68.9.109.211 at 19:53, 6 August 2012

2012-08-06T19:53:00Z

New page

'''Hardy's inequality''' is an [[inequality (mathematics)|inequality]] in [[mathematics]], named after [[G. H. Hardy]]. It states that if <math>a_1, a_2, a_3, \dots </math> is a [[sequence]] of [[non-negative]] [[real number]]s which is not identically zero, then for every real number ''p'' > 1 one has

:<math>\sum_{n=1}^\infty \left (\frac{a_1+a_2+\cdots +a_n}{n}\right )^p<\left (\frac{p}{p-1}\right )^p\sum_{n=1}^\infty a_n^p.</math>

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

:<math>\int_0^\infty \left (\frac{1}{x}\int_0^x f(t)\, dt\right)^p\, dx\le\left (\frac{p}{p-1}\right )^p\int_0^\infty f(x)^p\, dx.</math>

Equality holds [[if and only if]] ''f''(''x'') = 0 [[almost everywhere]].

Hardy's inequality was first published and proved (at least the discrete version with a worse constant) in 1920 in a note by Hardy.<ref name=Hardy1920>{{Cite doi|10.1007/BF01199965}}</ref> The original formulation was in an integral form slightly different from the above.

==See also==
* [[Carleman's inequality]]

==Notes==
<references />

==References==
*{{cite book
| last = Hardy
| first = G. H.
| coauthors = Littlewood. J.E.; Pólya, G.
| title = Inequalities, 2nd ed
| publisher = Cambridge University Press
| year = 1952
| pages =
| isbn = 0-521-35880-9
}}

*{{cite book
| last = Kufner
| first = Alois
| coauthors = Persson, Lars-Erik
| title = Weighted inequalities of Hardy type
| publisher = World Scientific Publishing
| year = 2003
| pages =
| isbn = 981-238-195-3
}}

==External links==
* {{springer|title=Hardy inequality|id=p/h046340}}

[[Category:Inequalities]]
[[Category:Real analysis]]

Leftover hash lemma - Revision history

68.9.109.211 at 19:53, 6 August 2012