Suppose $\Omega \subset \mathbb{C}$ is open, bounded, and connected, and its boundary $\partial \Omega$ is also connected. If it helps, we may assume $\partial \Omega$ is a Jordan curve or even that $\Omega$ is a disk.
Let $f \colon \overline{\Omega} \rightarrow \mathbb{C}$ be continuous with $f|_{\Omega} \colon \Omega \rightarrow \mathbb{C}$ holomorphic. My question is whether $f$ necessarily sends the boundary $\partial \Omega$ into the boundary of $f(\overline{\Omega})$. If $f$ happens to be injective, then this is part of Carathéodory's theorem. However, if $f$ is not injective, I was not able to verify whether $f(\partial \Omega) \subset \partial f(\overline{\Omega})$ must hold. (To be perfectly clear: that $f$ continuously extends to the boundary is assumed.)
Certainly, the pre-image of $\partial f(\overline{\Omega})$ lies in $\partial \Omega$ by the open mapping theorem, but this does not suffice. If it is true, I expect it to be a topological property, i.e. it suffices to assume $f$ is an open map and does not need analyticity, but I included it in the problem statement to be on the safe side. Any ideas or counter-examples would be greatly appreciated! Cheers.
