generalized Hölder inequality

Theorem Let 1r< and 1pj<, where j=1n1pj=1r. If fjLpj for 1jn, then

j=1nfjLr and


The usual Hölder inequalityMathworldPlanetmath has n=2 and r=1.

Let X be a finite set, say X={x1,,xm} and μ is the counting measure on X, so that μ({xi})=1 for all i. Let fj(xi)=aij0 for j=1,,n and take r=1. Then the inequality becomes:



Now let αj=1pj, and bij=aijpj. Then the inequality becomes:

Title generalized Hölder inequality
Canonical name GeneralizedHolderInequality
Date of creation 2013-03-22 16:54:35
Last modified on 2013-03-22 16:54:35
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 8
Author Mathprof (13753)
Entry type Theorem
Classification msc 46E30