Given any positive real numbers $a,b,c$, we have $(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$ [closed]
Solution 1:
Hint: you can use Cauchy Schwarz inequality to prove stronger inequality $$(a^2+2)(b^2+2)(c^2+2)\ge 3(a+b+c)^2\ge 9(ab+bc+ac)$$ Use Cauchy-Schwarz inequality we have $$(b^2+1)(1+c^2)\ge (b+c)^2, (b^2+c^2)(1+1)\ge (b+c)^2$$ so $$(b^2+2)(c^2+2)=(b^2+1)(1+c^2)+b^2+c^2+3\ge (b+c)^2+\dfrac{1}{2}(b+c)^2+3=3[1+\dfrac{(b+c)^2}{2}]$$
and use Cauchy-Schwarz inequality $$(a^2+2)[1+\dfrac{(b+c)^2}{2}]\ge (a+b+c)^2$$