Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Some Companions of Ostrowski's Inequality for Absolutely Continuous Functions and Applications

Sever Silvestru Dragomir

TL;DR

The paper develops companions to Ostrowski's inequality for absolutely continuous functions by first establishing a precise integral identity for the symmetry-based deviation and then deriving sharp, norm-dependent bounds when $f'\in L_p$ (including $L_\infty$). It then introduces a composite two-point quadrature rule based on $f$ evaluated at symmetric off-midpoint locations, with rigorous remainder estimates in terms of $f'$ in various Lebesgue spaces. Applications to PDFs are given, translating these bounds into expressions for the CDF and the expectation of a random variable on $[a,b]$, and providing concrete, sharp constants for practical use. Overall, the work extends Ostrowski-type results to broader functional classes and demonstrates concrete quadrature and probabilistic applications with explicit error control.

Abstract

Companions of Ostrowski's integral ineqaulity for absolutely continuous functions and applications for composite quadrature rules and for p.d.f.'s are provided.

Some Companions of Ostrowski's Inequality for Absolutely Continuous Functions and Applications

TL;DR

The paper develops companions to Ostrowski's inequality for absolutely continuous functions by first establishing a precise integral identity for the symmetry-based deviation and then deriving sharp, norm-dependent bounds when (including ). It then introduces a composite two-point quadrature rule based on evaluated at symmetric off-midpoint locations, with rigorous remainder estimates in terms of in various Lebesgue spaces. Applications to PDFs are given, translating these bounds into expressions for the CDF and the expectation of a random variable on , and providing concrete, sharp constants for practical use. Overall, the work extends Ostrowski-type results to broader functional classes and demonstrates concrete quadrature and probabilistic applications with explicit error control.

Abstract

Companions of Ostrowski's integral ineqaulity for absolutely continuous functions and applications for composite quadrature rules and for p.d.f.'s are provided.

Paper Structure

This paper contains 4 sections, 12 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Some Integral Inequalities
  3. A Composite Quadrature Formula
  4. Applications for P.D.F.'s

Key Result

Theorem 1

Let $f:\left[ a,b\right] \rightarrow \mathbb{R}$ be such that with $k\in (0,1],$ i.e., $f\in Lip_{M}\left( k\right) .$ Then, for each $x\in \left[ a,\frac{a+b}{2}\right] ,$ we have the inequality This inequality is sharp for each admissable $x.$ Equality is obtained if and only if $f=\pm Mf_{\ast }+c$ with $c\in \mathbb{R}$ and

Theorems & Definitions (24)

  • Theorem 1
  • Lemma 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Remark 2
  • Corollary 1
  • Theorem 3
  • proof
  • ...and 14 more