Faber polynomial coefficients of bi-subordinate functions

Abstract

A function is said to be bi-univalent in the open unit disk D if both the function and its inverse map are univalent in D. By the same token, a function is said to be bi-subordinate in D if both the function and its inverse map are subordinate to certain given function in D. The behavior of the coefficients of such functions are unpredictable and unknown. In this paper, we use the Faber polynomial expansions to find upper bounds for the n-th (n≥3) coefficients of classes of bi-subordinate functions subject to a gap series condition as well as determining bounds for the first two coefficients of such functions.