Proving a much stronger version of AM-GM for three variables Here, @MichaelRozenberg stated the following inequality without proof:

Theorem. For all non-negative $a,b,c\in\mathbb R$,
  $$a^6+b^6+c^6-3a^2b^2c^2\geq16(a-b)^2(a-c)^2(b-c)^2.$$

In my answer below I give a complete brute-force proof. However, more elegant proofs are welcome.

I will use the Buffalo method: By symmetry of the inequality, we can assume without loss of generality that $a\le b\le c$. So we can substitute $a=x,b=x+y,c=x+y+z$ for some non-negative reals $x,y,z$.
Putting this in our inequality and expanding results in
$$\color{green}{12 x^4 y^2 + 12 x^4 y z + 12 x^4 z^2 + 28 x^3 y^3 + 42 x^3 y^2 z + 54 x^3 y z^2 + 20 x^3 z^3 + 27 x^2 y^4 + 54 x^2 y^3 z + 87 x^2 y^2 z^2 + 60 x^2 y z^3 + 15 x^2 z^4 + 12 x y^5 + 30 x y^4 z + 60 x y^3 z^2 + 60 x y^2 z^3 + 30 x y z^4 + 6 x z^5} + \color{blue}{2 y^6 + 6 y^5 z - y^4 z^2 - 12 y^3 z^3 - y^2 z^4 + 6 y z^5 + z^6}\geq 0.$$
The green part is automatically greater or equal than $0$ because the $x,y,z$ are non-negative. Also,
$$\color{blue}{2 y^6 + 6 y^5 z - y^4 z^2 - 12 y^3 z^3 - y^2 z^4 + 6 y z^5 + z^6}=y^6+(y-z)^2(y+z)^2(y^2+6yz+z^2)\geq0.$$
This achieves a proof.
