I'm working on an exercise concerning about domain of holomorphy from Krantz's Function Theory of Several Complex Variables:
Let $\Omega \subset \mathbb{C}^{n}$ be a domain of holomorphy and let $\Omega_0 \subset \mathbb{C}^{k}$ be another domain of holomorphy. Assume that $f$ : $\Omega \rightarrow \mathbb{C}^{k}$ is a holomorphic mapping. Prove that preimage $f^{-1}(\Omega_0) \cap \Omega$ is a domain of holomorphy.
This proposition gives ample examples of domain of holomorphy. I focus on the definition of domain of holomorphy, but I have no idea how to deal with $\partial(f^{-1}(\Omega_0) \cap \Omega)$. Maybe I should consider the holomorphically convex property by Cartan-Thullen Theorem. Which is the proper way? Is there any relevant reference for this problem? Appreciate any help!