A question related to a family of functions defined on the unit disc

Question

Let $\mathcal{F}$ denote the set of all analytic functions $g:\mathbb{D}\to \{z\in \mathbb{C}\mid \text{Re}(z)>0, \text{Im}(z)>0\}$ such that $g(0)=1+i.$ I am trying to show that $\mathcal{F}$ is a normal family of functions. I know that using Montel's Theorem, it suffices to show that $\mathcal{F}$ is locally bounded. However I am not entirely sure to proceed. Any hint/help will be very useful. Thanks in advance.

Also, on a related note, is it necessary to assume that $g(0)=1+i$ to verify the normality of $\mathcal{F}$?

Answer 1

My gut instinct would be to construct a conformal map $h$ that sends the open set $\{z\in \mathbb{C}\mid \text{Re}(z)>0, \text{Im}(z)>0\}$ to the unit disk, and sends $1 + i$ to $0$.

Then I would apply Schwarz's lemma to the family $\{ h \circ g : g \in \mathcal F \}$. And I would go from there.

Regarding your second question - if you were to go with my suggestion, then the condition that the functions in $\mathcal F$ send $0$ to $1 + i$ is pretty essential in order for Schwarz's lemma to give you anything useful.

In fact, suppose you drop the condition that functions in $\mathcal F$ send $0$ to $1 + i$. Then you ought to be able to construct a sequence of functions in $\mathcal F$ that is not uniformly bounded on compact subsets of $\mathbb B$. I suggest you consider a sequence of the form $f_n := h^{-1} \circ \psi_{\alpha_n}$, where $ \psi_{\alpha_n} : = \frac{z+\alpha_n}{1+\bar{\alpha_n}z}$ and where $\alpha_n$ is a sequence of points in $\mathbb B$ that converges to the point on the boundary of $\mathbb B$ that gets mapped to infinity by $h^{-1}$.

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Edit: As per the comment below, I'll sketch out a few more steps in the application of Schwarz's lemma.

Let $\mathcal F ' := \{ h \circ g : g \in \mathcal F \}$.

1. I would start by proving that if $K$ is a compact subset of $\mathbb B$, then there exists an $r < 1$ such that $K \subset \overline{B(0, r)}$
2. Using Schwarz's lemma, I would deduce that if $f$ is any function in the family $\mathcal F'$, then $f(K) \subset \overline{B(0, r)}$.
3. I would then argue that $h^{-1}({ \overline{B(0, r)}})$ is a bounded subset of $\mathbb C$. (Think about how continuous functions act on compact sets...) Thus $h^{-1} \circ f(K)$ is contained in this bounded subset of $\mathbb C$, for any $f$ in the family $\mathcal F'$.