Ayanna Trujillo

2022-06-24

Let $\mathrm{\Omega }\subset \mathbb{C}$ be a bounded domain which does not contain any pole of a rational function $R$ and ${R}^{\prime }\left(z\right)\ne 0$ for all $z\in \mathrm{\Omega }$. Is it true that $R\left(z\right)$ is injective on $\mathrm{\Omega }$?

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Quinn Everett

Expert

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

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