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 an integrable function with non-negative values, then

$\int_0^\infty \left (\frac{1}{x}\int_0^x f(t)\, dt\right)^p\operatorname{ d}x\le\left (\frac{p}{p-1}\right )^p\int_0^\infty f(x)^p\, dx.$

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

Hardy's inequality was first published (without proof) in 1920 in a note by Hardy^[1]. The original formulation was in an integral form slightly different from the above.