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Hardy's Inequality

Let $\{a_n\}$ be a Nonnegative Sequence and $f(x)$ a Nonnegative integrable Function. Define

A_n=\sum_{k=1}^n a_k
\end{displaymath} (1)

F(x)=\int_0^x f(t)\,dt
\end{displaymath} (2)

and take $p>1$. For sums,
\sum_{n=1}^\infty \left({A_n\over n}\right)^p<\left({p\over p-1}\right)^p \sum_{n=1}^\infty (a_n)^p
\end{displaymath} (3)

(unless all $a_n=0$), and for integrals,
\int_0^\infty \left[{F(x)\over x}\right]^p\,dx < \left({p\over p-1}\right)^p \int_0^\infty [f(x)]^p\,dx
\end{displaymath} (4)

(unless $f$ is identically 0).


Hardy, G. H.; Littlewood, J. E.; and Pólya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 239-243, 1988.

Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities Involving Functions and Their Integrals and Derivatives. New York: Kluwer, 1991.

Opic, B. and Kufner, A. Hardy-Type Inequalities. Essex, England: Longman, 1990.

© 1996-9 Eric W. Weisstein