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STUDIA MATHEMATICA  Issue no. 2 / 2012  
Article: 
SOME PROPERTIES ON GENERALIZED CLOSETOSTAR FUNCTIONS. Authors: . 

Abstract: Let f(z) = a_{1}z+a_{1}z^{2 +}... , a_{1}≠ 0, be regular in z < 1 and have there no zeros except at the origin. Reade ([3]) and the Sakaguchi ([2]) showed that a necessary and sufficient condition for f(z) to be a member of the class C(k) is that f(z) has a representation of the form f(z) = s(z)(p(z))^{k} where s(z) is a regular function starlike with respect to the origin for z < 1, k is a positive constant, and p(z) is a regular function with positive real part in z < 1. The class of closetostar functions introduced by Reade ([4]) is equivalent to C(1). In this paper we define the class C(k, A,B) (−1 £ B < A £ 1, k is positive constant) which contains the functions of the form f(z) = s(z)(p(z))^{k} where s(z) is a regular Janowski starlike function, and p(z) is a regular functionwith positive real part in z < 1. The aim of this paper is to give some propertiesand distortion theorems for this class. Mathematics Subject Classification (2010): 30C45. Keywords: Distortion theorem, radius of starlikeness. 
