# Talk:Young's inequality

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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 ${\displaystyle x_{1}=a^{p}}$, ${\displaystyle x_{2}=a^{q}}$, ${\displaystyle w_{1}=1/p}$ and ${\displaystyle w_{2}=1/q}$ and apply weighted AM-GM, that's it. (${\displaystyle w_{1}x_{1}+w_{2}x_{2}=a^{p}/p+b^{q}/q\geq x_{1}^{w_{1}}x_{2}^{w_{2}}=ab}$. 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?

89.135.29.171 (talk) 06:00, 15 May 2010 (UTC)

To the second question: Write the stated inequality as g(b)≥ab-f(a) and note that the Legendre transform g(b) is in fact defined to be the maximum value of ab−f(a). Hanche (talk) 21:06, 15 May 2010 (UTC)

Thank you, it's clear. To my first question, I managed to find a book resource, Mitrinovic-Vasic:Analytic Inequalities,pp 48-50. It's interesting however that they don't mention Legendre transorm here.89.135.29.171 (talk) 06:26, 16 May 2010 (UTC)

## Inner product version

The generalization to inner products that was recently added works also for the general case. But more to the point, it is an immediate corollary of the scalar case and the Cauchy–Schwarz inequality. Hence I think it adds little value, and should be removed. Hanche (talk) 16:44, 28 October 2011 (UTC)

## Fork convolution inequality

I think that the convolution inequality is a quite different inequality: the convolution inequality is not a special case or a generalization of the inequality for products. I propose thus moving the section on the convolution inequality to a new article “Young's convolution inequality”. Jvohn (talk) 12:36, 20 March 2017 (UTC)