proof of Marty’s theorem

(i) Fix KΩ compactPlanetmathPlanetmath. We have:

2|f(z)|1+|f(z)|2 MKf,zK ($*$)

Let V be a region with K=V¯ and let γ:[a,b]V be the C1 curves connecting the points P,QΩ. Then we have:

dσ(f(P),f(Q)) =infγlσ(fγ)=infγab(fγ)(t)σ,fγ(t)𝑑t
(($*$)proof of Marty’s theorem)MKinfγab|γ(t)|𝑑t

Thus f is Lipschitz continuous and thus is equicontinuous. By the Ascoli-Arzelà Theorem we conclude that is normal.

(ii) Now assume to be normal. Define:

f(z) :=2|f(z)|1+|f(z)|2

Let KΩ be compact. To obtain contradictionMathworldPlanetmathPlanetmath assume {f:f} is not uniformly bounded on K. But then there exists a sequence {fn} such that:

maxzKfn(z) (n)

Since is normal for each PΩ let there be a neighbourhood UP such that {fn} convergesPlanetmathPlanetmath normally to a meromorphic function f. But from (1/fn)=fn we see that {fn} converges normally on UP. Since K is compact it can be covered by finitely many sets UP. We conclude that {fn} must be boundedPlanetmathPlanetmathPlanetmath on K and obtain a contradiction. ∎

Title proof of Marty’s theorem
Canonical name ProofOfMartysTheorem
Date of creation 2013-03-22 18:23:11
Last modified on 2013-03-22 18:23:11
Owner karstenb (16623)
Last modified by karstenb (16623)
Numerical id 4
Author karstenb (16623)
Entry type Proof
Classification msc 30D30