A new variant of the modified Bernstein-Kantorovich operators defined by Özarslan and Duman

Abstract

In the present article, we introduce a new function $ \zeta $ in the modified Bernstein-Kantorovich operators defined by Özarslan and Duman (http://dx.doi.org/10.1080/01630563.2015.1079219) and properties of the function $ \zeta $ is define as: this is an infinitely differentiable function in the closed interval $ [0, 1] \; \text{such that}\; \zeta(0) = 0, \; \zeta(1) = 1 $ and $ \zeta'(\varkappa) $. We discussed an approximation properties with the help of the Bohman-Korovkin's type theorem and scrutinize the rate of convergence with the aid of modulus of continuity, Lipschitz type functions. Furthermore, we study the rate of convergence by means of derivatives of bounded variation and an approximation theorem by using of the Bohman-Korovkin's type theorem in $ \mathcal{A}- $Statistical convergence.In lastly, in order to validate our theoretical results, we provide some numerical experiments that are produced by a MATHEMATICA compiler and show that a careful choice of the function $ \zeta(\varkappa) $ for the both operators and we see that the our operator gives the better approximation results as compare to the modified Bernstein-Kantorovich operators defined by Özarslan (doi.org/10.1080/01630563.2015.1079219).