<s>[INST] Inequality. $2(x^2+y^2+z^2)^2 \geq 3(x^2y^2+y^2z^2+z^2x^2)+3(x^3y+y^3z+z^3x)$ How do I prove that : 
$$2(x^2+y^2+z^2)^2 \geq 3(x^2y^2+y^2z^2+z^2x^2)+3(x^3y+y^3z+z^3x).$$
It seems to be easy , but I have no idea:) 
thanks :) 
 [/INST] We start the same way as uforoboa: Expanding the left hand side, you are trying to show that
$$2x^4 + 2y^4 + 2z^4 + 4(x^2y^2+y^2z^2+z^2x^2) \geq 3(x^2y^2+y^2z^2+z^2x^2)+3(x^3y+y^3z+z^3x)$$
This equivalent to 
$$2x^4 + 2y^4 + 2z^4 +x^2y^2+y^2z^2+z^2x^2 \geq 3(x^3y+y^3z+z^3x)$$
By AM-GM one has $x^3y \leq 1/2(x^4 + x^2y^2)$, $y^3z \leq 1/2(y^4 + y^2z^2)$, and $z^3x \leq 1/2(z^4 + z^2x^2)$. Inserting this into the above it suffices to show that
$$2x^4 + 2y^4 + 2z^4 +x^2y^2+y^2z^2+z^2x^2 \geq {3 \over 2}x^4 + {3 \over 2}y^4 + {3 \over 2}z^4 +{3 \over 2}x^2y^2+{3 \over 2}y^2z^2+{3 \over 2}z^2x^2$$
This is the same as showing 
$$x^4 + y^4 + z^4 \geq x^2y^2+y^2z^2+z^2x^2$$
This follows from AM-GM on each term on the right, or just expanding $(x^2 - y^2)^2 + (y^2 - z^2)^2 + (x^2 - z^2)^2 \geq 0$.
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