# $\{g_{n_k}\}_{k \in \Bbb N}$ 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

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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