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 , … is a sequence of non-negative real numbers which is not identically zero, then for every real number p > 1 one has

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

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.