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Some Grüss type inequalities in inner product spaces. (English) Zbl 1063.26012
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 $$ \vert \langle x,y\rangle -\langle x,e\rangle \langle e,y\rangle \vert \leq \frac{1}{4}\vert \Phi -\varphi \vert \cdot \vert \Gamma -\gamma \vert -[ \text{Re}\langle \Phi e-x,x-\varphi e\rangle ] ^{1/2}[ \text{Re}\langle \Gamma e-y,y-\gamma e\rangle ] ^{1/2}. $$

26D15Inequalities for sums, series and integrals of real functions
46C05Hilbert and pre-Hilbert spaces: geometry and topology
Full Text: EuDML