Show that $f$ is either injective or a constant function.

Let $\Omega$ be a domain in $\mathbb{C}$ and let $\{f_n\}_{n \in \mathbb{N}}$ be a sequence of injective functions that converge in $O(\Omega)$ to $f$ . Show that $f$ is either injective or a constant function.

How does the conclusion change if, instead of a domain, we allow $\Omega$ to be an arbitrary open set ?

I know that $f$ is holomorphic as an almost uniform limit. But I dont know how to proceed.

Solution 1:

Assume that the limit function $f$ is neither constant nor injective. Then $f$ takes some value $a$ in disjoint disks $B_1, B_2 \subset \Omega$. As in

• Number of roots of a sequence of a uniformly convergent holomorphic functions implies an upper bound for the number of roots of their limit

it follows from Rouché's theorem that there are $n_1, n_2$ such that for $n \ge n_j$, $f_n$ takes the value $a$ in $B_j$ (at least once). For $n \ge \max(n_1, n_2)$ this is a contradiction to $f_n$ being injective.

Rouché's theorem is applied separately to the two disks in $\Omega$, so the same conclusion holds if $\Omega$ is a (not necessarily connected) open set.