Applications of Q-Derivative Operator to Subclasses of Bi-Univalent Functions Involving Gegenbauer Polynomials

Abstract

In recent years, using the idea of analytic and bi-univalent functions, many ideas have been developed by different well-known authors, but the using Gegenbauer polynomials along with certain bi-univalent functions is very rare in the literature. We are essentially motivated by this recent research going on, here in our present investigation, we make use of certain q-derivative operator and Gegenbauer polynomials and define a new subclass of analytic and bi-univalent functions. We then obtain certain coefficient bounds, the Fekete-Szego inequalities and upper bounds for the second-order Hankel determinant for the defined functions class.