Coefficients of Meromorphic Bi-Bazilevic Functions

A function is said to be bi-Bazilevic in a given domain if both the function and its inverse map are Bazilevic there. Applying the Faber polynomial expansions to the meromorphic Bazilevic functions, we obtain the general coefficient bounds for bi-Bazilevic functions. We also demonstrate the unpredictability of the behavior of early coefficients of bi-Bazilevic functions. 1. Introduction Let denote the family of meromorphic functions of the form which are univalent in . The coefficients of , the inverse map of the function , are given by the Faber polynomial expansion: where and with is a homogeneous polynomial of degree in the variables (see [1], p. 349 or [2–4]). For , , , and , let denote the class of bi-Bazilevic functions of order and type (see Bazilevic [5]) if and only if Estimates on the coefficients of classes of meromorphic univalent functions were widely investigated in the literature. For example, Schiffer [4] obtained the estimate for meromorphic univalent functions with and Duren ([6] or [7]) proved that if for , then . Schober [8] considered the case where and obtained the estimate for the odd coefficients of the inverse function subject to the restrictions if is even or that if is odd. Kapoor and Mishra [9] considered the inverse function , where , and obtained the bound if . This restriction imposed on is a very tight restriction since the class shrinks for large values of . More recently, Hamidi et al. [10] (also see [11]) improved the coefficient estimate given by Kapoor and Mishra in [9]. The real difficulty arises when the bi-univalency condition is imposed on the meromorphic functions and its inverse . The unexpected and unusual behavior of the coefficients of meromorphic functions and their inverses prove the investigation of the coefficient bounds for bi-univalent functions to be very challenging. In this paper we extend the results of Kapoor and Mishra [9] and Hamidi et al. [10, 11] to a larger class of meromorphic bi-univalent functions, namely, . We conclude our paper with an examination of the unexpected behavior of the early coefficients of meromorphic bi-Bazilevic functions which is the best estimate yet appeared in the literature. 2. Main Results Applying a result of Airault [12] or [1, 3] to meromorphic functions of the form (1), for real values of we can write where and is a homogeneous polynomial of degree in the variables . A simple calculation reveals that the first three terms of may be expressed as In general, for any real number , an expansion of (e.g., see [2, equation (4)] or [3]) is given by where and . Here we