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Ahlfors function[edit]

For each compact , there exists a unique extremal function, i.e. such that , and . This function is called the Ahlfors function of K This can be proved by using a normal family argument involving Montel's theorem.

Proof of existence for a continuum[edit]

There is a relatively simple proof of the existence of an Ahlfors function, based on the Riemann mapping theorem, if we assume additionally that K is connected.

If K is compact and connected, we can assume (otherwise by Liouville's theorem and hence ). Then there exists a unique connected component U of that contains , where is the Riemann sphere.

The claim is that U is simply connected. To see this, consider first a smooth simple closed curve in and let be some point in . By the Jordan curve theorem (actually, since is smooth, one only needs easy versions of the Jordan curve theorem), contains a connected component, say that is disjoint from . Then . Moreover, since is smooth, the union is homeomorphic to

The Riemann mapping theorem now yields a biholomorphism such that and . (Here, denotes the unit disk in .) Defining for each , this defines a holomorphic map . In particular, , so that .

To prove the reverse inequality, let with and put . Then is analytic (since f and g are),

and so we may apply the Schwarz lemma to F. Hence, . Thus,

which gives us . Taking the supremum over all such f, we get . This concludes the proof.

Additional properties assuming finite connectivity[edit]

Let . If and E has n components, then the Ahlfors function is analytic across . Moreover, if is smooth, then .