# Talk:Young's inequality

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Mathematics rating:
 C Class
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Field: Number theory

## Jensen

What's with this application of Jensen inequality? That is just a generalization of concavity from sums to integrals. Since here we only use a 2term sum exactly as per the definition of concavity, there is absolutely no need to include Jensen in the proof. MarSch 14:21, 14 Mar 2005 (UTC)

You're right; I was very hasty. Although it is indeed a special case of J's inequality, it's a vacuous one. Michael Hardy 23:08, 14 Mar 2005 (UTC)

## Simpler proof

Why not prove it with AM-GM as hinted to in the introduction "Young's inequality is a special case of the inequality of weighted arithmetic and geometric means. "? Just set $x_1=a^p$, $x_2=a^q$, $w_1=1/p$ and $w_2=1/q$ and apply weighted AM-GM, that's it. ($w_1 x_1+w_2 x_2 =a^p/p+b^q/q\geq x_1^{w_1} x_2^{w_2}=a b$. How the proof can be done with exp/log can be inferred from the proofs on AM-GM is the respective article. 134.169.77.186 12:42, 21 August 2007 (UTC) (ezander)

The current proof is quite self-contained and discusses also the case of equality. Of course, it's just (strict) convexity in different disguises. Furthermore, the more general formulation from the German page should be included (which has an easy proof). Maybe I (or someone else) does this eventually and thereby rewrites the current proof anyway. Schmock 19:29, 22 August 2007 (UTC)

## Proof by picture in section "Generalization using Legendre transforms"

In the section "Generalization using Legendre transforms" of the article, there is a nice graphical proof of the generalized version of Young's inequality (i note, that PlanethMath calls Young's inequality just this generalized version). I have two questions about it.

1. Could somebody show a book reference for this version of Young' inequality and for this graphical proof?

2. It isn't quite clear for me that why is this just in the section "Generalization using Legendre transforms". What is here to do with Legendre transform?