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New Inequalities of Gruss Type for the Stieltjes Integral and Applications

Sever Silvestru Dragomir

TL;DR

This work extends Grüss-type inequalities from the weighted Čebyšev framework to Stieltjes integrals by introducing the generalized Čebyšev functional $T(f,g;u)$ for $f,g$ continuous and $u\in BV[a,b]$ with $u(b)\neq u(a)$. It proves sharp, $\tfrac{1}{2}$-constant bounds under various conditions on $f$ and $u$ (including bounded, monotone, Lipschitz, and Hölder cases) and develops a quadrature rule $S_n(f,g;u,I_n)$ with a sharp remainder bound for approximating $\int_a^b f(t)g(t)\,du(t)$. The paper then specializes these results to weighted functionals $T_w$ and to $D(f;u)$, providing identities, positivity criteria, and numerous corollaries, thereby broadening the applicability to numerical integration and analysis of Stieltjes-type integrals. Overall, the results yield tight error estimates and practical quadrature schemes for Stieltjes integrals with diverse integrators, impacting numerical analysis and approximation theory in this domain.

Abstract

Sharp inequalitieis of Gruss type for Stieltjes integrals with application in numerical integration are provided.

New Inequalities of Gruss Type for the Stieltjes Integral and Applications

TL;DR

This work extends Grüss-type inequalities from the weighted Čebyšev framework to Stieltjes integrals by introducing the generalized Čebyšev functional for continuous and with . It proves sharp, -constant bounds under various conditions on and (including bounded, monotone, Lipschitz, and Hölder cases) and develops a quadrature rule with a sharp remainder bound for approximating . The paper then specializes these results to weighted functionals and to , providing identities, positivity criteria, and numerous corollaries, thereby broadening the applicability to numerical integration and analysis of Stieltjes-type integrals. Overall, the results yield tight error estimates and practical quadrature schemes for Stieltjes integrals with diverse integrators, impacting numerical analysis and approximation theory in this domain.

Abstract

Sharp inequalitieis of Gruss type for Stieltjes integrals with application in numerical integration are provided.

Paper Structure

This paper contains 6 sections, 22 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Some Inequalities by Generalised Čebyšev Functional
  3. A Quadrature Formula
  4. Some Particular Cases
  5. Other Inequalities for Stieltjes Integral
  6. Some Results for Monotonic Integrators

Key Result

Theorem 1

Let $f,g:\left[ a,b\right] \rightarrow \mathbb{R}$ be continuous on $\left[ a,b\right]$ and $u:\left[ a,b\right] \rightarrow \mathbb{R}$ with $u\left( a\right) \neq u\left( b\right) .$ Assume also that there exists the real constants $m,M$ such that If $u$ is of bounded variation on $\left[ a,b\right] ,$ then we have the inequality where $\bigvee_{a}^{b}\left( u\right)$ denotes the total variati

Theorems & Definitions (44)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Corollary 1
  • proof
  • ...and 34 more