If , and , then
except when all the are zero. The constant in this inequality is best possible.
The Hardy inequalities for integrals are:
The inequalities are valid for all functions for which the right-hand sides are finite, except when vanishes almost-everywhere on . (In this case the inequalities turn into equalities.) The constants and are best possible.
The integral Hardy inequalities can be generalized to arbitrary intervals:
where , , and where the 's are certain constants.
Generalized Hardy inequalities are inequalities of the form
If and , inequality (1) holds if and only if
and (2) holds if and only if
|||G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934)|
|||S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)|
|||B. Muckenhoupt, "Hardy's inequality with weights" Studia Math. , 44 (1972) pp. 31–38|
Hardy inequality. L.D. Kudryavtsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hardy_inequality&oldid=13888