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A function analytic in the open unit disk D is said to be bi-univalent in D if both the function and its inverse map are univalent there. The bi-univalency condition imposed on the functions analytic in D 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.

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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.