On the Hilbert Inequality

Gao Mingzhe

Abstract: 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

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