Suppose $\Omega$ is open that contains the closed unit disc. If $f:\Omega \rightarrow \mathbb{C}$ is holomorphoc with $|f(z)| < 1$ for all $z$ with $|z|=1$. How many solutions are there of $f(z)=z^n$ if $n\in \mathbb{Z}$?
For the case that $n$ is postive, it is quite easy to see because of Rouché's theorem. I am, however, stuck on the case where $n$ is negative.