Durrmeyer-type generalization of $\mu$-Bernstein operators

Abstract

In the present manuscript, we consider $\mu$-Bernstein-Durrmeyer operators involving a strictly positive continuous function. Firstly,  we prove a Voronovskaja type, quantitative Voronovskaja type and Gr\"uss-Voronovskaja type asymptotic formula, the rate of convergence by means of the modulus of continuity and for functions in a Lipschitz type space.  Finally, we show that the numerical examples which describe the validity of the theoretical example and the effectiveness of the defined operators.