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
:
ight )^p ight )^psum_{n=1}^infty a_n^p.
An integral version of Hardy's inequality states if ''f'' an integrable function with non-negative values, then
:
ight)^poperatorname{ d}xleleft (rac{p}{p-1}
ight )^pint_0^infty f(x)^p, dx.
The 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 than the above.

Contents

See also

Notes

References

See also

★ Carleman's inequality

Notes

1. Hardy, G.H., ''Note on a Theorem of Hilbert'', Math. Z. '6' (1920), 314-317.

References

★ Inequalities, 2nd ed, , G. H., Hardy, Cambridge University Press, ,

★ Weighted inequalities of Hardy type, , Alois, Kufner, World Scientific Publishing, ,