# Family of analytic functions from unit disk to the plane minus a line

Let $\mathcal F$ be the family of analytic functions on the unit disk $\,\mathbb D=\{z: \lvert z\rvert<1 \},$ such that $f[\mathbb D] \subset \mathbb C\!\smallsetminus\!(-\infty, 0]$. Show that $\mathcal F$ is a normal family.

(Normal Family= relatively compact= every sequence contains a subsequence which converges uniformly on every compact subset of $\mathbb D$)

EDIT. I see that this is immediate from Montel's theorem, but can it be proved somewhat easily without that?

Hint. Note that $$\mathbb C\setminus (-\infty,0]$$ is a simply connected domain, and hence there exists a conformal map $$\varphi : \mathbb C\setminus (-\infty,0]\to\mathbb D.$$ Next observe that the family $$\mathcal G=\{\varphi\circ f : f\in\mathcal F\}$$ is bounded and hence a normal family. Next, if $$\{\varphi\circ f_n\}$$ converges on compact subsets of $$\mathbb D$$, then so does the sequence $$f_n=\varphi^{-1}\circ(\varphi\circ f_n), \quad n\in\mathbb N.$$