A solution to the problem of Rasa connected with Bernstein polynomials

TL;DR

This paper provides a complete positive solution to a longstanding inequality problem involving Bernstein polynomials, utilizing stochastic orderings and concentration inequalities, and extends the problem's scope.

Contribution

It introduces new methods using stochastic orderings and concentration inequalities to solve and extend a classical inequality problem related to Bernstein polynomials.

Findings

Solved the inequality problem in positive form

Proved stochastic orderings for binomial distributions

Extended the original problem using new methods

Abstract

During the Conference on Ulam's Type Stability (Rytro, Poland, 2014), Ioan Rasa recalled his 25-years-old problem concerning some inequality involving the Bernstein polynomials. We offer the complete solution (in positive). As a~tool we use stochastic orderings (which we prove for binomial distributions) as well as so-called concentration inequality. Our methods allow us to pose (and solve) the extended version of the problem in question.