# Marty’s theorem

###### Theorem (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.

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 $f$.

## References

• 1 Theodore B. Gamelin. . Springer-Verlag, New York, New York, 2001.
Title Marty’s theorem MartysTheorem 2013-03-22 14:18:39 2013-03-22 14:18:39 jirka (4157) jirka (4157) 7 jirka (4157) Theorem msc 30D30