A function is 0 if is almost bounded

Question

I am having problems with this excercise, if $f$ is analytic in $D \subset \mathbb{C}$ and:

$$|f(\cos(z))| \leq m|z|^n$$ for some $m,n \in \mathbb{N}$ and $z \in D$. Then show that $f \equiv 0$.

Please some hint could be very useful.

Answer 1

As Maik Pickl said in the comments: let $D=\{\sigma+it: \sigma>1\}$ (because the branch points are at $\pm 1$) and $f$ be any branch of $\arccos(z)$ on $D$. Then $f$ is clearly nonzero, but $|f(\cos(z))|=|z|\leq m|z|^n$, not just for some choice of $m$ and $n$ but in fact for every choice of $m,n\in\Bbb N$!