Some properties of starlike harmonic mappings

A fundamental result of this paper shows that the transformation

defines a function in whenever is , and we will give an application of this fundamental result.

MSC: 30C45, 30C55.

Let Ω be the family of functions which are regular in and satisfy the conditions , for all ; denote by the family of functions

regular in , such that is in if and only if

for some function and every .

Next, let and be regular functions in , if there exists such that for all , then we say that is subordinated to and we write , then .

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 as the sum of a regular function and the conjugate of a regular function . With the convention that , the representation is unique. The power series expansions of and are denoted by

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

Equivalently [1], the inequality holds for all . This shows, in particular, that , so there is no loss of generality in supposing that and . The class of all sense-preserving harmonic mappings of the disc with and will be denoted by . Thus contains the standard class S of regular univalent functions. Although the regular part of a function is locally univalent, it will become apparent that it need not be univalent. The class of functions with will be denoted by . At the same time, we note that is a normal family and is a compact normal family [2].

Finally, let be an element (or ). If f satisfies the condition

then f is called harmonic starlike function. The class of such functions is denoted by (or ). Also, let be an element (or ). If f satisfies the condition

then f is called a convex harmonic function. The class of convex harmonic functions is denoted by (or ).

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

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

If, then there exist anglesαandβsuch that

for all.

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

Ifis a starlike function and ifandare the regular functions defined by, , , thenis a convex function.

Lemma 2.1Letbe an element of, then

where

Proof Using Theorem 1.2, we write

On the other hand, since

is regular and satisfies the condition , with , the function

is an element of [4]. Therefore, we have

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

Corollary 2.2Letbe an element of, then

Proof Since , then and the second dilatation 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 2.3Letbe an element of, then

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

Theorem 2.4Ifis inandais in, then

is likewise in.

Proof For ρ real, , let

then we have

Letting and in (2.11) and after the straightforward calculations, we obtain

and we conclude that

is in for every admissible ρ. From the compactness of [2] and (2.11), we infer that is in . We also note that this theorem is a generalization of the theorem of Libera and Ziegler [3]. □

Corollary 2.5Letbe an element of, then

Proof Using Theorem 2.4, we have

If we apply Corollary 2.3 to and by taking

, and after straightforward calculations, we get (2.13) and (2.14). □

The authors declare that they have no competing interests.

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

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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