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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 whenever is , and we will give an application of this fundamental result.
MSC:30C45, 30C55.
1 Introduction
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 angles α and β such that
for all .
Theorem 1.2 ([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
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 2.2 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
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 2.5 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). □
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 for which 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