Subclasses of Bi-Univalent Functions Defined by Frasin Differential Operator

Let Omega denote the class of functions f (z) = z + a(2)z(2) + a(3)z(3) + ... belonging to the normalized analytic function class A in the open unit disk U = {z : vertical bar z vertical bar < 1}, which are bi-univalent in U, that is, both the function f and its inverse f(-1) are univalent in U. In this paper, we introduce and investigate two new subclasses of the function class Omega of bi-univalent functions defined in the open unit disc U, which are associated with a new differential operator of analytic functions involving binomial series. Furthermore, we find estimates on the Taylor-Maclaurin coefficients vertical bar a(2)vertical bar and vertical bar a(3)vertical bar for functions in these new subclasses. Several (known or new) consequences of the results are also pointed out.