Marty’s theorem

Theorem (Marty).

A set F of meromorphic functions is a normal family on a domain GC if and only if the spherical derivatives are uniformly bounded (uniformly over 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 planeMathworldPlanetmath. That is, if σ is the spherical metric then we will say fnf normally if σ(fn(z),f(z)) converges to 0 uniformly on compact subsets.

A related theorem can be stated.


If fn(z)f(z) uniformly in the spherical metric on compact subsets of GC then fn(z)f(z) uniformly on compact subsets of G.

Here f denotes the spherical derivative of f.


  • 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