Prove that if x, y, and z are positive real numbers, then

I haven't done much with inequalities so please explain your steps, thanks!

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- Jan 4th 2008, 07:54 AMDivideBy0classic inequality
Prove that if x, y, and z are positive real numbers, then

I haven't done much with inequalities so please explain your steps, thanks! - Jan 4th 2008, 10:47 AMIsomorphism
This is Nesbitts inequality, I know many proofs for this. Nesbitt's inequality - Wikipedia, the free encyclopedia

The above link proves it using AM-GM and Rearrangement inequality

I know a few more too :D

I will post them if you want - Jan 4th 2008, 11:43 AMPaulRS
Define

THus: ; ;

So we have to prove:

Let: which is clearly convex in (use the second derivative)

It follows from Jensen's Inequality that:

Therefore:

Now remember that: and the assertion follows

Jensen's inequality - Wikipedia, the free encyclopedia - Jan 4th 2008, 04:48 PMDivideBy0
Thanks!

- Jan 5th 2008, 08:12 PMThePerfectHacker
It can also be done with AM-HM, that is how I learned it.