MathNet 2016-part-02 — 184 problems (EN–VI) Question 162

Let $a$, $b$ and $c$ are positive numbers such that $a^2 + b^2 + c^2 = 3$. Prove that
$$
ab + bc + ca + 3\sqrt{\frac{a^3 + b^3 + c^3}{a + b + c}} \le 6$$