Prove that $\sum_{cyc} {\frac{y-x}{y^2-1}}$ >0

Question

If $x,y,z$ are real numbers, each greater than 1, then show that

$\frac{y-x}{y^2-1}$+$\frac{z-y}{z^2-1}$+$\frac{x-z}{x^2-1}\gt 0$

It is not the actual problem,I deducted the actual problem in those stage. Then I did not find any way. Seeing this problem ,I tried to use rearrangement inequality but found nothing good. Please help me.

Answer 1

First, I think your inequality is "$\leq$" instead of "$>$". The inequality is equivalent to

$$ \frac{x-1}{x^2-1} + \frac{y-1}{y^2-1} + \frac{z-1}{z^2-1} < \frac{z-1}{x^2-1} + \frac{x-1}{y^2-1} + \frac{y-1}{z^2-1}. $$

Then use rearrangement inequality on $x-1, y-1, z-1$ and $(x^2-1)^{-1}, (y^2-1)^{-1}, (z^2-1)^{-1}$.