Coefficient estimates for m-fold symmetric bi-subordinate functions

Abstract

A function is said to be bi-univalent in the open unit disk U if both the function and its inverse map are univalent in U. By the same token, a function is said to be bisubordinate in U if both the function and its inverse map are subordinate to a given function in U. In this paper, we consider the m-fold symmtric transform of such functions and use their Faber polynomial expansions to find upper bounds for their n-th (n ≥ 3) coefficients subject to a given gap series condition. We also determine bounds for the first two coefficients of such functions with no restrictions imposed.