Keywords:harmonic starlike function; growth theorem; distortion theorem
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
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 .
For the aim of this paper, we will need the following lemma and theorem.
Lemma 1.1 (, p.51])
Theorem 1.2 (, p.108])
2 Main results
Proof Using Theorem 1.2, we write
On the other hand, since
is an element of . Therefore, we have
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). □
then we have
and we conclude that
Proof Using Theorem 2.4, we have
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.