Let $f:\mathbb{D}=\{z\in\mathbb{C}\mid |z|<1\}\rightarrow\mathbb{C}$ be a local diffeomorphism (i.e. an immersion) from an open disk in the plane to the plane.
The only situation I can image where $f$ is not injective is that $f$ sends $\mathbb{D}$ to a "self-overlapping'' region, in which case $f$ can not have continuous injective boundary values. But it seems non-trivial to me whether nice boundary values can guarantee injectivity:
Question. Assume that $f$ extends to a continuous map $\overline{\mathbb{D}}\rightarrow\mathbb{C}$ such that the boundary values $f|_{\partial\mathbb{D}}$ is injective and continuous, so that (by Jordan Curve Theorem) it maps $\partial\mathbb{D}$ homeomorphically to a Jordan curve which is the boundary of a simply connected domain $\Omega\subset\mathbb{C}$. Then it is true that $f$ is a homeomorphism from $\mathbb{D}$ to $\Omega$?
Answer
Yes, this is true. The uses that the map $f : \mathbb{D} \to \Omega$ is a proper local diffeomorphism. It follows that $f$ is a covering map, by applying a theorem of elementary topology says that every proper local homeomorphism from a locally compact space to a Hausdorff space is a covering map. Since $\mathbb{D}$ is simply connected, every covering map defined on $\mathbb{D}$ is a homeomorphism.
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