Ostrowski-type inequalities in abstract distance spaces

TL;DR

This paper develops generalized Ostrowski-type inequalities within abstract distance spaces, providing sharp bounds for deviations of Lipschitz mappings and unifying many existing estimates in a broad theoretical framework.

Contribution

It introduces a new framework for Ostrowski-type inequalities in abstract distance spaces with partially ordered values, unifying and extending existing estimates.

Findings

Derived sharp inequalities for Lipschitz mappings in abstract distance spaces.

Unified various known estimates under a general theoretical framework.

Extended Ostrowski inequalities to spaces with partially ordered metric values.

Abstract

For non-empty sets X we define notions of distance and pseudo metric with values in a partially ordered set that has a smallest element $\theta $. If $h_X$ is a distance in $X$ (respectively, a pseudo metric in $X$), then the pair $(X,h_X)$ is called a distance (respectively, a pseudo metric) space. If $(T,h_T)$ and $(X,h_X)$ are pseudo metric spaces, $(Y,h_Y)$ is a distance space, and $H(T,X)$ is a class of Lipschitz mappings $f\colon T\to X$, for a broad family of mappings $\Lambda\colon H (T,X)\to Y$, we obtain a sharp inequality that estimates the deviation $h_Y(\Lambda f(\cdot),\Lambda f(t))$ in terms of the function $h_T(\cdot, t)$. We also show that many known estimates of such kind are contained in our general result.