A sharpening of a problem on Bernstein polynomials and convex functions

TL;DR

This paper provides an elementary proof of a 2017 conjecture involving Bernstein polynomials and convex functions, simplifying previous stochastic methods.

Contribution

It offers a new, elementary proof of a Bernstein polynomial inequality conjecture, improving understanding and accessibility.

Findings

Confirmed the conjecture with an elementary proof

Simplified the proof technique compared to stochastic approaches

Enhanced the theoretical foundation of inequalities involving Bernstein polynomials

Abstract

We present an elementary proof of a conjecture proposed by I. Rasa in 2017 which is an inequality involving Bernstein basis polynomials and convex functions. It was affirmed in positive by A. Komisarski and T. Rajba very recently by the use of stochastic convex orderings.