<s>[INST] Inequality with three variables Let $a,b,c\ge 0$,show that
$$\sqrt{a^3+2}+\sqrt{b^3+2}+\sqrt{c^3+2}\ge \sqrt{\dfrac{9+3\sqrt{3}}{2}(a^2+b^2+c^2)}$$
 [/INST] I think the following will help.
By Holder $$\left(\sum_{cyc}\sqrt{a^3+2}\right)^2\sum_{cyc}\frac{(a^2+1+\sqrt3)^3}{a^3+2}\geq(a^2+b^2+c^2+3+3\sqrt3)^3.$$
Thus, it's enough to prove that
$$(a^2+b^2+c^2+3+3\sqrt3)^3\geq\frac{9+3\sqrt3}{2}(a^2+b^2+c^2)\sum_{cyc}\frac{(a^2+1+\sqrt3)^3}{a^3+2},$$ 
which looks true, but I have no the proof. 
At least, we deleted radicals of variables and the inequality is a polynomial inequality already!  
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