Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Median Principle for Inequalities and Applications

Sever Silvestru Dragomir

TL;DR

The paper introduces the Median Principle to convert inequalities with bounds in terms of the $L^{\infty}$-norm of derivatives into bounds expressed through the derivative range $M_r-m_r$, thereby sharpening Gr\üss and Ostrowski-type inequalities. It provides a formal mechanism for shifting the function by the midrange of the derivative bounds and, via polynomial perturbations, yields bounds that depend on the variation of derivatives rather than their worst-case magnitude. The authors develop a unified treatment across 0th-, 1st-, and $n$th-degree inequalities, giving explicit, sharp constants and perturbed forms, including Ostrowski and trapezoid-type results and multi-function variants. These results have broad applicability to integral inequalities involving BV functions, absolute continuity, and Peano kernels, offering practical improvements in estimating quadrature errors and related integral bounds.

Abstract

The median principle is applied for different integral inequalities of Gruss and Ostrowski type.

The Median Principle for Inequalities and Applications

TL;DR

The paper introduces the Median Principle to convert inequalities with bounds in terms of the -norm of derivatives into bounds expressed through the derivative range , thereby sharpening Gr\üss and Ostrowski-type inequalities. It provides a formal mechanism for shifting the function by the midrange of the derivative bounds and, via polynomial perturbations, yields bounds that depend on the variation of derivatives rather than their worst-case magnitude. The authors develop a unified treatment across 0th-, 1st-, and th-degree inequalities, giving explicit, sharp constants and perturbed forms, including Ostrowski and trapezoid-type results and multi-function variants. These results have broad applicability to integral inequalities involving BV functions, absolute continuity, and Peano kernels, offering practical improvements in estimating quadrature errors and related integral bounds.

Abstract

The median principle is applied for different integral inequalities of Gruss and Ostrowski type.

Paper Structure

This paper contains 5 sections, 13 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. The Median Principle
  3. Inequalities of the $0^{\text{th}}-$Degree
  4. Inequalities of the $1^{\text{st}}-$Degree
  5. Inequalities of the $n^{\text{th}}-$Degree

Key Result

Theorem 1

Let $f:\left[ a,b\right] \subset \mathbb{R}\rightarrow \mathbb{R}$ be a function so that $f^{\left( n-1\right) }$ is absolutely continuous and $f^{\left( n\right) }\in L_{\infty }\left[ a,b\right] .$ Assume that the following inequality holds where $L\left( \cdot ,\cdot ,\cdots ;a,b\right) :\mathbb{R}^{\left( n+1\right) }\rightarrow \mathbb{R}$ is a general function, $R:[0,\infty )\rightarrow \ma

Theorems & Definitions (25)

  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Remark 2
  • Theorem 3
  • proof
  • ...and 15 more