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Some properties of starlike harmonic mappings
Journal of Inequalities and Applications volume 2012, Article number: 163 (2012)
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.
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 , 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 .
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 (, p.51])
If , then there exist angles α and β such that
for all .
Theorem 1.2 (, p.108])
If is a starlike function and if and are the regular functions defined by , , , then is a convex function.
2 Main results
Lemma 2.1 Let be an element of , then
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 . Therefore, we have
After simple calculations from (2.3), we get (2.1). □
Corollary 2.2 Let be 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 2.3 Let be an element of , then
Proof Using Theorem 1.2 and Corollary 2.2, we obtain (2.7) and (2.8). □
Theorem 2.4 If is in and a is 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
Corollary 2.5 Let be 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). □
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The authors are very grateful to the referees for their valuable comments and suggestions. They were very helpful for our paper.
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.
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Cite this article
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
- harmonic starlike function
- growth theorem
- distortion theorem