On two inequalities of \v{C}eby\v{s}ev

TL;DR

This paper derives sharp bounds for the ebyv functional involving various classes of functions, including convex, Lipschitz, and functions in $L_p$-spaces, expanding the theoretical understanding of these inequalities.

Contribution

It introduces new sharp bounds for the ebyv functional for multiple types of functions, including convex, Lipschitz, and those with bounded variation.

Findings

Sharp bounds for ebyv functional with convex functions

Bounds involving functions in $L_p$-spaces

Results for convex and concave function pairs

Abstract

In this work, several sharp bounds for the \v{C}eby\v{s}ev functional involving various type of functions are proved. In particular, for the \v{C}eby\v{s}ev functional of two absolutely continuous functions whose first derivatives are both convex, convex and belong to $L_p$-spaces, convex and bounded variation, convex and Lipschitz mappings new sharp bounds are presented. Other related results regarding two convex and concave functions are given.