| Version 3 |
Version 2 |
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\theoremstyle{theorem} |
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\newtheorem*{thm}{Theorem} |
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| \begin{thm}[Marty] |
\begin{thm}[Marty] |
| A set ${\mathcal{F}}$ of meromorphic functions is a normal family |
A set ${\mathcal{F}}$ of meromorphic functions is a normal family |
| on a domain $G \subset {\mathbb{C}}$ if and only if the spherical |
on a domain $G \subset {\mathbb{C}}$ if and only if the spherical |
| derivatives are uniformly bounded |
derivatives are uniformly bounded |
| on each compact subset of $G$. |
on each compact subset of $G$. |
| \end{thm} |
\end{thm} |
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| Here normal convergence (convergence on compact subsets) is given using the |
Here normal convergence (convergence on compact subsets) is given using the |
| spherical metric and not the standard metric of the complex plane. That is if |
spherical metric and not the standard metric of the complex plane. That is if |
| $\sigma$ is the spherical metric then we will say $f_n \to f$ normally |
$\sigma$ is the spherical metric then we will say $f_n \to f$ normally |
| if $\sigma(f_n(z),f(z))$ converges to 0 uniformly on compact subsets. |
if $\sigma(f_n(z),f(z))$ converges to 0 uniformly on compact subsets. |
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| A related theorem can be stated. |
A related theorem can be stated. |
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| \begin{thm} |
\begin{thm} |
| If $f_n(z) \to f(z)$ uniformly in the spherical metric on compact subsets of |
If $f_n(z) \to f(z)$ uniformly in the spherical metric on compact subsets of |
| $G \subset {\mathbb{C}}$ then $f_n^\sharp(z) \to f^\sharp(z)$ uniformly |
$G \subset {\mathbb{C}}$ then $f_n^\sharp(z) \to f^\sharp(z)$ uniformly |
| on compact subsets of $G$. |
on compact subsets of $G$. |
| \end{thm} |
\end{thm} |
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| Here $f^\sharp$ denotes the spherical derivative of $f$. |
Here $f^\sharp$ denotes the spherical derivative of $f$. |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{Gamelin:complex} |
\bibitem{Gamelin:complex} |
| Theodore~B.\@ Gamelin. |
Theodore~B.\@ Gamelin. |
| {\em \PMlinkescapetext{Complex Analysis}}. |
{\em \PMlinkescapetext{Complex Analysis}}. |
| Springer-Verlag, New York, New York, 2001. |
Springer-Verlag, New York, New York, 2001. |
| \end{thebibliography} |
\end{thebibliography} |
