Zeitschrift für Analysis und ihre Anwendungen Vol. 18, No. 4, pp. 11171122 (1999) 

On the Hilbert InequalityGao MingzheXiangxi Educ. Coll., Dept. Math., Jishou, Hunan 416000, P.R. ChinaAbstract: It is shown that the Hilbert inequality for double series can be improved by introducing the positive real number ${1 \over \pi^2}\big({s^2(a) \over \a\^2} + {s^2(b) \over \b\^2}\big)$ where $s(x) = \sum_{n=1}^\infty {x_n \over n}$ and $\x\^2 = \sum_{n=1}^\infty x_n^2 \ (x = a, b)$. 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 Hilbert integral inequality is also proved. Keywords: hilbert inequality, binary quadratic form, exponential integral, inner product Classification (MSC2000): 26D, 46C Full text of the article:
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