Two inequalities involving the rearrangement inequality Well, there are two more inequalities I'm struggling to prove using the Rearrangement Inequality (for $a,b,c>0$):
$$ 
a^4b^2+a^4c^2+b^4a^2+b^4c^2+c^4a^2+c^4b^2\ge 6a^2b^2c^2
$$
and 
$$a^2b+ab^2+b^2c+bc^2+ac^2+a^2c\ge 6abc
$$
They seems somewhat similar, so I hope there'a an exploitable link between them. They fall easily under Muirhead, yet I cannot figure out how to prove them using the Rearrangement Inequality. 
Any hints greatly appreciated. 

Without loss of generality, $a\le b\le c$. Then $ab\le ac\le bc$. Denote $x_1=x_2=ab, x_3=x_4=ac, x_5=x_6=bc$, and $y_1=y_2=a,y_3=y_4=b,y_5=y_6=c$. Then both $x_i$ and $y_i$ are monotonic, and $x_1y_6+...+x_6y_1=6abc$, so it is the least sum among all rearrangements. It is not hard to find a rearrangement to get left-hand side of the second inequality. The first is the same, just change all the letters by their squares.
