Post Made Community Wiki by Ben Webster
1

# Applications of Hardy's inequality

Every so often I would encounter Hardy's inequality:

Theorem 1 (Hardy's inequality). If $p>1$, $a_n \geq 0$, and $A_n=a_1+a_2+\cdots+a_n$, then $$\sum_{n=1}^\infty \left(\frac{A_n}{n}\right)^p < \left(\frac{p}{p-1}\right)^p \sum_{n=1}^\infty a_n^p,$$ unless $(a_n)_{n=1}^\infty$ is identically zero. The constant is the best possible.

and its integral version:

Theorem 2 (Hardy's integral inequality). If $p>1$, $f(x) \geq 0$, and $F(x) = \int_0^x f(t) \ dt$, then $$\int_0^\infty \left(\frac{F}{x}\right)^p \ dx < \left(\frac{p}{p-1}\right)^p \int_0^\infty f^p(x) \ dx,$$ unless $f \equiv 0$. The constant is the best possible.

with a comment or two emphasizing how important and fundamental they are. Nevertheless, I have yet to see a good application of the above inequalities. So...

Could you give an application of Theorem 1 or Theorem 2 that you think is particularly useful or instructive?