generalization of Young inequality

It’s straightforward to extend Young inequalityMathworldPlanetmathPlanetmath ( to an arbitrary finite number of : provided that ai>0, ci>0 and i=1n1ci=1r,


In fact,

(i=1nai)r = exp[log(i=1nai)r]
= exp[ri=1nlogai]
= exp[ri=1n1cilog(aici)]
= exp[i=1n1cilog(aici)1r]
(by Jensen’s inequalityMathworldPlanetmath and monotonicity of exp) exp[log(i=1n1ciaici1r)]

Remark: in the case

1ci=1 i

one obtains:


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,  i=1nwi=W=1r, xi=ai1wi we have:


This expression shows that Young inequality is nothing else than geometric-arithmentic weighted mean ( 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