Hardy's Inequality

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

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

and

$$F(x)=\int_0^x f(t)\,dt$$ (2)

and take $p>1$. For sums,

$$\sum_{n=1}^\infty \left({A_n\over n}\right)^p<\left({p\over p-1}\right)^p \sum_{n=1}^\infty (a_n)^p$$ (3)

(unless all $a_n=0$), and for integrals,

$$\int_0^\infty \left[{F(x)\over x}\right]^p\,dx < \left({p\over p-1}\right)^p \int_0^\infty [f(x)]^p\,dx$$ (4)

(unless $f$ 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.