Hardy’s inequality

Suppose $p>1$ and $\{a_{n}\}$ is a sequence of nonnegative real numbers. Let $A_{n}=\sum_{i=1}^{n}a_{i}$ . Then

\sum_{n\geq 1}\left(\frac{A_{n}}{n}\right)^{p}<\left(\frac{p}{p-1}\right)^{p}% \sum_{n\geq 1}{a_{n}}^{p},

unless all the $a_{n}$ are zero. The constant is best possible.

This theorem has an integral analogue: Suppose that $p>1$ and $f\geq 0$ on $(0,\infty)$ . Let $F(x)=\int_{0}^{x}f(t)dt$ . 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 best possible.

References

1 G.H. Hardy, J.E. Littlewood and G.Pólya, Inequalities, Cambridge University Press, Cambridge, 2nd edition, 1952, pp. 239-240.