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Methods of geometric function theory. II. (English. Russian original) Zbl 0917.30002

St. Petersbg. Math. J. 9, No. 5, 889-930 (1998); translation from Algebra Anal. 9, No. 5, 1-50 (1997).
In these two survey papers the author provides an overview of the methods of geometric function theory. Each paper is dedicated to the memory of G. M. Goluzin, and each paper has its own extensive bibliography. The author succeeds in capturing the interplay between the geometric properties of \(f(D)\) and the analytic properties of \(f\). Elementary aspects and classical versions of each method are presented first. Then the author progresses to applications, which, in the author’s view, most clearly reflect the present state of geometric function theory and the possibilities offered by its methods. Following a general historical overview, Part I deals with quadratic differentials for conformal and quasiconformal mappings, the area method and the method of contour integration, Loewner’s parametric method (including the history of the Bieberbach conjecture), and variational methods. Part II repeats the definitions provided in Part I and then continues with the method of extremal metric, the method of symmetrization, and the method of extreme points.

MSC:

30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
30C75 Extremal problems for conformal and quasiconformal mappings, other methods
30C35 General theory of conformal mappings

Citations:

Zbl 0917.30001
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