Školjka

Prove that for all positive real numbers $a, b, c$ such that $abc = 1$ the following inequality holds: $$\frac{a}{2b + c^2} + \frac{b}{2c + a^2} + \frac{c}{2a + b^2} \leqslant \frac{a^2 + b^2 +c^2}{3}$$