<s>[INST] $x,y,z \geqslant 0$ and $x^2+y^2+z^2+xyz=4$, prove $x^{\frac85}+y^{\frac85}+z^{\frac85} \geqslant 3$ $x,y,z \geqslant 0$ and $x^2+y^2+z^2+xyz=4$, prove
$$x^{\frac85}+y^{\frac85}+z^{\frac85} \geqslant 3$$
1) The equality occurs only at $x,y,z=1$. Let's assume $F=x^n+y^n+z^n$, I noticed that $n=1$ then $F \leqslant 3$ and $n=2$ then $F \geqslant 3$. I believe $n=\frac85=1.6$ is a very sharp inequality.
2) I try trig substitution but cannot remove the radical . $n=\frac85$  is too much for algebraic manipulation.
 [/INST] Let $x=2\cos\alpha$ and $y=2\cos\beta$, where $\{\alpha,\beta\}\subset\left[0,\frac{\pi}{2}\right]$.
Hence, $z=2\cos\gamma$, where $\alpha+\beta+\gamma=\pi$ and $\gamma\in\left[0,\frac{\pi}{2}\right]$.
Let $f(x)=\left(\cos x\right)^{\frac{8}{5}}$.
Since $f''(x)=\frac{-8(1+4\cos2x)}{25\sqrt[5]{\cos^2x}}\geq0$ for all $x\in\left[\frac{\pi}{3},\frac{\pi}{2}\right)$, by  Vasc's RCF Theorem 
it remains to prove our inequality for $\beta=\alpha$ and $\gamma=\pi-2\alpha$ or
for $y=x$ and $z=2-x^2$, where $0\leq x\leq\sqrt2$.
Id est, it remains to prove that $2x^{\frac{8}{5}}+(2-x^2)^{\frac{8}{5}}\geq3$, which is obvious.
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