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On bounded univalent functions that omit two given values

Dimitrios Betsakos (1999)

Colloquium Mathematicae

Let a,b ∈ z: 0<|z|<1 and let S(a,b) be the class of all univalent functions f that map the unit disk into {a,bwith f(0)=0. We study the problem of maximizing |f’(0)| among all f ∈ S(a,b). Using the method of extremal metric we show that there exists a unique extremal function which maps onto a simply connnected domain D 0 bounded by the union of the closures of the critical trajectories of a certain quadratic differential. If a<0

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