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Zeitschrift für Analysis und ihre Anwendungen

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Volume 18, Issue 4, 1999, pp. 1117–1122
DOI: 10.4171/ZAA/932

Published online: 1999-12-31

On the Hubert Inequality

Gao Mingzhe[1]

(1) Xiangxi Education College, Hunan, China

It is shown that the Hilbert inequality for double series can be improved by introducing the positive real number $\frac{1}{\pi^2} (\frac{s^2(b)}{\|a\|^2} + \frac{s^2(b)}{\|b\|^2})$ where $s(x) = \sum^{\infty}_{n=1} \frac{x_n}{n}$ and $\|x\|^2 = \sum^{\infty}_{n=1} x^2_n (x = a, b)$. The coefficient $\pi$ of the classical Hilbert inequality is proved not to be the best possible if $\|a\|$ or $\|b\|$ is finite. A similar result for the Hubert integral inequality is also proved.

Keywords: Hubert inequality, binary quadratic form, exponential integral, inner product

Mingzhe Gao: On the Hubert Inequality . Z. Anal. Anwend. 18 (1999), 1117-1122. doi: 10.4171/ZAA/932