How can I prove the this inequality?

Question

$$ a^6+b^6+c^6+3a^2 b^2 c^2 \geq 2(a^3 b^3 + b^3 c^3 +c^3 a^3)$$ $\forall a,b,c \in \mathbb{R}$

Can this be done with just weighted AM-GM?

Answer 1

No! Even for positive variables $a^2b^2c^2$ bothers.

By the way, your inequality is obviously true by Schur and Muirhead.

Answer 2

By Schur's and followed by AM-GM inequalites: $x^3+y^3+z^3 + 3xyz \ge xy(x+y) + yz(y+z) + zx(z+x) \ge 2xy\sqrt{xy} + 2yz\sqrt{yz} + 2zx\sqrt{zx}$, with $x = a^2,y=b^2,z=c^2$, the inequality follows.