# On Pompeiu-Chebyshev functional and its generalization

## TL;DR

This paper generalizes the Chebyshev functional, establishes new inequalities using Pompeiu's mean value theorem, and applies these results to bounds in inequalities like CBS and Hardy type inequalities.

## Contribution

It introduces a broad generalization of the Chebyshev functional, derives new inequalities, and extends classical results with applications to reverse CBS and Hardy inequalities.

## Key findings

- New inequalities of Gruss type established
- Bounds for reverse CBS inequality derived
- Hardy type inequalities on [a,b] introduced

## Abstract

In this work, a generalization of Chebyshev functional is presented. New inequalities of Gruss type via Pompeiu's mean value theorem are established. Improvements of some old inequalities are proved. A generalization of pre-Gruss inequality is elaborated. Some remarks to further generalization of Chebyshev functional are presented. As applications, bounds for the reverse of CBS inequality are deduced. Hardy type inequalities on bounded real interval [a,b] under some other circumstances are introduced. Other related ramified inequalities for differentiable functions are also given.