classic inequality

**DivideBy0** · January 4th 2008, 06:54 AM

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

$\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y} \geq \frac{3}{2}$

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

**Isomorphism** · January 4th 2008, 09:47 AM

Originally Posted by DivideBy0

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

$\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y} \geq \frac{3}{2}$

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

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

I will post them if you want

**PaulRS** · January 4th 2008, 10:43 AM

Define $x+y+z=\lambda>0$

THus: $y+z=\lambda-x$ ; $x+z=\lambda-y$ ; $x+y=\lambda-z$

So we have to prove: $\frac{x}{\lambda-x}+\frac{y}{\lambda-y}+\frac{z}{\lambda-z}\geq{\frac{3}{2}}$

Let: $f(x)=\frac{x}{\lambda-x}=\frac{\lambda}{\lambda-x}-1$ which is clearly convex in $(0; \lambda)$ (use the second derivative)

It follows from Jensen's Inequality that: $\frac{1}{3}\cdot{\left(f(x)+f(y)+f(z)\right)}\geq{ f\left(\frac{x+y+z}{3}\right)}$

Therefore: $\frac{1}{3}\cdot{\left(\frac{x}{\lambda-x}+\frac{y}{\lambda-y}+\frac{z}{\lambda-z}\right)}\geq{\frac{\left(\frac{x+y+z}{3}\right)} {\lambda-\left(\frac{x+y+z}{3}\right)}}$

Now remember that: $x+y+z=\lambda$ and the assertion follows

Jensen's inequality - Wikipedia, the free encyclopedia

**DivideBy0** · January 4th 2008, 03:48 PM

Thanks!

**ThePerfectHacker** · January 5th 2008, 07:12 PM

It can also be done with AM-HM, that is how I learned it.

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