Prove Inequality $p^n+q^n+r^n \geq p^m+q^m+r^m$ when $pqr=1, n > m \geq 0$ Prove Inequality $$p^n+q^n+r^n \geq p^m+q^m+r^m$$ when $pqr=1, n &gt; m \geq 0, p,q,r &gt;0$


Help me to prove this thm... I think it use AM-GM, But It work not useful to me...
Hint : With $\alpha = \frac13\left(1-\frac{m}n\right)$ weighted AM-GM gives $(1-2\alpha) p^n +\alpha q^n + \alpha r^n \geqslant p^{n(1-2\alpha)}\cdot q^{n\alpha}\cdot r^{n \alpha} = p^m \cdot(pqr)^{n\alpha}$
$RHS=(p^m+q^m+r^m)(pqr)^\frac{n-m}{3}=\dfrac{1}{2}\sum_{sym} p^\frac{n+2m}{3}q^\frac{n-m}{3}r^\frac{n-m}{3} \\ LHS=\dfrac{1}{2}\sum_{sym} p^nq^0r^0 \\ \because (n,0,0)\succ(\frac{n+2m}{3},\frac{n-m}{3},\frac{n-m}{3}) \\ \therefore LHS \ge RHS$

By Muirhead's Inequality.