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I am trying to prove young's inequality for integrals $$ ab \leq \int_0^a \! f(x) \, \mathrm{d}x + \int_0^b \! f^{-1}(x) \, \mathrm{d}x. $$ Can you help me please?

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6  
This isn't true in general. Wikipedia (en.wikipedia.org/wiki/…) gives the assumptions under which it holds, and the image in that article suggests a proof. (Also I think putting the $-1$ in parentheses is unusual; it looks a bit as if you're trying to denote the reciprocal instead of the inverse.) –  joriki Mar 4 '11 at 12:39
    
@joriki: the parentheses might have been my fault. I had fixed the tex code. Will remove them. –  Rasmus Mar 5 '11 at 0:27

3 Answers 3

up vote 4 down vote accepted

You can find a couple of short proofs in the Journal of Inequalities in Pure and Applied Mathematics.

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You can find some proofs here:

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A nice presentation is given in the classic book Introduction to Inequalities, by Beckenbach and Bellman. I have cited this book in detail in answer to another question about inequalities elsewhere on this site. Here is the link: Geometric mean never exceeds arithmetic mean

Regards, Mike Jones

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