An Extension of Rasa's Conjecture to q-Monotone Functions

We extend an inequality involving the Bernstein basis polynomials and convex functions on [0, 1]. The inequality was originally conjectured by Rasa about thirty years ago, but was proved only recently. Our extension provides an inequality involving q-monotone functions, q is an element of 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 q >= 2, possess a (q - 2)nd derivative in (0, 1), which is convex there. We also discuss some other linear positive approximation processes.