On Coefficients of a Certain Subclass of Starlike and Bi-starlike Functions

In this paper we investigate a subclass M(alpha) of the class of starlike functions in the unit disk vertical bar z vertical bar < 1. M(alpha), pi/2 <= alpha < pi, is the set of all analytic functions f in the unit disk vertical bar z vertical bar < 1 with the normalization f(0) = f'(0) - 1 = 0 that satisfy the condition 1 + alpha - pi/2 sin alpha < Re {zf'(z)/f(z)} < 1 + alpha/2 sin alpha (z is an element of Delta). The class M(alpha) was introduced by Kargar et al. [Complex Anal. Oper. Theory 11: 1639-1649, 2017]. In this paper some basic geometric properties of the class M(alpha) are investigated. Among others things, coefficients estimates and bound are given for the Fekete-Szego functional associated with the k-th root transform [f(z(k))](1/k). Also a certain subclass of bi-starlike functions is introduced and the bounds for the initial coefficients are obtained.