$\{g_{n_k}\}_{k \in \Bbb N}$ normally convergent $\implies \{g^2_{n_k}\}_{k \in \Bbb N}$ normally convergent (as meromorphic functions)?

by reflexive   Last Updated April 15, 2019 11:20 AM - source

Let $\mathscr F$ be a family of one-to-one holomorphic functions on a simply-connected domain $D \subset \Bbb C$ such that $\mathscr F$ omits 0. Show that $\mathscr F$ is a normal family (when considered as a family of meromorphic functions).

My attempt :

Let $\mathscr F=\{f_i\}_{i \in I}$ . Then since, $\mathscr F$ be a family of holomorphic functions on $D$ , $\mathscr F$ omits $\infty \in \hat{\Bbb C}$ . Thus $\mathscr F$ omits $0,\infty \in \hat{\Bbb C}$ . So if we can find any other point in $\hat{\Bbb C}$ that $\mathscr F$ omits, we are done by Montel's theorem on Meromorphic functions .

Now using that $D$ is simply-connected domain in $\Bbb C$ and $\mathscr F$ is actually a family of holomorphic functions on $D$ omitting 0, we get for each $f_i$, a $g_i$ holomorphic on $D$ such that $f_i = {g_i}^2$ .

Now looking at a point say 1, if $\mathscr{F}$ omits 1 we are done. Otherwise, we have the possibility that some $f_i$'s assume 1 . Looking at corresponding $g_i$'s we get that $g_i =\pm 1$ , so those which take $-1$, replacing them by $-g_i$ and unaltering the others, we obtain $\mathscr G=\{g_i\}_{i\in I}$ of meromorphic functions on $D$ that omit $0,-1,\infty \in \hat{\Bbb C}$ , thus $\mathscr G$ is a normal family of meromorphic functions on $D \ldots(*)$

Now taking any sequence $\{f_n\}_{n \in \Bbb N}\subset \mathscr F$ . Look at the corresponding $\{g_n\}_{n \in \Bbb N}\subset \mathscr G$ . By $(*), \exists \{g_{n_k}\}_{k \in \Bbb N}$ normally convergent. Then does it imply that $\{g^2_{n_k}\}_{k \in \Bbb N}$ i.e. $\{f_{n_k}\}_{k \in \Bbb N}$ normally convergent (as meromorphic functions i.e. with respect to the chordal metric), which would give our desired result!

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