On the Hubert Inequality

Abstract

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