Coefficient Estimates for Starlike and Convex Classes of -fold Symmetric Bi-univalent Functions

Abstract

In an article of Pommerenke [10] he remarked that, for an -fold symmetric functions in the class , the well known lemma stated by Caratheodary for a one fold symmetric functions in still holds good. Exploiting this concept, we introduce certain new subclasses of the bi-univalent function class in which both and −1 are -fold symmetric analytic with their derivatives in the class of analytic functions. Furthermore, for functions in each of the subclasses introduced in this paper, we obtain the coefficient bounds for | +1 | and | 2 +1 |. We remark here that the concept of -fold symmetric bi-univalent is not in the literature and the authors hope it will make the researchers interested in these type of investigations in the forseeable future. By the working procedure and the difficulty involved in these procedures, one can clearly conclude that there lies an unpredictability in finding the coefficients of a -fold symmetric bi-univalent functions.