Hardy's inequality

Hardy's inequality is an inequality in mathematics, named after G. H. Hardy. It states that if \( a_1, a_2, a_3, \dots \) is a sequence of non-negative real numbers which is not identically zero, then for every real number p > 1 one has

\( \sum_{n=1}^\infty \left (\frac{a_1+a_2+\cdots +a_n}{n}\right )^p<\left (\frac{p}{p-1}\right )^p\sum_{n=1}^\infty a_n^p. \)

An integral version of Hardy's inequality states if f is an integrable function with non-negative values, then

\( \int_0^\infty \left (\frac{1}{x}\int_0^x f(t)\, dt\right)^p\, dx\le\left (\frac{p}{p-1}\right )^p\int_0^\infty f(x)^p\, dx. \)

Equality holds if and only if f(x) = 0 almost everywhere.

Hardy's inequality was first published (without proof) in 1920 in a note by Hardy.[1] The original formulation was in an integral form slightly different from the above.
See also

Carleman's inequality

Notes

^ Hardy, G.H., Note on a Theorem of Hilbert, Math. Z. 6 (1920), 314–317.

References

Hardy, G. H.; Littlewood. J.E.; Pólya, G. (1952). Inequalities, 2nd ed. Cambridge University Press. ISBN 0521358809.

Kufner, Alois; Persson, Lars-Erik (2003). Weighted inequalities of Hardy type. World Scientific Publishing. ISBN 9812381953.

Ribarič, M. (1973), "On some inequalities for convex functions", Mathematica Balkanica 3: 435–442.

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