Marty's theorem
MartysTheorem
2004-04-16 20:45:58
2006-06-18 21:06:16
Theorem
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\begin{thm}[Marty]
A set ${\mathcal{F}}$ of meromorphic functions is a normal family
on a domain $G \subset {\mathbb{C}}$ if and only if the spherical
derivatives are uniformly bounded (uniformly over ${\mathcal{F}}$)
on each compact subset of $G$.
\end{thm}
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
$\sigma$ 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.
A related theorem can be stated.
\begin{thm}
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
on compact subsets of $G$.
\end{thm}
Here $f^\sharp$ denotes the spherical derivative of $f$.
\begin{thebibliography}{9}
\bibitem{Gamelin:complex}
Theodore~B.\@ Gamelin.
{\em \PMlinkescapetext{Complex Analysis}}.
Springer-Verlag, New York, New York, 2001.
\end{thebibliography}