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
Canonical name	MartysTheorem
Date of creation	2013-03-22 14:18:39
Last modified on	2013-03-22 14:18:39
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	7
Author	jirka (4157)
Entry type	Theorem
Classification	msc 30D30