Show a certain subset of $\mathcal{O}(\mathbb{H})$ is a normal family

Question

I need to show that $\mathcal{F}=\{f\in \mathcal{O}(\mathbb{H}): |f(z)|\neq 5 \forall z \in \mathbb{H}\}$ is a normal family. Here $\mathbb{H}$ is the upper half plane and $\mathcal{O}(\mathbb{H})$ is all of the holomorphic functions on $\mathbb{H}$. Since $\mathbb{H}$ is connected and each $f\in \mathcal{F}$ is continuous we see that either $f(\mathbb{H})\subset \{z\in \mathbb{C}: |z|<5\}$ or $f(\mathbb{H})\subset \{z\in \mathbb{C}: |z|>5\}$. Let's split $\mathcal{F}$ into the two sub families just mentioned: $\mathcal{F_{1}}$ and $\mathcal{F_{2}}$, respectively. I'm aiming to use Montel's theorem so I've got to establish a uniform bound on each compact $K\subset\mathbb{H}$. This trivial for $\mathcal{F_{1}}$ but I don't know how to do this for $\mathcal{F_{2}}$. Any hints or answers would be greatly appreciated.

Answer 1

I assume by Montel's theorem, you here mean its simplest form: a family of functions that is (locally) bounded is normal.

(This is the "normal families version" of Liouville's theorem, and the easiest to prove.)

As you mention, the family $\mathcal{F}_1$ is normal by this theorem.

To show that $\mathcal{F}_2$ is also normal, consider the family of functions $1/f$, for $f\in\mathcal{F}_2$.