Betsakos, Dimitrios On bounded univalent functions that omit two given values. (English) Zbl 0937.30012 Colloq. Math. 80, No. 2, 253-258 (1999). 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 \(D_o =\mathbb D \setminus [-1,a] \setminus [b,1]\). Reviewer: D.Betsakos (Kezani) Cited in 2 Documents MSC: 30C75 Extremal problems for conformal and quasiconformal mappings, other methods 30C85 Capacity and harmonic measure in the complex plane Keywords:conformal radius; quadratic differential; symmetrization; harmonic measure PDFBibTeX XMLCite \textit{D. Betsakos}, Colloq. Math. 80, No. 2, 253--258 (1999; Zbl 0937.30012) Full Text: DOI EuDML