Some general coefficient estimates for a new class of analytic and bi-univalent functions defined by a linear combination

In the present paper, we introduce and investigate a new class of analytic and bi-univalent functions f (z) in the open unit disk U. For this purpose, we make use of a linear combination of the following three functions: f(z)/z, f'(z) and z f''(z) for a function belonging to the normalized univalent function class S. By applying the technique involving the Faber polynomials, we determine estimates for the general Taylor-Maclaurin coefficient of functions belonging to the analytic and bi-univalent function class which we have introduced here. We also demonstrate the not-too-obvious behaviour of the first two Taylor-Maclaurin coefficients of such functions.