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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)^na_k
(1)

and

F(x)=int_0^xf(t)dt
(2)

and take p>1. For sums,

sum_(n=1)^infty((A_n)/n)^p<(p/(p-1))^psum_(n=1)^infty(a_n)^p
(3)

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

int_0^infty[(F(x))/x]^pdx<(p/(p-1))^pint_0^infty[f(x)]^pdx
(4)

(unless f is identically 0).

See also

Carleman's Inequality

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References

Broadbent, T. A. A. "A Proof of Hardy's Convergence Theorem." J. London Math. Soc. 3, 232-243, 1928.Elliot, E. B. "A Simple Exposition of Some Recently Proved Facts as to Convergency." J. London Math. Soc. 1, 93-96, 1926.Grandjot, K. "On Some Identities Relating to Hardy's Convergence Theorem." J. London Math. Soc. 3, 114-117, 1928.Hardy, G. H. "Note on a Theorem of Hilbert." Math. Z. 6, 314-317, 1920.Hardy, G. H. "Notes on Some Points in the Integral Calculus. LX." Messenger Math. 54, 150-156, 1925.Hardy, G. H.; Littlewood, J. E.; and Pólya, G. "Hardy's Inequality." §9.8 in Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 239-243, 1988.Kaluza, T. and Szegö, G. "Über Reihen mit lauter positiven Gliedern." J. London Math. Soc. 2, 266-272, 1927.Knopp, K. "Über Reihen mit positiven Gliedern." J. London Math. Soc. 3, 205-211, 1928.Landau, E. "A Note on a Theorem Concerning Series of Positive Terms." J. London Math. Soc. 1, 38-39, 1926.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.

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Hardy's Inequality

Cite this as:

Weisstein, Eric W. "Hardy's Inequality." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HardysInequality.html

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