Some properties of starlike harmonic mappings
© Aydogĝan et al.; licensee Springer 2012
Received: 10 April 2012
Accepted: 6 July 2012
Published: 23 July 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.
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 .
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 .
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])
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
After simple calculations from (2.3), we get (2.1). □
which is given in (2.4) and (2.5). □
Proof Using Theorem 1.2 and Corollary 2.2, we obtain (2.7) and (2.8). □
is likewise in .
, and after straightforward calculations, we get (2.13) and (2.14). □
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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