On the q-Monotonicity Preservation of Durrmeyer-Type Operators

Abstract

We prove that various Durrmeyer-type operators preserve q-monotonicity in [0, 1] or [0,infty ) as the case may be. Recall that a 1-monotone function is nondecreasing, a 2-monotone one is convex, and for q>2, a q-monotone function possesses a convex (q-2)nd derivative in the interior of the interval. The operators are the Durrmeyer versions of Bernstein (including genuine Bernstein–Durrmeyer), Szász and Baskakov operators. As a byproduct we have a new type of characterization of continuous q-monotone functions by the behavior of the integrals of the function with respect to measures that are related to the fundamental polynomials of the operators.