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Zbl 1056.26009
Dragomir, S.S.
Reverses of Schwarz, triangle and Bessel inequalities in inner product spaces.
(English)
[J] JIPAM, J. Inequal. Pure Appl. Math. 5, No. 3, Paper No. 76, 18 p., electronic only (2004). ISSN 1443-5756/e

The author obtains certain reverse inequalities for some classical inequalities in inner product spaces, as Schwarz, triangle, Bessel, etc. inequalities. We quote the following result: Let $H$ be an inner product space, and let $x,a\in H$ such that $\Vert x-a\Vert\leq r$ (where $r>0$). Then, if $\Vert a\Vert> r$, then $0\leq \Vert x\Vert^2 \Vert a\Vert^2- \vert (x,a)\vert^2\leq\Vert x\Vert^2 \Vert a\Vert^2- (\text{Re} (x,a))^2\leq r^2\Vert x\Vert^2$. Similar inequalities are valid, if $\Vert a\Vert=r$, or $\Vert a\Vert< r$. The results are then applied to the case of integral inequalities.
[József Sándor (Cluj-Napoca)]
MSC 2000:
*26D15 Inequalities for sums, series and integrals of real functions
46C05 Geometry and topology of inner product spaces

Keywords: inequalities in inner product spaces; integral inequalities

Cited in: Zbl 1196.15020 Zbl 1060.46016

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