Hardy's inequality

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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.

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

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