Ostrowski-type inequalities in abstract distance spaces

V.F. Babenko; V.V. Babenko; O.V. Kovalenko

doi:10.15421/242416

Ostrowski-type inequalities in abstract distance spaces

V.F. Babenko (Oles Honchar Dnipro National University), https://orcid.org/0000-0001-6677-1914
V.V. Babenko (Drake University), https://orcid.org/0000-0003-4859-4437
O.V. Kovalenko (Oles Honchar Dnipro National University), https://orcid.org/0000-0002-0446-1125

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.

Keywords

Ostrowski-type inequality; distance space; pseudo metric space; Lipschitz function; modulus of continuity

MSC 2020

41A65; 41A17; 41A44

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