A problem of I. Ra\c{s}a on Bernstein polynomials and convex functions

TL;DR

This paper provides an elementary proof of a conjecture by I. Ra c{s}a involving Bernstein polynomials and convex functions, extending the results to related operators.

Contribution

It offers a new elementary proof of Ra c{s}a's conjecture and extends the inequality to Mirakyan-Favard-Sz\'asz and Baskakov operators.

Findings

Confirmed the inequality for Bernstein polynomials using elementary methods.

Extended the inequality to Mirakyan-Favard-Sz\'asz operators.

Extended the inequality to Baskakov operators.

Abstract

We present an elementary proof of a conjecture by I. Ra\c{s}a which is an inequality involving Bernstein basis polynomials and convex functions. It was affirmed in positive very recently by the use of stochastic convex orderings. Moreover, we derive the corresponding results for Mirakyan-Favard-Sz\'asz operators and Baskakov operators.