New bounds for the companion of Ostrowski's inequality and applications

TL;DR

This paper derives new bounds for the companion of Ostrowski's inequality under various integrability conditions of the derivatives, improving existing estimates and demonstrating applications in numerical integration and probability.

Contribution

It introduces sharper bounds for the companion of Ostrowski's inequality for different derivative spaces, with some bounds proven to be optimal.

Findings

New bounds for the companion of Ostrowski's inequality established.

Some bounds are proven to be sharp and optimal.

Applications to quadrature rules and probability density functions provided.

Abstract

In this paper we establish some new bounds for the companion of Ostrowski's inequality for the case when $f'\in L^1[a,b]$, $f"\in L^2[a,b]$ and $f'\in L^2[a,b]$, respectively. We point out that the results in the first and third cases are sharp and that some of these new estimations can be better than the known results. Some applications to composite quadrature rules, and to probability density functions are also given.