I have the following question and I have no idea how I can solve it. Let $F$ be family of functions $f$ in $A(D)$ where $D=\{x\in\mathbb{C}, \vert x\vert<1\}$ is a unit disc and $A(D)$ denotes the set of analytic functions on $D$.
If:
$$\begin{cases}
f(0) &=2i\\
\vert f(z)\vert&>1
\end{cases}$$
for all $z\in D$, then prove that $F$ is a normal family in $A(D)$.
I tried to use montel's but this family of functions is not locally bounded and the same issue with Ascoli-Arzela theorem.
Could you please give me a hint. Thanks.
Best Answer
We define $$ G:= \bigg\{ \frac{1}{f}\ \ \bigg|\ \ f \in F \bigg\} $$
Easy to see $G \subset A(D)$ is a normal family (since uniformly bounded).
Let $f_n$ be a sequence in $F$.
Define $g_n := \dfrac{1}{f_n}$ for all $n \in \mathbb N$.
Now, since $G$ is normal, there is a subsequence of $g_n$ (let us say $g_{n_k}$) and a function $g$ such that $g_{n_k} \to g$ uniformly on $D$.
Check that $f_{n_k} := \frac{1}{g_{n_k}}$ converges uniformly to $f := \frac{1}{g}$ on $D$. (use the fact that $g_n$ are bounded)