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2019 Extension of Dunkl--Williams inequality and characterizations of inner product spaces
J. Rooin, S. Rajabi, M.S. Moslehian
Rocky Mountain J. Math. 49(8): 2755-2777 (2019). DOI: 10.1216/RMJ-2019-49-8-2755

Abstract

Given $ p, q \in \mathbb {R} $, we generalize the classical Dunkl--Williams inequality for $p$-angular and $q$-angular distances in inner product spaces. We extend the Hile inequality for arbitrary $p$-angular and $q$-angular distances and study some geometric aspects of a generalization of Dunkl--Williams inequality. Introducing power refinements, we show significant power refinements of the generalized Dunkl--Williams inequality under some mild conditions. Among other things, we give new characterizations of inner product spaces with regard to $p$-angular and $q$-angular distances. In particular, we prove that if $ p, q, r \in \mathbb {R} $, $ q \neq 0$ and $ 0\leq p/q\lt 1 $, then $ X $ is an inner product space if and only if for every $ x, y \in X \setminus \{0\} $, $$ \bigl \lVert \lVert x \rVert ^{p-1} x - \lVert y \rVert ^{p-1}y \bigr \rVert \leq \frac {2^{1/r}\bigl \lVert \lVert x \rVert ^{q-1} x - \lVert y \rVert ^{q-1}y \bigr \rVert }{\bigl [\lVert x \rVert ^{r(q-p)} + \lVert y \rVert ^{r(q-p)}\bigr ]^{\frac {1}{r}}}. $$

Citation

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J. Rooin. S. Rajabi. M.S. Moslehian. "Extension of Dunkl--Williams inequality and characterizations of inner product spaces." Rocky Mountain J. Math. 49 (8) 2755 - 2777, 2019. https://doi.org/10.1216/RMJ-2019-49-8-2755

Information

Published: 2019
First available in Project Euclid: 31 January 2020

zbMATH: 07163197
MathSciNet: MR4058348
Digital Object Identifier: 10.1216/RMJ-2019-49-8-2755

Subjects:
Primary: 47A30
Secondary: 26D15., 46C15

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

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Vol.49 • No. 8 • 2019
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