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

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

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

See also

• Carleman's inequality