Preservation properties by Bernstein-type operators. A probabilistic approach

In this paper, we are concerned with preservation properties of first- and second-order by Bernstein-type operators which preserve monotone functions. We obtain characterizations of the preservation of nondecreasing right-continuous functions, first- and second-order modulus of smoothness, Lipschitz classes of first- and second-order, uniform and absolute continuity, and convexity. These kinds of problems lead us to consider the notions of dual and derived operators. We give a simple unified approach based on stochastic orders and probabilistic coupling techniques, in the sense that we represent the operators under consideration in terms of stochastic processes. The preceding results are illustrated by considering well-known Bernstein-type operators, such as generalized Bernstein-Kantorovich, generalized Szász-Kantorovich, Gamma, and Beta operators.