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I am trying to prove young's inequality for integrals $ ab \leq \int\nolimits_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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    @joriki: the parentheses might have been my fault. I had fixed the tex code. Will remove them.2011-03-05

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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