# 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.
Title Hardy’s inequality HardysInequality 2013-03-22 17:04:32 2013-03-22 17:04:32 Mathprof (13753) Mathprof (13753) 8 Mathprof (13753) Theorem msc 26D15