Hardy's inequality

Hardy's inequality is an inequality in mathematics, named after G. H. Hardy. It states that if ${\displaystyle a_{1},a_{2},a_{3},\dots }$ is a sequence of non-negative real numbers, then for every real number p > 1 one has

${\displaystyle \sum _{n=1}^{\infty }\left({\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}\right)^{p}\leq \left({\frac {p}{p-1}}\right)^{p}\sum _{n=1}^{\infty }a_{n}^{p}.}$

If the right-hand side is finite, equality holds if and only if ${\displaystyle a_{n}=0}$ for all n.

An integral version of Hardy's inequality states the following: if f is a measurable function with non-negative values, then

${\displaystyle \int _{0}^{\infty }\left({\frac {1}{x}}\int _{0}^{x}f(t)\,dt\right)^{p}\,dx\leq \left({\frac {p}{p-1}}\right)^{p}\int _{0}^{\infty }f(x)^{p}\,dx.}$

If the right-hand side is finite, 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.