Proving Cauchy inequality involving four expression Show that $$(a^2 + b^2 + c^2) (a^2b^2 +b^2c^2 +c^2a^2) \geq (a^2b + b^2c + c^2a)(ab^2 + bc^2 + ca^2)$$
i should prove this inequality by making it a Cauchy form inequality(as teacher stated).
my problem : this inequality is involving 4 parentheses but in other question i proved previously there was one or two in left side and one in right side.
things i have done so far:
$(a^2 + b^2 + c^2) (a^2b^2 +b^2c^2 +c^2a^2) \geq (a^2b + b^2c + c^2a)^2$ 

It's $$\sum_{cyc}(a^4b^2+a^4c^2-a^3b^3-a^4bc)\geq0,$$
which is just Muirhead:
$$\sum_{cyc}(a^4b^2+a^4c^2-2a^3b^3)+\sum_{cyc}(a^4b^2+a^4c^2-2a^4bc)\geq0,$$
which is true because $(4,2,0)\succ(3,3,0)$ and $(4,2,0)\succ(4,1,1)$.
