Hardy inequality

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for series

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


[1] G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934)
[2] S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)
[3] B. Muckenhoupt, "Hardy's inequality with weights" Studia Math. , 44 (1972) pp. 31–38
How to Cite This Entry:
Hardy inequality. L.D. Kudryavtsev (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098