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Covering theorems for univalent functions. (English) Zbl 0624.30007

This paper consists essentially of two parts. In the first the author studies conformal mappings f of the unit disc into a general plane domain W for which \(f(0)=a\), \(a\in W\), and asks for the region of values of f(r), \(0<r<1\). The results obtained are of a qualitative nature and the method is essentially an exercise in applying the fundamental concepts of the reviewer [Ann. Math., II. Ser. 66, 440-453 (1957; Zbl 0082.063)]. The second part studies the region of values of f(r) when f is a member of the family \({\mathcal S}_ 0\) consisting of regular univalent functions in the unit disc omitting the value 0 and with \(f(0)=1\). The corresponding region of values was found by Z. Lewandowski, R. Libera and E. Złotkiewicz [Ann. Univ. Mariae Curie-Skłodowska, Lect. A 31, 75-84 (1977; Zbl 0441.30022)]. (If one replaces the unit disc by a general schlichtartig domain the corresponding problem was later solved by the reviewer [Complex Variables 3, 169-172 (1984; Zbl 0521.30012)].) The author studies in a precise manner certain properties of the region of values indicated above. In particular he obtains bounds for l.u.b. arg f(r) and min \({\mathcal R}f(r)\). For some values of r the bounds are sharp. His final result treats an extremal problem for the module of a doubly- connected domain. The results ascribed to Schiffer on p. 38 were obtained much earlier by H. Grötzsch [Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl. 1933, 501-515 (1933)] actually in a more general context.
Reviewer: J.A.Jenkins

MSC:

30C25 Covering theorems in conformal mapping theory
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C55 General theory of univalent and multivalent functions of one complex variable
30C75 Extremal problems for conformal and quasiconformal mappings, other methods
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References:

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