On a Stancu form Szász-Mirakjan-Kantorovich operators based on shape parameter $\lambda$

Abstract

This paper deals with some approximation properties of Stancu-Kantorovich variant of Sz\'{a}sz-Mirakjan operators based on B\'{e}zier basis functions with shape parameter $\lambda\in\lbrack-1,1]$. We compute several preliminary results such as moments and central moments. Later, we introduce a Korovkin-type convergence theorem and discuss the order of approximation in terms of the modulus of continuity and for the elements belong to Lipschitz-type class and Peetre's $K$-functional, respectively. Also, we prove a Voronovskaya type asymptotic theorem. Lastly, we present the comparison of the convergence of constructed operators to the certain functions with some graphical illustrations for different values of $m$, $\alpha$, $\beta$ and $\lambda$ parameters.