Maurice Heins

A note on a theorem of Radó concerning the

Let Gz denote a region in the s-plane and let w =f(z) be a function of z denned for z<Z.Gz which has the following properties: (1) w =f(z) is analytic and single-valued for zC.Gz, (2) zQ.Gz implies that f(z)CGz, (3) to each point WoC.Gz there correspond m and only m points z^ (ft = 1,2, • • • ,m) contained in Gz such that ƒ (zf ) =w0 (£ = 1,2, • • -,m) where following the usual convention we count the 4 according to their multiplicities. Then w=f(z) is said to define a (1, m) conformai map ofGz onto itself. Such maps have been studied by Fatou and Julia for the case where Gz is simply-connected, and by Radó 4 who treated multiply-connected regions as well. Among the results which Radó established is the following theorem: