A differential version of the Chebyshev-Markov-Stieltjes inequalities

TL;DR

This paper introduces a novel differential form of the Chebyshev-Markov-Stieltjes inequalities applicable to a wide class of weight functions, providing effective estimates under certain regularity conditions.

Contribution

It presents a new differential version of the classical inequalities, extending their applicability to broader weight functions with regularity assumptions.

Findings

Differential inequalities hold for broad weight functions.

Effective estimates are obtained under regularity conditions.

The differential version appears to be a novel contribution.

Abstract

We show that a differential version of the classical Chebyshev-Markov-Stieltjes inequalities holds for a broad family of weight functions. Such a differential version appears to be new. Our results apply to weight functions which are bounded away from zero and piecewise absolutely continuous and yield effective estimates when the weight satisfies additional regularity conditions.