Does there exist an entire function on $\mathbb{C}$ mapping an open disc to an annulus?
The reason I ask this is because I want to to answer this question: Suppose $f$ is entire, and suppose there exists an open disc $U$ and $\delta > 0$ such that for all $z\in U$, $|f(z)|>\delta > 0$. Does there exist a branch cut for $\log$ such that $\log f(z)$ is analytic on all of $U$? (Note that $f$ is not necessarily one-to-one).
A problem arises if $U$ is mapped to an annulus.
EDIT: I guess the answer to the first question should be no, since if $f$ is entire and nonconstant, it is an open map. Therefore, all interior points of $U$ remain interior for $f(U)$. Thus $\partial f(U) = f(\partial U)$. On the other hand, since $\partial U$ is connected, $f(\partial U)$ cannot be the boundary of an annulus (which consists of two connected components). What do you think?
PS. For a set $S$, by $\partial S$ I of course mean the boundary of $S$.