parmenides51 wrote: Let $x,y,z>0$ Prove that $$\frac{x^2+2}{\sqrt{z^2+xy}}+\dfrac{y^2+2}{\sqrt{x ^2+yz}}+\dfrac{z^2+2}{\sqrt{y^2+zx}}\geq 6$$. (nooonuii) Let $x=\sqrt2a$, $y=\sqrt2b$, $z=\sqrt2c$ and $x^2+y^2+z^2=t$. Thus, we need to prove that $$\sum_{cyc}\frac{a^2+1}{\sqrt{c^2+ab}}\geq3\sqrt2.$$Now, by Holder and Vasc's inequality we obtain: $$\sum_{cyc}\frac{a^2+1}{\sqrt{c^2+ab}}=\sqrt{\frac{\left(\sum\limits_{cyc}\frac{a^2+1}{\sqrt{c^2+ab}}\right)^2\sum\limits_{cyc}(a^2+1)(c^2+ab)}{\sum\limits_{cyc}(a^2+1)(c^2+ab)}}\geq$$$$\geq\sqrt{\frac{\left(\sum\limits_{cyc}(a^2+1)\right)^3}{\sum\limits_{cyc}(a^2b^2+a^3b+a^2+ab)}}\geq\sqrt{\frac{(3t+3)^3}{6t^2+6t}}=\frac{3(t+1)}{\sqrt{2t}}\geq3\sqrt2.$$