## Hardy's Inequality

Let be a Nonnegative Sequence and a Nonnegative integrable Function. Define

 (1)

and
 (2)

and take . For sums,
 (3)

(unless all ), and for integrals,
 (4)

(unless is identically 0).

References

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.