Young inequality

Question

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?

This isn't true in general. Wikipedia (http://en.wikipedia.org/wiki/Young's_inequality#Standard_version_for_increasing_functions) 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 04 '11 at 12:39
@joriki: the parentheses might have been my fault. I had fixed the tex code. Will remove them. — Rasmus, Mar 05 '11 at 00:27

score 5 · Accepted Answer · answered Mar 05 '11 at 13:28

5

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

answered Mar 05 '11 at 13:28

Derek Jennings

13,531

score 1 · Answer 2 · answered Aug 29 '11 at 01:15

1

You can find some proofs here:

http://mathdl.maa.org/images/upload_library/22/Ford/ElmerTolsted.pdf

answered Aug 29 '11 at 01:15

score 0 · Answer 3 · edited Apr 13 '17 at 12:19

0

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

edited Apr 13 '17 at 12:19

Community

1

answered May 04 '11 at 20:30

Mike Jones

4,460

Young inequality

3 Answers3

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