Cardinality of Picard exceptional values of holomorphic function without an isolated singularity

Question

Picard's theorem says that if $f\colon D\to \mathbb{C}$ is an entire nonconstant function or holomorphic with an essential isolated singularity, then $\mathbb{C}\setminus f({D})$ is either empty or a singleton. In the meromorphic case this set of exceptional values is of cardinality at most two (including $\infty$).

Q: What can we say about $\mathbb{C}\setminus f({D})$ if $f\colon D\to \mathbb{C}$, $D\subset \mathbb{C}$ is nonempty open and connected, is holomorphic:

1. If $n\ge 3$, then does there exist $f$ such that $\mathbb{C}\setminus f({D})$ is of cardinality $n$?

2. Does there exist $f$ such that $\mathbb{C}\setminus f({D})$ is of infinite but countable cardinality?

3. Does there exist $f$ such that $\mathbb{C}\setminus f({D})$ is a meagre set or of Lebesgue measure zero but of uncountable cardinality?

Answer 1

It is clear that $f(D)$ is connected.

If $D$ is the unit disc, then every closed set whose complement is connected can be exceptional. Let $F$ be a closed set with connected complement containing at least 2 points. If $C\backslash D$ is not simply commented, then the universal covering map $f:D\to C\backslash F$ takes every value infinitely many times, and does not take values in $F$. If $C\backslash F$ is simply connected, construction is easier. If $F$ is one point, again we have easy examples.

Same happens when $D$ is any simply connected set, other that $C$, by the Riemann mapping theorem. So the answer to all your questions is "yes" if $D$ is simply connected but different from $C$.

If $E=C\backslash D$ is a finite set, then the number of exceptional values at every point of $E$ is at most one, and these exceptional values can be arbitrarily prescribed. So we can obtain any finite set of cardinality at most that of $E$.

If $D$ is a complement of a Cantor set $E$, then the answer is more complicated and not all possibilities were studied. Carleson gave an example of a cantor $E$, or zero capacity such that every analytic function outside $E$, for which $E$ is singular, can omit at most 5 values, and 5 is best possible for this $E$.