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This inequality doesn't hold in general. It is false for instance for $m=7$, $n=8$: Set $X=3/4$, $Y=1$, and $Z=9/7$. Then $XY+YZ+ZX=3$, however $\frac{X^8}{X^7+Y^7}+\frac{Y^8}{Y^7+Z^7}+\frac{Z^8}{Z^7+X^7}<\frac{3}{2}$.