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

In this paper, we introduce a new family of generalized Bernstein operators based on q integers, called (α,q) -Bernstein operators, denoted by Tn,q,α(f) . We investigate a Kovovkin-type approximation theorem, and obtain the rate of convergence of Tn,q,α(f) 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 $f(x)$ . We also obtain the monotonicity with n and q of Tn,q,α(f) .