Prove that $3(a^3+b^3+c^3) > (a+b+c)(a^2+b^2+c^2)$ Let $a,b,c$ be positive, not equal. Prove that $3(a^3+b^3+c^3) > (a+b+c)(a^2+b^2+c^2)$.
I know the proof by subtracting LHS by RHS and then doing some arrangement.
But isn't there any inequality which can be used in $(1+1+1)(a^3+b^3+c^3) >(a+b+c)(a^2+b^2+c^2)$, or an easy proof?

You can express the inequality as
$$ \frac{a^3+b^3+c^3}{3} > \frac{a+b+c}{3} \frac{a^2+b^2+c^2}{3}$$
or
$$  [m_3(a,b,c)]^3  > [m_1(a,b,c)] \; [m_2(a,b,c)]^2 $$
where $m_p$ is the generalized power (Hölder) $p$-mean. Applying the (important) property that $m_p > m_q$ whenever $p>q$ (and the values are not all equal), you're done.
