# Some properties of starlike harmonic mappings

## Abstract

A fundamental result of this paper shows that the transformation

$F=\frac{az\left(h\left(\frac{z+a}{1+\overline{a}z}\right)+\overline{g\left(\frac{z+a}{1+\overline{a}z}\right)}\right)}{\left(h\left(a\right)+\overline{g\left(a\right)}\right)\left(z+a\right)\left(1+\overline{a}z\right)}$

defines a function in ${S}_{H{S}^{\ast }}^{0}$ whenever $f=h\left(z\right)+\overline{g\left(z\right)}$ is ${S}_{H{S}^{\ast }}^{0}$, and we will give an application of this fundamental result.

MSC:30C45, 30C55.

## 1 Introduction

Let Ω be the family of functions $\varphi \left(z\right)$ which are regular in $\mathbb{D}$ and satisfy the conditions $\varphi \left(0\right)=0$, $|\varphi \left(z\right)|<1$ for all $z\in \mathbb{D}$; denote by $\mathcal{P}$ the family of functions

$p\left(z\right)=1+{p}_{1}z+{p}_{2}{z}^{2}+\cdots$

regular in $\mathbb{D}$, such that $p\left(z\right)$ is in $\mathcal{P}$ if and only if

$p\left(z\right)=\frac{1+\varphi \left(z\right)}{1-\varphi \left(z\right)}$
(1.1)

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

Next, let ${s}_{1}\left(z\right)=z+{c}_{2}{z}^{2}+{c}_{3}{z}^{3}+\cdots$ and ${s}_{2}\left(z\right)=z+{d}_{2}{z}^{2}+{d}_{3}{z}^{3}+\cdots$ be regular functions in $\mathbb{D}$, if there exists $\varphi \left(z\right)\in \mathrm{\Omega }$ such that ${s}_{1}\left(z\right)={s}_{2}\left(\varphi \left(z\right)\right)$ for all $z\in \mathbb{D}$, then we say that ${s}_{1}\left(z\right)$ is subordinated to ${s}_{2}\left(z\right)$ and we write ${s}_{1}\left(z\right)\prec {s}_{2}\left(z\right)$, then ${s}_{1}\left(\mathbb{D}\right)\subset {s}_{2}\left(\mathbb{D}\right)$.

Moreover, univalent harmonic functions are generalizations of univalent regular functions; the point of departure is the canonical representation

$f=h\left(z\right)+\overline{g\left(z\right)},\phantom{\rule{1em}{0ex}}g\left(0\right)=0$
(1.2)

of a harmonic function f in the unit disc $\mathbb{D}$ as the sum of a regular function $h\left(z\right)$ and the conjugate of a regular function $g\left(z\right)$. With the convention that $g\left(0\right)=0$, the representation is unique. The power series expansions of $h\left(z\right)$ and $g\left(z\right)$ are denoted by

$h\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}{a}_{n}{z}^{n},\phantom{\rule{2em}{0ex}}g\left(z\right)=\sum _{n=1}^{\mathrm{\infty }}{b}_{n}{z}^{n}.$
(1.3)

If f is a sense-preserving harmonic mapping of $\mathbb{D}$ onto some other region, then, by Lewy theorem, its Jacobian is strictly positive, i.e.,

${J}_{f\left(z\right)}=|{h}^{\prime }\left(z\right){|}^{2}-|{g}^{\prime }\left(z\right){|}^{2}>0.$
(1.4)

Equivalently [1], the inequality $|{g}^{\prime }\left(z\right)|<|{h}^{\prime }\left(z\right)|$ holds for all $z\in \mathbb{D}$. This shows, in particular, that ${h}^{\prime }\left(z\right)\ne 0$, so there is no loss of generality in supposing that $h\left(0\right)=0$ and ${h}^{\prime }\left(0\right)=1$. The class of all sense-preserving harmonic mappings of the disc with ${a}_{0}={b}_{0}=0$ and ${a}_{1}=1$ will be denoted by ${S}_{H}$. Thus ${S}_{H}$ contains the standard class S of regular univalent functions. Although the regular part $h\left(z\right)$ of a function $f\in {S}_{H}$ is locally univalent, it will become apparent that it need not be univalent. The class of functions $f\in {S}_{H}$ with ${g}^{\prime }\left(0\right)=0$ will be denoted by ${S}_{H}^{0}$. At the same time, we note that ${S}_{H}$ is a normal family and ${S}_{H}^{0}$ is a compact normal family [2].

Finally, let $f=h\left(z\right)+\overline{g\left(z\right)}$ be an element ${S}_{H}$ (or ${S}_{H}^{0}$). If f satisfies the condition

$\frac{\partial }{\partial \theta }\left(Argf\left(r{e}^{i\theta }\right)\right)=Re\left(\frac{z{h}^{\prime }\left(z\right)-\overline{z{g}^{\prime }\left(z\right)}}{h\left(z\right)+\overline{g\left(z\right)}}\right)>0$
(1.5)

then f is called harmonic starlike function. The class of such functions is denoted by ${S}_{H{S}^{\ast }}$ (or ${S}_{H{S}^{\ast }}^{0}$). Also, let $f=h\left(z\right)+\overline{g\left(z\right)}$ be an element ${S}_{H}$ (or ${S}_{H}^{0}$). If f satisfies the condition

$\frac{\partial }{\partial \theta }\left(\frac{\partial }{\partial \theta }\left(Argf\left(r{e}^{i\theta }\right)\right)\right)=Re\left(\frac{z{\left(z{h}^{\prime }\left(z\right)\right)}^{\prime }-\overline{z{\left(z{g}^{\prime }\left(z\right)\right)}^{\prime }}}{z{h}^{\prime }\left(z\right)+\overline{z{g}^{\prime }\left(z\right)}}\right)>0,$
(1.6)

then f is called a convex harmonic function. The class of convex harmonic functions is denoted by ${S}_{HC}$ (or ${S}_{HC}^{0}$).

For the aim of this paper, we will need the following lemma and theorem.

Lemma 1.1 ([2], p.51])

If $f=h\left(z\right)+\overline{g\left(z\right)}\in {S}_{HC}$, then there exist angles α and β such that

$Re\left[\left({e}^{i\alpha }{h}^{\prime }\left(z\right)+{e}^{-i\alpha }{g}^{\prime }\left(z\right)\right)\left({e}^{i\beta }-{e}^{-i\beta }{z}^{2}\right)\right]>0$
(1.7)

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

Theorem 1.2 ([2], p.108])

If $f=h\left(z\right)+\overline{g\left(z\right)}\in {S}_{H}$ is a starlike function and if $H\left(z\right)$ and $G\left(z\right)$ are the regular functions defined by $z{H}^{\prime }\left(z\right)=h\left(z\right)$, $z{G}^{\prime }\left(z\right)=-g\left(z\right)$, $H\left(0\right)=G\left(0\right)=0$, then $F=H\left(z\right)+\overline{G\left(z\right)}$ is a convex function.

## 2 Main results

Lemma 2.1 Let $f=h\left(z\right)+\overline{g\left(z\right)}$ be an element of ${S}_{HC}^{0}$, then

$\frac{G\left(\alpha ,\beta ,-r\right)}{{\left(1+{r}^{2}\right)}^{2}}\le |{h}^{\prime }\left(z\right)+{e}^{-2i\alpha }{g}^{\prime }\left(z\right)|\le \frac{G\left(\alpha ,\beta ,r\right)}{{\left(1-{r}^{2}\right)}^{2}},$
(2.1)

where

$\begin{array}{c}G\left(\alpha ,\beta ,r\right)=2cos\left(\alpha +\beta \right)r+\sqrt{1+\left[2cos\left(\alpha +\beta \right)\right]{r}^{2}+{r}^{4}},\hfill \\ \phantom{\rule{1em}{0ex}}cos\left(\alpha +\beta \right)>0.\hfill \end{array}$

Proof Using Theorem 1.2, we write

$\begin{array}{c}p\left(z\right)=\left({e}^{i\alpha }{h}^{\prime }\left(z\right)+{e}^{-i\alpha }{g}^{\prime }\left(z\right)\right)\left({e}^{i\beta }-{e}^{-i\beta }{z}^{2}\right),\phantom{\rule{1em}{0ex}}Rep\left(z\right)>0,\hfill \\ p\left(0\right)=\left({e}^{i\alpha }{h}^{\prime }\left(0\right)+{e}^{-i\alpha }{g}^{\prime }\left(0\right)\right)\left({e}^{-i\beta }-{e}^{i\beta }{0}^{2}\right)=cos\left(\alpha +\beta \right)+isin\left(\alpha +\beta \right).\hfill \end{array}$

On the other hand, since

$p\left(z\right)=\left[cos\left(\alpha +\beta \right)+isin\left(\alpha +\beta \right)\right]+{p}_{1}z+{p}_{2}{z}^{2}+\cdots$

is regular and satisfies the condition $Rep\left(z\right)>0$, with $cos\left(\alpha +\beta \right)>0$, the function

${p}_{1}\left(z\right)=\frac{1}{cos\left(\alpha +\beta \right)}\left[p\left(z\right)-isin\left(\alpha +\beta \right)\right]$
(2.2)

is an element of $\mathcal{P}$ [4]. Therefore, we have

$|{p}_{1}\left(z\right)-\frac{1+{r}^{2}}{1-{r}^{2}}|\le \frac{2r}{1-{r}^{2}}.$
(2.3)

After simple calculations from (2.3), we get (2.1). □

Corollary 2.2 Let $f=h\left(z\right)+\overline{g\left(z\right)}$ be an element of ${S}_{HC}^{0}$, then

$\frac{G\left(\alpha ,\beta ,-r\right)}{{\left(1+{r}^{2}\right)}^{2}\left(1-r\right)}\le |{h}^{\prime }\left(z\right)|\le \frac{G\left(\alpha ,\beta ,r\right)}{{\left(1-r\right)}^{3}{\left(1+r\right)}^{2}},$
(2.4)
$\frac{|w\left(z\right)|G\left(\alpha ,\beta ,-r\right)}{{\left(1+{r}^{2}\right)}^{2}\left(1-r\right)}\le |{g}^{\prime }\left(z\right)|\le \frac{rG\left(\alpha ,\beta ,r\right)}{{\left(1-r\right)}^{3}{\left(1+r\right)}^{2}}.$
(2.5)

Proof Since $f\in {S}_{HC}^{0}$, then ${g}^{\prime }\left(z\right)={h}^{\prime }\left(z\right)w\left(z\right)$ and the second dilatation $w\left(z\right)$ satisfies the condition of Schwarz lemma, then the inequality (2.1) can be written in the form

$\frac{G\left(\alpha ,\beta ,-r\right)}{|1+{e}^{-2i\alpha }w\left(z\right)|{\left(1+{r}^{2}\right)}^{2}\left(1-r\right)}\le |{h}^{\prime }\left(z\right)|\le \frac{G\left(\alpha ,\beta ,r\right)}{|1+{e}^{-2i\alpha }w\left(z\right)|{\left(1-{r}^{2}\right)}^{2}}$
(2.6)

which is given in (2.4) and (2.5). □

Corollary 2.3 Let $f=h\left(z\right)+g\left(z\right)$ be an element of ${S}_{CH}^{0}$, then

$\frac{rG\left(\alpha ,\beta ,-r\right)}{{\left(1+{r}^{2}\right)}^{2}\left(1-r\right)}\le |h\left(z\right)|\le \frac{rG\left(\alpha ,\beta ,r\right)}{{\left(1-r\right)}^{3}{\left(1+r\right)}^{2}},$
(2.7)
$\frac{|w\left(z\right)|rG\left(\alpha ,\beta ,-r\right)}{{\left(1+{r}^{2}\right)}^{2}\left(1-r\right)}\le |g\left(z\right)|\le \frac{{r}^{2}G\left(\alpha ,\beta ,r\right)}{{\left(1-r\right)}^{3}{\left(1+r\right)}^{2}}.$
(2.8)

Proof Using Theorem 1.2 and Corollary 2.2, we obtain (2.7) and (2.8). □

Theorem 2.4 If $f=h\left(z\right)+\overline{g\left(z\right)}$ is in ${S}_{H{S}^{\ast }}^{0}$ and a is in $\mathbb{D}$, then

$F=\frac{az\left(h\left(\frac{z+a}{1+\overline{a}z}\right)+\overline{g\left(\frac{z+a}{1+\overline{a}z}\right)}\right)}{\left(h\left(a\right)+\overline{g\left(a\right)}\right)\left(z+a\right)\left(1+\overline{a}z\right)}$
(2.9)

is likewise in ${S}_{H{S}^{\ast }}^{0}$.

Proof For ρ real, $0<\rho <1$, let

${F}_{\rho }=\frac{az\left(h\left(\rho \left(\frac{z+a}{1+\overline{a}z}\right)\right)+\overline{g\left(\rho \left(\frac{z+a}{1+\overline{a}z}\right)\right)}\right)}{\left(h\left(\rho a\right)+\overline{g\left(\rho a\right)}\right)\left(z+a\right)\left(1+\overline{a}z\right)}$
(2.10)

then we have

(2.11)

Letting $z={e}^{i\theta }$ and $w=\rho \left(\frac{z+a}{1+\overline{a}z}\right)$ in (2.11) and after the straightforward calculations, we obtain

$Re\left(\frac{z{F}_{z}-\overline{z}{F}_{\overline{z}}}{F}\right)=\frac{1-|a{|}^{2}}{|a+{e}^{i\theta }{|}^{2}}Re\left(\frac{w{h}^{\prime }\left(w\right)-\overline{w{\rho }^{\prime }\left(w\right)}}{h\left(w\right)+\overline{\rho \left(w\right)}}\right)>0,$
(2.12)

and we conclude that

${F}_{\rho }=\frac{az\left(h\left(\rho \left(\frac{z+a}{1+\overline{a}z}\right)\right)+\overline{g\left(\rho \left(\frac{z+a}{1+\overline{a}z}\right)\right)}\right)}{\left(h\left(\rho a\right)+\overline{g\left(\rho a\right)}\right)\left(z+a\right)\left(1+\overline{a}z\right)}$

is in ${S}_{H{S}^{\ast }}^{0}$ for every admissible ρ. From the compactness of ${S}_{H{S}^{\ast }}^{0}$ [2] and (2.11), we infer that $F={lim}_{\rho \to 1}{F}_{\rho }$ is in ${S}_{H{S}^{\ast }}^{0}$. We also note that this theorem is a generalization of the theorem of Libera and Ziegler [3]. □

Corollary 2.5 Let $f=h\left(z\right)+\overline{g\left(z\right)}$ be an element of ${S}_{H{S}^{\ast }}^{0}$, then

(2.13)
(2.14)

Proof Using Theorem 2.4, we have

$\left\{\begin{array}{c}F=\frac{a.z.h\left(\frac{z+a}{1+\overline{a}z}\right)}{\left(h\left(a\right)+\overline{g\left(a\right)}\right)\left(z+a\right)\left(1+\overline{a}z\right)}+\frac{a.z.\overline{g\left(\frac{z+a}{1+\overline{a}z}\right)}}{\left(h\left(a\right)+\overline{g\left(a\right)}\right)\left(z+a\right)\left(1+\overline{a}z\right)}\hfill \\ \phantom{F}=H\left(z\right)+\overline{G\left(z\right)}.\hfill \end{array}$
(2.15)

If we apply Corollary 2.3 to $H\left(z\right)$ and $G\left(z\right)$ by taking

$u=\frac{z+a}{1+\overline{a}z}\phantom{\rule{1em}{0ex}}⇔\phantom{\rule{1em}{0ex}}z=\frac{u-a}{1+\overline{a}u}$

$a=ku$, $-1 and after straightforward calculations, we get (2.13) and (2.14). □

## References

1. Clunie J, Sheil-Small T: Harmonic univalent functions. Ann. Acad. Sci. Fenn., Ser. A 1 Math. 1984, 9: 3–25.

2. Duren P: Harmonic Mappings in the Plane. Cambridge University Press, Cambridge; 2004.

3. Libera RJ, Ziegler MR: Regular functions $f\left(z\right)$ for which $z{f}^{\prime }\left(z\right)$ is α -spirallike. Trans. Am. Math. Soc. 1972, 166: 361–370.

4. Nehari Z: Conformal Mapping. Dover, New York; 1975.

## Acknowledgements

The authors are very grateful to the referees for their valuable comments and suggestions. They were very helpful for our paper.

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Correspondence to Melike Aydog̃an.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally and significantly to writing this paper. All authors read and approved the final manuscript.

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Aydog̃an, M., Yemisci, A. & Polatog̃lu, Y. Some properties of starlike harmonic mappings. J Inequal Appl 2012, 163 (2012). https://doi.org/10.1186/1029-242X-2012-163