Let ∑ denote the class of functions f(z)=z + ∑_n=2^∞ a_nzⁿ 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|.

Tbilisi Mathematical Journal, Vol. 7(2) (2014), pp. 1-10