On the Hubert Inequality

Abstract

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.