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Hilbert's Inequality

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

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

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

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

\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}

unless all ${a_n}$ or all $b_n$ are 0. If $f(x)$ and $g(x)$ are Nonnegative integrable functions, then the integral form is

\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}.

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


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

© 1996-9 Eric W. Weisstein