Suppose that $a,b$ and $c$ are positive real numbers such that $ab+bc+ca=3$. Prove that $$(a^8+1)(b^8+1)(c^8+1)(a^4+1)(b^4+1)(c^4+1) \geq (a^2+1)^2(b^2+1)^2(c^2+1)^2$$ When does equality hold?
I used Hölder and got equality at $ a = b = c = 1 $
The rest is analogous. I hope for a solution
By C-S $$\prod_{cyc}((a^8+1)(a^4+1))\geq\prod_{cyc}(a^6+1)^2.$$ Thus, it's enough to prove that $$\sum_{cyc}(a^6+1)\geq\prod_{cyc}(a^2+1)$$ or $$\prod_{cyc}(a^4-a^2+1)\geq1.$$ Now, by AM-GM, C-S and since $$(a+b)(a+c)(b+c)\geq\frac{8}{9}(ab+ac+bc)(a+b+c)$$ it's $$\sum_{cyc}c(a-b)^2\geq0,$$ we obtain: $$\prod_{cyc}(a^4-a^2+1)\geq\prod_{cyc}\left(\frac{a^2+1}{2}\right)^2=\frac{1}{64}\prod_{cyc}((a^2+1)(1+b^2))\geq\frac{1}{64}\prod_{cyc}(a+b)^2\geq$$ $$\geq\frac{1}{64}\left(\frac{8}{9}(a+b+c)(ab+ac+bc)\right)^2=\frac{1}{9}(a+b+c)^2\geq\frac{1}{9}\cdot3(ab+ac+bc)=1.$$