An improved Popoviciu-type inequality for a new Bernstein-type operator

TL;DR

This paper improves a Popoviciu-type inequality for a new Bernstein-type operator derived from Pólya's urn model, demonstrating a smaller constant than the classical Bernstein operator, with a new factorial inequality as a key tool.

Contribution

It introduces a refined Popoviciu inequality for a novel Bernstein-type operator, utilizing a new factorial inequality to establish a smaller constant than the classical case.

Findings

The new operator satisfies a Popoviciu inequality with a smaller constant.

A new inequality for the rising factorial is proved and applied.

The constant in the inequality is proven to be smaller than that of the classical Bernstein operator.

Abstract

Recently we introduced a new Bernstein-type operator using P\'olya's urn model with negative replacement, and we showed that it satisfies a Popoviciu-type inequality with a constant slightly larger than that of the corresponding inequality for the classical Bernstein operator. In the present paper we prove an inequality for the rising factorial (of independent interest), and we use it in order to show that the constant in the Popoviciu inequality for the new operator is in fact smaller than the corresponding constant for the Bernstein operator.