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Zbl 1063.26012
Dragomir, S.S.
Some Grüss type inequalities in inner product spaces.
(English)
[J] JIPAM, J. Inequal. Pure Appl. Math. 4, No. 2, Paper No. 42, 10 p., electronic only (2003). ISSN 1443-5756/e

The follwing refinement of the Grüss inequality is proved. Let $( H,\langle .,.\rangle )$ be a real or complex inner product space and $e\in H,\;\Vert e\Vert =1.$ If $\varphi ,\gamma ,\Phi ,\Gamma$ are real or complex numbers and $x,y$ are vectors in $H$ such that $$\text{Re}\langle \Phi e-x,x-\varphi e\rangle \geq 0\text{ and } \text{Re}\langle \Gamma e-y,y-\gamma e\rangle \geq 0$$ hold, then we have the inequality $$\vert \langle x,y\rangle -\langle x,e\rangle \langle e,y\rangle \vert \leq \frac{1}{4}\vert \Phi -\varphi \vert \cdot \vert \Gamma -\gamma \vert -[ \text{Re}\langle \Phi e-x,x-\varphi e\rangle ] ^{1/2}[ \text{Re}\langle \Gamma e-y,y-\gamma e\rangle ] ^{1/2}.$$
MSC 2000:
*26D15 Inequalities for sums, series and integrals of real functions
46C05 Geometry and topology of inner product spaces

Keywords: Grüss inequality; inner product spaces; integral inequalities; discrete inequalities

Cited in: Zbl 1221.26031

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