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Marty's theorem (Theorem)
Theorem 1 (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$.

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.

Theorem 2   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$.

Here $ f^\sharp$ denotes the spherical derivative of $ f$.

Bibliography

1
Theodore B. Gamelin. Complex Analysis. Springer-Verlag, New York, New York, 2001.



"Marty's theorem" is owned by jirka.
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Cross-references: converges, complex plane, standard metric, spherical metric, normal convergence, compact subset, bounded, spherical derivatives, domain, normal family, functions, meromorphic
There is 1 reference to this entry.

This is version 4 of Marty's theorem, born on 2004-04-16, modified 2006-06-18.
Object id is 5774, canonical name is MartysTheorem.
Accessed 1817 times total.

Classification:
AMS MSC: 30D30 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Meromorphic functions, general theory)
Pending Errata and Addenda
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