In this paper we introduce a bivariate of q-Durrmeyer variant of generalized Bernstein operators by using Pólya distribution. The convergence rate of these operators is examined by means of the Lipschitz class and the modulus of continuity. Furthermore, we obtain a Voronovskaja type symptotic formula, error estimation in terms of the partial modulus of continuity and Peetre's K-functional.

In this article, we establish an extension of the bivariate generalization of the q -Bernstein type operators involving parameter λ and extension of GBS (Generalized Boolean Sum) operators of bivariate q -Bernstein type. For the first operators, we state the Volkov-type theorem and we obtain a Voronovskaja type and investigate the degree of approximation by means of the Lipschitz type space. For the GBS type operators, we establish their degree of approximation in terms of the mixed modulus of smoothness. The comparison of convergence of the bivariate q -Bernstein type operators based on parameters and its GBS type operators is shown by illustrative graphics using MATLAB software.

Using methods from the theory of differential subordinations, we obtain several results that describe the relation between two classes univalent functions. Examples that demonstrate these results are provided.

The class𝒰(λ,μ)of normalized analytic functions that satisfy|(z/f(z))1+μ·f′(z)−1|<λfor allzin the open unit disk is studied and sufficient conditions for anα-convex function to be in𝒰(λ,μ)are given.