How to prove the inequality $ (1+a+ab)(1+b+bc)(1+c+ca) \leq (1+a+a^2)(1+b+b^2)(1+c+c^2)?$ For $a,b,c>0$  prove the inequality
$$
(1+a+ab)(1+b+bc)(1+c+ca) \leq  (1+a+a^2)(1+b+b^2)(1+c+c^2).
$$
I know  that I should use the multiplicative rearrangement inequality  but I am not sure how  to choose the involved sequences correctly.  Any ideas?

HINT: use that $$x^2+y^2+z^2\geq xy+xz+yz$$ for all real $x,y,z$ is hold.
And your inequality can be written as
$$(ab)^2+(ac)^2+(bc)^2-a^2bc-ab^2c-abc^2+a^2b+ac^2+b^2c-3abc+a^2+b^2+c^2-ab-ac-bc\geq 0$$
