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Determining of an extremal domain fo the functions from the \(S\)-class. (English) Zbl 1114.30301

Summary: Let \(S\) be the class of analytic functions of the form \(f(z)=z+a_2z^2+\dots\), \(f(0)=0\), \(f^\prime(0)=1\) defined on the unit disk \(| z|<1\). Petru T. Mocanu raised the question of the determination max Re \(f(z)\) when Re \( f^\prime(z)=0\), \(| z|=r\), \(r>0\) given. For solving the problem we shall use the variational method of Schiffer-Goluzin [G. M. Goluzin, Geometrische Theorie der Funktionen einer komplexen Veränderlichen. Moskau-Leningrad: Staatsverlag für technisch-theoretische Literatur (1952; Zbl 0049.05902)].

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)

Citations:

Zbl 0049.05902
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