How can you prove the inequality $2x^3 (x^3 + 8y^3) + 2y^3 (y^3 + 8z^3) + 2z^3 (z^3 + 8x^3) ≥ 9x^4 (y^2 + z^2) + 9y^4 (z^2 + x^2) + 9z^4 (x^2 + y^2)$ I changed the RHS as $\displaystyle \sum_{cyc} 9x^4(S-x^2) $ for $S = x^2 + y^2 + z^2$ 
Then I thought I could apply Jensen's inequality for $f(x) = 9x^4(S-x^2) = -9x^6 + 9Sx^4$ in the RHS as follows (i think $f(x)$ is concave):
$f(\frac{x +y +z}{3}) \leq \frac{f(x) + f(y) + f(z)}{3}$
$\Rightarrow 3(-9(\frac{x +y +z}{3})^6+9S(\frac{x +y +z}{3})^4) \leq RHS$
$\Rightarrow -\frac{(x +y +z)^6}{27}+S(\frac{(x +y +z)^4}{3}) \leq RHS$
So I think what it remains to be proved is that 
$LHS \geq -\frac{(x +y +z)^6}{27}+S(\frac{(x +y +z)^4}{3})$
But I have no clue how to go on :P

In fact, we have $$ LHS-RHS=\sum\limits_{cyc}{\left(x-y\right)^4\left(x^2+4xy+y^2\right)} \ge 0 $$
