}|r\right)}^{2}}\\ \phantom{\rule{1em}{0ex}}\le {J}_{f\left(z\right)}\le \frac{{\left[\left(1+r\right)cos\alpha +\left(1-r\right)sin\alpha \right]}^{2}{\left(F\left(cos\alpha ,r\right)\right)}^{2}\left(1-{r}^{2}\right)\left(1-{|{b}_{1}|}^{2}\right)}{{\left(1-|{b}_{1}|r\right)}^{2}},\end{array}$

where $F\left(cos\alpha ,r\right)$ is given by (1.6) for all $|z|=r<1$, $|\alpha |<\pi /2$ and ${J}_{f\left(z\right)}$ is the Jacobian of f defined by ${J}_{f\left(z\right)}={|{h}^{\prime }\left(z\right)|}^{2}-{|{g}^{\prime }\left(z\right)|}^{2}$ for all $z\in \mathbb{D}$.

Proof The proof of this corollary is a simple consequence of Lemma 1.1 and Lemma 1.4. □

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Correspondence to Emel Yavuz Duman.

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Yavuz Duman, E., Polatoğlu, Y. & Kahramaner, Y. An investigation on a new class of harmonic mappings. J Inequal Appl 2013, 478 (2013). https://doi.org/10.1186/1029-242X-2013-478