AlastorMoody wrote: For every positive integer $m$ and for all non-negative real numbers $x,y,z$ denote \begin{align*} K_m =x(x-y)^m (x-z)^m + y (y-x)^m (y-z)^m + z(z-x)^m (z-y)^m \end{align*} Prove that $K_m \geq 0$ for every odd positive integer $m$ Let $M$ $= \prod_{cyc} (x-y)^2$. Prove, $K_7+M^2 K_1 \geq M K_4$ $$x(x-y)^m (x-z)^m + y (y-x)^m (y-z)^m + z(z-x)^m (z-y)^m\geq 0$$ Schur's inequality states that for all non-negative $a,b,c \in \mathbb{R}$ and $r>0$: $${a^r(a-b)(a-c)+b^r(b-a)(b-c)+c^r(c-a)(c-b) \geq 0}$$