Hardy's inequality

Hardy's inequality is an inequality in mathematics, named after G. H. Hardy. It states that if $a\_1,\; a\_2,\; a\_3,\; dots$ is a sequence of non-negative real numbers which is not identically zero, then for every real number "p" > 1 one has

:

An integral version of Hardy's inequality states if "f" an integrable function with non-negative values, then

:$int\_0^infty\; left\; (frac\{1\}\{x\}int\_0^x\; f(t),\; dt\; ight)^poperatorname\{\; d\}xleleft\; (frac\{p\}\{p-1\}\; ight\; )^pint\_0^infty\; f(x)^p,\; dx.$

Equality holds if and only if "f"("x") = 0 almost everywhere.

Hardy's inequality was first published (without proof) in 1920 in a note by Hardy [Hardy, G.H., "Note on a Theorem of Hilbert", Math. Z. 6 (1920), 314–317.] . The original formulation was in an integral form slightly different from the above.

ee also

* Carleman's inequality

Notes

References

*cite book
last = Hardy
first = G. H.
coauthors = Littlewood. J.E.; Pólya, G.
title = Inequalities, 2nd ed
publisher = Cambridge University Press
date = 1952
pages =
isbn = 0521358809

*cite book
last = Kufner
first = Alois
coauthors = Persson, Lars-Erik
title = Weighted inequalities of Hardy type
publisher = World Scientific Publishing
date = 2003
pages =
isbn = 9812381953