Hilbert's Inequality

Given a Positive Sequence $\{a_n\}$ ,

$\begin{displaymath} \sqrt{\sum_{j=-\infty}^\infty \,\left\vert{\sum_{\scriptstyl... ...rt^2} \leq \pi\sqrt{\sum_{n=-\infty}^\infty \vert a_n\vert^2}, \end{displaymath}$

where the ${a_n}$ s are Real and ``square summable.''

Another Inequality known as Hilbert's applies to Nonnegative sequences $\{a_n\}$ and $\{b_n\}$ ,

$\begin{displaymath} \sum_{m=1}^\infty \sum_{n=1}^\infty {a_mb_n\over m+n}<\pi\cs... ...^p}\right)^{1/p}\left({\sum_{n=1}^\infty {b_n}^q}\right)^{1/q} \end{displaymath}$

unless all ${a_n}$ or all

are 0. If

and

are Nonnegative integrable functions, then the integral form is

$\begin{displaymath} \int_0^\infty \int_0^\infty {f(x)g(y)\over x+y}\,dx\,dy < \p... ...}\right)^{1/p}\left({\int_0^\infty [g(x)]^q\,dx}\right)^{1/q}. \end{displaymath}$

The constant $\pi\csc(\pi/P)$ is the best possible, in the sense that counterexamples can be constructed for any smaller value.

References

Hardy, G. H.; Littlewood, J. E.; and Pólya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 308-309, 1988.