Non-zero derivative and injectivity of a rational function

Question

Let $\Omega\subset\mathbb{C}$ be a bounded domain which does not contain any pole of a rational function $R$ and $R'(z)\neq0$ for all $z\in\Omega$. Is it true that $R(z)$ is injective on $\Omega$?

Answer 1

No; consider $R(z)=z^n$ for some $n\geq 2$ and $\Omega$ a neighborhood of a sufficiently long arc of the unit circle $\mathbb{S}^1\subset\mathbb{C}$.