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Zbl 0943.46011
Dragomir, Sever Silvestru
A generalization of Grüss's inequality in inner product spaces and applications.
(English)
[J] J. Math. Anal. Appl. 237, No.1, 74-82 (1999). ISSN 0022-247X

The following generalization of the classical Grüss integral inequality is proved:\par 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 $\text{Re}(\Phi e-x,x- \varphi e)\ge 0$, and $\text{Re}(\Gamma e- y,y-\gamma e)\ge 0$, then $|(x,y)- (x,e)(e,y)|\le{1\over 4}|\Phi- \varphi|\cdot|\Gamma- \gamma|$. The constant $1/4$ is the best possible.\par Some applications of this result for positive linear functionals and integrals are given.
[I.Vidav (Ljubljana)]
MSC 2000:
*46C05 Geometry and topology of inner product spaces

Keywords: inner product spaces; Grüss integral inequality; positive linear functionals and integrals

Cited in: Zbl 1065.26022

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