Original scientific paper

On the linear combination of the representations of starlikeness and convexity

Nikola Tuneski
Roza Aceska

Abstract

Let be the class of analytic functions in the unit disk U = {z : |z| < 1} that are normalized with f(0) = f'(0) - 1 = 0. Also, let S*[A,B], -1 ≤ B < A ≤ 1, be the class of functions f , such that zf'(z) / f(z) (1 + Az) / (1 + Bz), where "" denotes the usual subordination. In this paper we investigate the linear combination of the analytic representations of starlikeness and convexity and give sharp sufficient conditions over the differential operator

a zf'(z) / f(z) + b (1 + zf''(z) / f'(z))

that imply f S*[A,B]. In that purpose we use the method of differential subordinations. Several corollaries and examples for different choices of A, B, a and b are given and comparison with previous known results is done.

Keywords

Starlike function; starlike function of order α; criteria; differential subordination; Jack lemma

Hrčak ID:

1249

URI

https://hrcak.srce.hr/1249

Publication date:

29.11.2004.

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