Initial faber polynomial coefficient estimates for a subclasses of m-fold symmetric and bi-univalent functions

A function is said to be univalent (or schlicht) if it never takes the same value twice: f (z1)  f (z2 ) if 1 2 z  z . A function f in the class of analytic functions is said to be bi-univalent in the open unit disc U z : z 1 if both f and 1 f  are univalent in U . The Faber polynomials for analytic functions of the complex plane are of a basic interest of polynomial approximations. The main purpose of this thesis is to introduce a new subclass of bi-univalent functions which are m-fold symmetric analytic functions in the open unit disc. Furthermore, using the Faber polynomial expansion, general and initial upper bounds of coefficients for analytic, m-fold symmetric, biunivalent functions are obtained in this thesis.