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generalization of Young inequality


It’s straightforward to extend Young inequalityMathworldPlanetmathPlanetmath (http://planetmath.org/YoungInequality) to an arbitrary finite number of : provided that ai>0, ci>0 and ni=11ci=1r,

(ni=1ai)rrni=1aciici

In fact,

(ni=1ai)r = exp[log(ni=1ai)r]
= exp[rni=1logai]
= exp[rni=11cilog(acii)]
= exp[ni=11cilog(acii)1r]
(by Jensen’s inequalityMathworldPlanetmath and monotonicity of exp) exp[log(ni=11ciacii1r)]
=rni=1aciici

Remark: in the case

1ci=1 i

one obtains:

(ni=1ai)1n1nni=1ai

that is, the usual arithmetic-geometric mean inequality, which suggests Young inequality could be regarded as a generalization of this classical result. Actually, let’s consider the following restatement of Young inequality. Having defined: wi=1ci,  ni=1wi=W=1r, xi=a1wii we have:

(ni=1xwii)1W1Wni=1wixi

This expression shows that Young inequality is nothing else than geometric-arithmentic weighted mean (http://planetmath.org/ArithmeticMean) inequality.

Title generalization of Young inequality
Canonical name GeneralizationOfYoungInequality
Date of creation 2013-03-22 15:43:08
Last modified on 2013-03-22 15:43:08
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 25
Author Andrea Ambrosio (7332)
Entry type Result
Classification msc 46E30