Hardy’s inequality

Suppose p>1 and {an} is a sequence of nonnegative real numbers. Let An=i=1nai. Then


unless all the an are zero. The constant is best possible.

This theorem has an integral analogue: Suppose that p>1 and f0 on (0,). Let F(x)=0xf(t)𝑑t. Then


unless f0. The constant is best possible.


  • 1 G.H. Hardy, J.E. Littlewood and G.Pólya, InequalitiesMathworldPlanetmath, Cambridge University Press, Cambridge, 2nd edition, 1952, pp. 239-240.
Title Hardy’s inequality
Canonical name HardysInequality
Date of creation 2013-03-22 17:04:32
Last modified on 2013-03-22 17:04:32
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 8
Author Mathprof (13753)
Entry type Theorem
Classification msc 26D15