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.

By Cauchy inequality we have:
$$(a^2 + 1)(1+b^2) \ge (a + b)^2$$
$$(a^2 + 1)(1+c^2) \ge (a + c)^2$$
$$(c^2 + 1)(1+b^2) \ge (c + b)^2$$
Multiply these and take square root and thus the conclusion.
