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I have encountered an inequality in the book Foundations of Modern Probability by Kallenberg. In Lemma 1.29, chapter 1, where the author proves the Holder inequality, here is one step.

Let $p >0$ and $q>0$ and that $p^{-1}+q^{-1} =1$. Also let $f, g$ be two functions then

$|fg| \leq \int_{0}^{|f|} x^{p-1} dx + \int_{0}^{|g|} y^{q-1} dy$

This seems to be a simple inequality as the only hint given by the author is "by calculus". But for some reason I cannot wrap my mind around it. Any help is welcomed.

Ran Wang
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  • Look at the picture here: https://en.wikipedia.org/wiki/Young%27s_inequality_for_products#Standard_version_for_increasing_functions –  Aug 02 '17 at 20:28
  • @ByronSchmuland Uggggggggg, I didn't think at all about the graphic representation......... If you can make the comment into an answer I will accept it. Thank you very much. – Ran Wang Aug 02 '17 at 20:43
  • See also: https://math.stackexchange.com/questions/2253569/prove-youngs-inequality –  Aug 02 '17 at 20:45
  • And https://math.stackexchange.com/questions/24994/young-inequality –  Aug 02 '17 at 20:46

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There is a "proof without words" picture at the Wikipedia page for "Young's inequality".