Let $f$ be analytic in a neighborhood of $0$ and $f'(0) \neq 0$. Use Rouche's theorem to show that there is a neighborhood of $U$ of $0$ such that $f$ is a bijection in $U$.
What does Rouche's theorem have to do with this problem? I am not sure how there are any zeros or poles in $U$ to begin with. Is $z=0$ suppose to be a pole? Any hints would be appreciated.