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

$\int _{0}^{\infty }\left({\frac {1}{x}}\int _{0}^{x}f(t)\,dt\right)^{p}\,dx\leq \left({\frac {p}{p-1}}\right)^{p}\int _{0}^{\infty }f(x)^{p}\,dx.$

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

Hardy's inequality was first published and proved (at least the discrete version with a worse constant) 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

1. Hardy, G. H. (1920). "Note on a theorem of Hilbert". Mathematische Zeitschrift. 6 (3–4): 314–317. doi:10.1007/BF01199965.

References

• Hardy, G. H.; Littlewood J.E.; Pólya, G. (1952). Inequalities, 2nd ed. Cambridge University Press. ISBN 0-521-35880-9.

• Kufner, Alois; Persson, Lars-Erik (2003). Weighted inequalities of Hardy type. World Scientific Publishing. ISBN 981-238-195-3.

External links

• Hazewinkel, Michiel, ed. (2001), "Hardy inequality", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4