# Hardy inequality

From Encyclopedia of Mathematics

*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:

and

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

(1) |

(2) |

If and , inequality (1) holds if and only if

and (2) holds if and only if

#### References

[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: http://www.encyclopediaofmath.org/index.php?title=Hardy_inequality&oldid=13888

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098