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On bounded univalent functions that omit two given values. (English) Zbl 0937.30012

Let \(a,b \in \{z: 0<|z|<1\}\) and let \(S(a,b)\) be the class of all univalent functions \(f\) that map the unit disk \(\mathbb D\) into \(\mathbb D \setminus \{a,b\}\) with \(f(0)=0\). The paper studies the problem of maximizing \(|f'(0)|\) among all \(f\in S(a,b)\). It is shown that there exists a unique extremal function which maps \(\mathbb D\) onto a simply connected domain \(D_o\) bounded by the union of the closures of the critical trajectories of a certain quadratic differential. If \(a<0<b\), then \(D_o =\mathbb D \setminus [-1,a] \setminus [b,1]\).

MSC:

30C75 Extremal problems for conformal and quasiconformal mappings, other methods
30C85 Capacity and harmonic measure in the complex plane
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