generalization of Young inequality
It’s straightforward to extend Young inequality (http://planetmath.org/YoungInequality) to an arbitrary finite number of : provided that ai>0, ci>0 and ∑ni=11ci=1r,
(n∏i=1ai)r≤rn∑i=1aciici |
In fact,
(n∏i=1ai)r | = | exp[log(n∏i=1ai)r] | ||
= | exp[rn∑i=1logai] | |||
= | exp[rn∑i=11cilog(acii)] | |||
= | exp[∑ni=11cilog(acii)1r] | |||
(by Jensen’s inequality and monotonicity of exp) | ≤ | exp[log(∑ni=11ciacii1r)] | ||
=rn∑i=1aciici |
Remark: in the case
1ci=1 ∀i |
one obtains:
(n∏i=1ai)1n≤1nn∑i=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:
(n∏i=1xwii)1W≤1Wn∑i=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 |
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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 |