<s>[INST] 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?
 [/INST] By Holder and C-S for non-negatives $a$, $b$ and $c$ we obtain:
$$(3(a^3+b^3+c^3))^2=(1+1+1)^2(a^3+b^3+c^3)(a^3+b^3+a^3)\geq(a+b+c)^3(a^3+b^3+c^3)=$$
$$=(a+b+c)(a^3+b^3+c^3)(a+b+c)^2\geq(a^2+b^2+c^2)^2(a+b+c)^2$$
and we are done!
</s>