Bezier variant of the Bernstein–Durrmeyer type operators

Abstract

In the present paper, we introduce the Bezier-variant of Durrmeyer modification of the Bernstein operators based on a function \(\tau \), which is infinite times continuously differentiable and strictly increasing function on [0, 1] such that \(\tau (0)=0\) and \(\tau (1)=1\). We give the rate of approximation of these operators in terms of usual modulus of continuity and K-functional. Next, we establish the quantitative Voronovskaja type theorem. In the last section we obtain the rate of convergence for functions having derivative of bounded variation.