Let $P$ be a polynomial; we ask about the existence of a non-constant analytic function $f : \Bbb{C}\longrightarrow \Bbb{C}$ such for all $z \in \Bbb{C}, f(z) = f(P(z))$. Clearly when $P$ is linear we can find such $f$. What happens when $P$ is not linear ? Any suggestion would be helpful.



A functional equation concerning analytic functions
Here is a more conventional proof:-) Let $M(r)=\max\{|f(z)|:|z|=r\}$. Maximum principle
implies that this function is strictly increasing (unless $f$ is constant).
This gives a contradiction because $|P(z)|>|z|$ when $z$ is sufficiently large, and $P$
is of degree greater than $1$.


