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# Some properties of starlike harmonic mappings

*Journal of Inequalities and Applications*
**volume 2012**, Article number: 163 (2012)

## Abstract

A fundamental result of this paper shows that the transformation

defines a function in {S}_{H{S}^{\ast}}^{0} whenever f=h(z)+\overline{g(z)} 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 (z) which are regular in \mathbb{D} and satisfy the conditions \varphi (0)=0, |\varphi (z)|<1 for all z\in \mathbb{D}; denote by \mathcal{P} the family of functions

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

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

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

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

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

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.*,

Equivalently [1], the inequality |{g}^{\prime}(z)|<|{h}^{\prime}(z)| holds for all z\in \mathbb{D}. This shows, in particular, that {h}^{\prime}(z)\ne 0, so there is no loss of generality in supposing that h(0)=0 and {h}^{\prime}(0)=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(z) 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}(0)=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(z)+\overline{g(z)} be an element {S}_{H} (or {S}_{H}^{0}). If *f* satisfies the condition

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(z)+\overline{g(z)} be an element {S}_{H} (or {S}_{H}^{0}). If *f* satisfies the condition

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(z)+\overline{g(z)}\in {S}_{HC}, *then there exist angles* *α* *and* *β* *such that*

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

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

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

## 2 Main results

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

*where*

*Proof* Using Theorem 1.2, we write

On the other hand, since

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

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

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

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

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

which is given in (2.4) and (2.5). □

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

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

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

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

*Proof* For *ρ* real, 0<\rho <1, let

then we have

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

and we conclude that

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(z)+\overline{g(z)} *be an element of* {S}_{H{S}^{\ast}}^{0}, *then*

*Proof* Using Theorem 2.4, we have

If we apply Corollary 2.3 to H(z) and G(z) by taking

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

## References

Clunie J, Sheil-Small T: Harmonic univalent functions.

*Ann. Acad. Sci. Fenn., Ser. A 1 Math.*1984, 9: 3–25.Duren P:

*Harmonic Mappings in the Plane*. Cambridge University Press, Cambridge; 2004.Libera RJ, Ziegler MR: Regular functions f(z) for which z{f}^{\prime}(z) is

*α*-spirallike.*Trans. Am. Math. Soc.*1972, 166: 361–370.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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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

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DOI: https://doi.org/10.1186/1029-242X-2012-163