In this paper, the \(q\)-derivative operator and the principle of subordination were employed to define a subclass \(\mathcal{B}_q(\tau,\lambda,\phi)\) of analytic and bi-univalent functions in the open unit disk \(\mathcal{U}\). For functions \(f(z)\in\mathcal{B}_q(\tau,\lambda,\phi)\), we obtained early coefficient bounds and some Fekete-Szegö estimates for real and complex parameters.

We use the concept of q-differentiation to define a class Eq(β, δ) of analytic and univalent functions. The investigations thereafter includes coefficient estimates, inclusion property and some conditions for membership of some analytic functions to be in the class Eq(β, δ). Our results generalize some known and new ones.

In this paper, we considered a family of analytic and univalent functions having positive real parts in the unit disk and defined by a q-difference operator. The coefficients, the Fekete-Szego estimates and the second Hankel determinants were established for the family of functions. Our family of functions generalized some earlier known ones and by varying some parameters, our results also generalized some known ones.

In this work, we investigate the (p, q)-Gegenbauer polynomials for a class of analytic and bi-univalent functions defined in the open unit disk, with respect to subordination. We give an elementary proof to establish some estimates for the coefficient bounds for functions in the new class. We conclude the study by giving a result of the Fekete-Szegö theorem. A corollary was given to show some results of some subclasses of our new class.

By making use of $q$-derivative and $q$-integral operators, we define a class of analytic and bi-univalent functions in the unit disk $|z|<1$. Subsequently, we investigate some properties such as some early coefficient estimates and then obtain the Fekete-Szeg\"o inequality for both real and complex parameters. Further, some interesting corollaries are discussed.