<s>[INST] Prove that $(a^2 + 1)(b^2 + 1)(c^2 + 1) \ge (a + b)(b + c)(c + a)$ for $a, b, c \in \mathbb{R}$ How to prove that $(a^2 + 1)(b^2 + 1)(c^2 + 1) \ge (a + b)(b + c)(c + a)$ for $a, b,  c \in \mathbb{R}$ ? I have tried AM-GM but with no effect.
 [/INST] By C-S we obtain:
$$\prod_{cyc}(a^2+1)=\sqrt{\prod_{cyc}(a^2+1)(1+b^2)}\geq\sqrt{\prod_{cyc}(a+b)^2}=$$
$$\geq\left|\prod_{cyc}(a+b)\right|\geq\prod_{cyc}(a+b)$$
and we are done!
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