A generalization of Grüss's inequality in inner product spaces and applications
type: Item
instance of: scholarly article
title: A generalization of Grüss's inequality in inner product spaces and applications
zbMATH Open document ID: 0943.46011
author: Silvestru Sever Dragomir
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-12-05T00:00:00Z
review text: The following generalization of the classical Grüss integral inequality is proved: Let \(X\) be a real or complex inner product space and \(e\in X\), \(\|e\|= 1\). If \(\varphi\), \(\gamma\), \(\Phi\), \(\Gamma\) are (real or complex) numbers and \(x,y\in X\) vectors such that \(\text{Re}(\Phi e-x,x- \varphi e)\geq 0\), and \(\text{Re}(\Gamma e- y,y-\gamma e)\geq 0\), then \(|(x,y)- (x,e)(e,y)|\leq{1\over 4}|\Phi- \varphi|\cdot|\Gamma- \gamma|\). The constant \(1/4\) is the best possible. Some applications of this result for positive linear functionals and integrals are given.
Mathematics Subject Classification ID: 46C05
zbMATH DE Number: 1348137
zbMATH Keywords: inner product spaces
zbMATH Keywords: Grüss integral inequality
zbMATH Keywords: positive linear functionals and integrals
reviewed by: Ivan Vidav
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6452
OpenAlex ID: W2052082751
cites work: Q5287501
DOI: 10.1006/JMAA.1999.6452
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description: scientific article; zbMATH DE number 1348137