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If $f$ is a meromorphic function of the region $G,$ then define $f^{\sharp}: G \to \mathbb{R}$ by

$$f^\sharp = \left\{\begin{array}{ll} \frac{2|f'(z)|}{1+|f(z)|^2}, &\text{if } z \text{ is not a pole of } f, \\\ \lim_{w \to z} \frac{2|f'(w)|}{1+|f(w)|^2}, &\text{if } z \text{ is a pole of } f.\end{array}\right\}$$

Prove that:

A family $\mathcal{F} \subseteq \mathbb{M}(G)$ (meromorphic functions on $G$) is normal in $C(G, \mathbb{C}_{\infty})$ (continuous functions from $G$ to the Riemann sphere) if and only if the family $\mathcal{F}^{\sharp}=\{f^{\sharp}\; : \; f \in \mathcal{F} \}$ is locally bounded.

Remark: $C(G, \mathbb{C}_{\infty})$ has a metric $\rho$ which is induced from the chordal metric $d$ of $ \mathbb{C}_{\infty}$ obtained via the stereographic projection. That is, for $f,g \in C(G, \mathbb{C}_{\infty})$ we have

$$\rho(f,g)=\sum_{n=1}^{\infty} 2^{-n} \frac{\rho_n(f,g)}{1+\rho_n(f,g)}, \;\;\;\; \rho_n(f,g)=\sup\{ d(f(z),g(z)) \; : \; z \in K_n\}$$

where $K_n$ is an increasing sequence of compact sets that exhaust $G$ and satisfy $K_n \subseteq int(K_{n+1}).$

And the chordal metric $d$ on $\mathbb{C}_{\infty}$ is defined as follows;

For $z, z' \in \mathbb{C},$

$$d(z,z')=\frac{2|z-z'|}{[(1+|z|^2)(1+|z'|^2)]^{1/2}}, \;\;\;\; d(z, \infty)=\frac{2}{(1+|z|^2)^{1/2}}.$$

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