Shape-preserving properties of a new family of generalized Bernstein operators

Abstract

In this paper, we introduce a new family of generalized Bernstein operators based on q integers, called $(\alpha ,q)$ -Bernstein operators, denoted by $T_{n,q,\alpha }\left(f\right)$ . We investigate a Kovovkin-type approximation theorem, and obtain the rate of convergence of $T_{n,q,\alpha }\left(f\right)$ to any continuous functions f. The main results are the identification of several shape-preserving properties of these operators, including their monotonicity- and convexity-preserving properties with respect to $f\left(x\right)$ . We also obtain the monotonicity with n and q of $T_{n,q,\alpha }\left(f\right)$ .

Keywords: Basis function; Bernstein operators; Monotonicity; Shape-preserving; q-integers.