This question accross to this question from SE which there some answers but they r n't

enough to me hop to see MO what can they say about it .

let $m,n$ be integers, show that if $ n>m\geq 0 $ :

$$\frac{x^n}{x^m+y^m}+\frac{y^n}{y^m+z^m}+\frac{z^n}{z^m+x^m}\geq \frac{3} {2}\left(\frac{1}{\sqrt{3}}\right)^{n-m}$$

where real $x,y,z > 0 $ and $xy + yz + zx = 1$

Note : The question is Already montioned here in journal k.s competition prolem 111.

Thank you for your help .