Question 139
Let $a, b$ and $c$ be positive real numbers such that $\min\{ ab, bc, ca\} \ge 1$. Prove that
$$
\sqrt[3]{(a^2 + 1)(b^2 + 1)(c^2 + 1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.
$$
$$
\sqrt[3]{(a^2 + 1)(b^2 + 1)(c^2 + 1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.
$$