# Muirhead inequality for convex orders and a problem of I. Ra\c{s}a on Bernstein polynomials

## TL;DR

This paper provides a concise proof of Rsa's conjecture involving Bernstein polynomials and convex functions, introduces new stochastic convex order conditions, and extends inequalities to various distributions and convolution polynomials.

## Contribution

It offers a novel, short proof of Rsa's inequality, introduces a useful stochastic convex order criterion, and generalizes Muirhead inequalities for convolution polynomials.

## Key findings

- A new proof of Rsa's inequality for Bernstein polynomials.
- A sufficient condition for stochastic convex order relations.
- Extension of inequalities to other distributions and convolution polynomials.

## Abstract

We present a new, very short proof of a conjecture by I. Ra\c{s}a, which is an inequality involving basic Bernstein polynomials and convex functions. It was affirmed positively very recently by J. Mrowiec, T. Rajba and S. W\k{a}sowicz (2017) by the use of stochastic convex orders, as well as by Abel (2017) who simplified their proof. We give a useful sufficient condition for the verification of some stochastic convex order relations, which in the case of binomial distributions are equivalent to the I. Ra\c{s}a inequality. We give also the corresponding inequalities for other distributions. Our methods allow us to give some extended versions of stochastic convex orderings as well as the I. Ra\c{s}a type inequalities. In particular, we prove the Muirhead type inequality for convex orders for convolution polynomials of probability distributions.