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

Let Σ denote the class of functions f(z)=z+∑ₙ₌₂∞aₙzⁿ 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 Σ 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 |aₘ₊₁| and |a₂ₘ₊₁|.