Hardy's inequality


Hardy's inequality is an inequality in mathematics, named after G. H. Hardy. It states that if is a sequence of non-negative real numbers, then for every real number p > 1 one has
If the right-hand side is finite, equality holds if and only if for all n.
An integral version of Hardy's inequality states the following: if f is a measurable function with non-negative values, then
If the right-hand side is finite, equality holds if and only if f = 0 almost everywhere.
Hardy's inequality was first published and proved in 1920 in a note by Hardy. The original formulation was in an integral form slightly different from the above.

Multidimensional version

In the multidimensional case, Hardy's inequality can be extended to -spaces, taking the form
where, and where the constant is known to be sharp.

Proof of the inequality

,
which is less or equal than by Minkowski's integral inequality. Finally, by another change of variables, the last expression equals
.

and, for, there holds

and thus
.