Initial coefficient bounds for a subclass of m-old symmetric bi-univalent functions

Let Sigma denote the class of functions f(z) = z + Sigma(infinity)(n=2) a(n)z(n) belonging to the normalized analytic function class A in the open unit disk U, which are bi-univalent in U, that is, both the function f and its inverse f(-1) are univalent in U. The usual method for computation of the coefficients of the inverse function f(-1)(z) by means of the relation f(-1)(f(z))= z is too difficult to apply in the case of m-fold symmetric analytic functions in U. Here, in our present investigation, we aim at overcoming this difficulty by using a general formula to compute the coefficients of f(-1)(z) in conjunction with the residue calculus. As an application, we introduce two new subclasses of the bi-univalent function class ? in which both f(z) and f(-1)(z) are m-fold symmetric analytic functions with their derivatives in the class P of analytic functions with positive real part in U. For functions in each of the subclasses introduced in this paper, we obtain the coefficient bounds for |a(m+1)| and |a(2m+1)|