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A characterization of a nonlinear integral triangle inequality

Ahmed A. Abdelhakim

Abstract

Let $(E,\|.\|)$ be a Banach space and let $(Ω,μ)$ be a Lebesgue measure space. We characterize, for all $p>0$, measurable functions $u:Ω\rightarrow \mathbb{R}$ for which \begin{equation*} \left\| \int_Ω f\,dμ\right\|^{p}\,\leq\,\int_Ω u \| f \|^{p}\,dμ.\tag{I} \end{equation*} We characterize $u$ for the reverse of (I) as well. The discrete counterpart of this problem is also solved.

A characterization of a nonlinear integral triangle inequality

Abstract

Let be a Banach space and let be a Lebesgue measure space. We characterize, for all , measurable functions for which \begin{equation*} \left\| \int_Ω f\,dμ\right\|^{p}\,\leq\,\int_Ω u \| f \|^{p}\,dμ.\tag{I} \end{equation*} We characterize for the reverse of (I) as well. The discrete counterpart of this problem is also solved.

Paper Structure

This paper contains 3 sections, 7 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. The integral inequality
  3. The discrete inequality

Key Result

Theorem 1.1

Let $(X,\|.\|)$ be a normed space, $x_{i}\in X,$$i=1,...,n$ and $\lambda:=(\lambda_{1},...,\lambda_{n})\in \mathbb{R}^{n}$. Then $\,\|\sum_{i=1}^{n} x_{i}\|^{p} \leq \sum_{i=1}^{n}{{\lambda_{i}} \| x_{i}\|^{p}}\,$ if and only if While the reverse $\,\|\sum_{i=1}^{n} x_{i}\|^{p} \geq \sum_{i=1}^{n}{{\lambda_{i}}\| x_{i}\|^{p}}\,$ holds if and only if $\lambda_{i}<0$ or

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 3.1
  • ...and 3 more