Coefficient Estimates for Certain Classes of Bi-Univalent Functions

A function analytic in the open unit disk is said to be bi-univalent in if both the function and its inverse map are univalent there. The bi-univalency condition imposed on the functions analytic in makes the behavior of their coefficients unpredictable. Not much is known about the behavior of the higher order coefficients of classes of bi-univalent functions. We use Faber polynomial expansions of bi-univalent functions to obtain estimates for their general coefficients subject to certain gap series as well as providing bounds for early coefficients of such functions. 1. Introduction Let denote the class of functions which are analytic in the open unit disk and normalized by Let denote the class of functions that are univalent in and let be the class of functions that are analytic in and satisfy the condition in . By the Caratheodory lemma (e.g., see [1]) we have . For and we let denote the family of analytic functions so that We note that is the class of bounded boundary turning functions and also that if . For , the class and was first defined and investigated by Ding et al. [2]. It is well known that every function has an inverse satisfying for and for , according to Kobe One Quarter Theorem (e.g., see [1]). A function is said to be bi-univalent in if both and . Finding bounds for the coefficients of classes of bi-univalent functions dates back to 1967 (see Lewin [3]). But the interest on the bounds for the coefficients of classes of bi-univalent functions picked up by the publications of Brannan and Taha [4], Srivastava et al. [5], Frasin and Aouf [6], Ali et al. [7], and Hamidi et al. [8]. The bi-univalency condition imposed on the functions makes the behavior of their coefficients unpredictable. Not much is known about the behavior of the higher order coefficients of classes of bi-univalent functions, as Ali et al. [7] also remarked that finding the bounds for when is an open problem. Here in this paper we let and and use the Faber polynomial coefficient expansions to provide bounds for the general coefficients of such functions with a given gap series. We also obtain estimates for the first two coefficients and of these functions as well as providing an estimate for their coefficient body . The bounds provided in this paper prove to be better than those estimates provided by Srivastava et al. [5] and Frasin and Aouf [6]. 2. Main Results Using the Faber polynomial expansion of functions of the form (1), the coefficients of its inverse map may be expressed as, [9], where such that with is a homogeneous polynomial in the variables [10]. In