<s>[INST] Show that $2(a^3+b^3+c^3)>a^2(b+c)+b^2(c+a)+c^2(a+b)>6abc$ If $a,b,c$ are positive real numbers, not all equal, then prove that  

$$2(a^3+b^3+c^3)>a^2(b+c)+b^2(c+a)+c^2(a+b)>6abc$$  

How can I show this?
 [/INST] Hint: $a^3 + b^3 > a^2b + ab^2$,
and use AM-GM inequality for $6$ terms for the other one.
</s>