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On a Hardy operator inequality

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Abstract

The famous Hardy inequality asserts that if f is a non-negative p-integrable \((p>1)\) function on \((0,\infty )\), then

$$\begin{aligned} \int _{0}^{\infty }\left( \frac{1}{x}\int _{0}^{x}f(t)dt\right) ^pdx\le \left( \frac{p}{p-1}\right) ^p\int _{0}^{\infty }f(x)^pdx. \end{aligned}$$

We present an external form of the Hardy inequality for Hilbert space operators. Moreover, utilizing the operator log-convex functions, a refinement of the operator Hardy inequality is also given.

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Acknowledgements

The author would like to thank the referee for useful comments. This research was in part supported by a grant from University of Bojnord (No. 95/368/12483).

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  1. Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, P.O. Box 1339, Bojnord, 94531, Iran

    Mohsen Kian

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  1. Mohsen Kian
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Correspondence to Mohsen Kian.

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Kian, M. On a Hardy operator inequality. Positivity 22, 773–781 (2018). https://doi.org/10.1007/s11117-017-0543-4

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  • DOI: https://doi.org/10.1007/s11117-017-0543-4

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