An Extension of Raşa’s Conjecture to q-Monotone Functions

Abstract

We extend an inequality involving the Bernstein basis polynomials and convex functions on [0, 1]. The inequality was originally conjectured by Raşa about thirty years ago, but was proved only recently. Our extension provides an inequality involving q-monotone functions, qin mathbb N. In particular, 1-monotone functions are nondecreasing functions, and 2-monotone functions are convex functions. In general, q-monotone functions on [0, 1], for qge 2, possess a (q-2)nd derivative in (0, 1), which is convex there. We also discuss some other linear positive approximation processes.