×

A generalization of Grüss’s inequality in inner product spaces and applications. (English) Zbl 0943.46011

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.

MSC:

46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Mitrinović, D. S.; Pečarić, J. E.; Fink, A. M., Classical and New Inequalities in Analysis (1993), Kluwer Academic: Kluwer Academic Dordrecht, Norwell · Zbl 0771.26009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.