8. (1) It is easy to see that the inequality to be proved is equivalent to $2\left(a^{3}+b^{3}+c^{3}\right) \geqslant a^{2} b+a^{2} c+b^{2} a+b^{2} c+$ $c^{2} a+c^{2} b$. Since $a, b, c$ are positive real numbers, we have
$$a^{3}+b^{3} \geqslant a^{2} b+a b^{2}, b^{3}+c^{3} \geqslant b^{2} c+b c^{2}, c^{3}+a^{3} \geqslant c^{2} a+c a^{2}$$

Adding the three inequalities yields the desired result.