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Hilbert's Inequality

Given a positive sequence {a_n},

sqrt(sum_(j=-infty)^infty|sum_(n=-infty; n!=j)^infty(a_n)/(j-n)|^2)<=pisqrt(sum_(n=-infty)^infty|a_n|^2),
(1)

where the a_ns are real and "square summable."

Another inequality known as Hilbert's applies to nonnegative sequences {a_n} and {b_n},

sum_(m=1)^inftysum_(n=1)^infty(a_mb_n)/(m+n)<picsc(pi/p)(sum_(m=1)^inftya_m^p)^(1/p)(sum_(n=1)^inftyb_n^q)^(1/q)
(2)

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^inftyint_0^infty(f(x)g(y))/(x+y)dxdy<picsc(pi/p) ×(int_0^infty[f(x)]^pdx)^(1/p)(int_0^infty[g(x)]^qdx)^(1/q).
(3)

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

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References

Hardy, G. H.; Littlewood, J. E.; and Pólya, G. "Hilbert's Double Series Theorem" and "On Hilbert's Inequality." §9.1 and Appendix III in Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 226-227 and 308-309, 1988.

Referenced on Wolfram|Alpha

Hilbert's Inequality

Cite this as:

Weisstein, Eric W. "Hilbert's Inequality." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HilbertsInequality.html

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