Let $f: U\to f(U)\subset \Bbb C$ be a holomorphic and nonconstant ( thus in particular an open map), and such that it can be extended continuously on $\overline{U}$.
Where $U$ is a bounded domain of $\Bbb C$ i.e an open and connectedness bounded set. Well I want to know if it's true that the boundary of $U$ is mapped onto ( surjective) the boundary $ f(U)$.
Well at least we know that the image of the boundary of $U$, is contained in the boundary of $f(U)$, i.e $f(\partial(U)) \subset \partial (f(U))$. And that is clear from the fact that $f$ is an open map. (I only used that). I want to know the other containment.