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 [Formula: see text] -Bernstein operators, denoted by [Formula: see text] . We investigate a Kovovkin-type approximation theorem, and obtain the rate of convergence of [Formula: see text] 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 [Formula: see text] . We also obtain the monotonicity with n and q of [Formula: see text] .