Hardy's inequality

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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

\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.

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

\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.

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.[1] The original formulation was in an integral form slightly different from the above.

See also[edit]


  1. ^ Hardy, G. H. (1920). "Note on a theorem of Hilbert". Mathematische Zeitschrift 6 (3–4): 314–317. doi:10.1007/BF01199965. 


  • Hardy, G. H.; Littlewood J.E.; Pólya, G. (1952). Inequalities, 2nd ed. Cambridge University Press. ISBN 0-521-35880-9. 
  • Kufner, Alois; Persson, Lars-Erik (2003). Weighted inequalities of Hardy type. World Scientific Publishing. ISBN 981-238-195-3. 

External links[edit]