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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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