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.