How to prove the inequality $(1+a+a^2)(1+b+b^2)(1+c+c^2) \leq (1+a+b^2)(1+b+c^2)(1+c+a^2)?$ For  $a,b,c>0$ prove the inequality
$$
(1+a+a^2)(1+b+b^2)(1+c+c^2) \leq (1+a+b^2)(1+b+c^2)(1+c+a^2).
$$
Seems the  rearrangement inequality must help but I can't  do it. Any ideas?

Hint:  You can use Karamata's inequality and the concavity of $\log$ function, with the fact that $(1+a+a^2, 1+b+b^2, 1+c+c^2) \succ (1+a+b^2, 1+b+c^2, 1+c+a^2)$ to conclude that:
$$\sum_{cyc} \log(1+a+a^2) \le \sum_{cyc} \log(1+a+b^2)$$
P.S: The same approach provides a simple proof for the more general statement Darij made in his post.
