The best possibility of the bound for the Kantorovich inequality and some remarks
type: Item
instance of: scholarly article
title: The best possibility of the bound for the Kantorovich inequality and some remarks
zbMATH Open document ID: 0934.26010
DOI: 10.1155/S1025583497000222
author: J. Ning
published in: Journal of Inequalities and Applications
publication date: 2000-04-10T00:00:00Z
full work available at URL: https://eudml.org/doc/120188
review text: Let \((\Omega,{\mathcal F},p)\) be a probability measure space. For any complex-valued random variable \(X\) with \(0< m\leq|X|\leq M\), a.s., the following inequality holds \[ \Biggl|\int X dp \int {1\over X} dp\Biggr|\leq {(m+ M)^2\over 4mM}. \] The equality is attained iff there exists \(0\leq\theta\leq 2\pi\) such that \[ p(\{X= me^{i\theta}\})= p(\{X= Me^{i\theta}\})= \begin{cases} {1\over 2}, &\text{if }m<M,\\ 1, & \text{if }m= M.\end{cases} \] {}.
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 47B15
zbMATH DE Number: 1086032
zbMATH Keywords: Kantorovich inequality
zbMATH Keywords: Schwarz inequality
zbMATH Keywords: conditional expectation
zbMATH Keywords: normal operator
reviewed by: J. Pečarić
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1086032