Marty’s theorem

Theorem (Marty).

A set ${\\mathcal{F}}$ of meromorphic functions is a normal familyon a domain $G\\subset{\\mathbb{C}}$ if and only if the sphericalderivatives are uniformly bounded (uniformly over ${\\mathcal{F}}$ )on each compact subset of $G$ .

Here normal convergence (convergence on compact subsets) is given using thespherical metric and not the standard metric of the complex plane. That is, if $\\sigma$ is the spherical metric then we will say $f_{n}\\to f$ normallyif $\\sigma(f_{n}(z),f(z))$ 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 $f_{n}^{\\sharp}(z)\\to f^{\\sharp}(z)$ uniformlyon 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.

单词	MartysTheorem
释义	Marty’s theorem Theorem (Marty). A set ${\\mathcal{F}}$ of meromorphic functions is a normal familyon a domain $G\\subset{\\mathbb{C}}$ if and only if the sphericalderivatives are uniformly bounded (uniformly over ${\\mathcal{F}}$ )on each compact subset of $G$ . Here normal convergence (convergence on compact subsets) is given using thespherical metric and not the standard metric of the complex plane. That is, if $\\sigma$ is the spherical metric then we will say $f_{n}\\to f$ normallyif $\\sigma(f_{n}(z),f(z))$ 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 $f_{n}^{\\sharp}(z)\\to f^{\\sharp}(z)$ uniformlyon 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.
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