One generalization of the class of helical functions
The paper under review offers a highly technical generalization of a certain class of schlicht functions that were intensely studied during the twentieth century, namely analytic, univalent and normalized functions on the unit disc of the complex plane. For a fairly general introduction to and overview of univalent functions and their properties see [Univalent functions. New York etc.: Springer (1983; Zbl 0514.30001)] by \textit{P. L. Duren} and [Univalent Functions. Vol. I, II. Tampa, FL: Mariner Publishing Co. (1983; Zbl 1041.30500; Zbl 1041.30501)] by \textit{A. W. Goodman}. In this paper, the boundaries of the domains onto which the considered analytic functions map the unit disc are one-parameter families of curves which happen to be generalizations of the logarithmic spirals studied by \textit{F. W. Gehring} [``Spirals and the universal Teichmüller space'', Acta Math. 141, 99--113 (1978; Zbl 0393.30015)]. The proof of the univalence of the studied families relies on arguments similar to those in [\textit{B. N. Rakhmanov}, Dokl. Akad. Nauk SSSR, n. Ser. 97, 973--976 (1954; Zbl 0056.30002)]. One of the enumerated classes of univalent functions thus obtained coincides with the well-known family of starlike functions [\textit{D. J. Hallenbeck} and \textit{T. H. MacGregor}, Linear problems and convexity techniques in geometric function theory. Boston etc.: Pitman Adavanced Publishing Program (1984; Zbl 0581.30001)]. Another class coincides with the family of functions studied in [Rakhmanov, loc. cit.]. Yet another is a subclass of the family of functions that map the unit circle onto a quasiconformal curve [\textit{L. V. Ahlfors}, Lectures on quasiconformal mappings. Providence, RI: American Mathematical Society (AMS) (2006; Zbl 1103.30001); \textit{M. Fait, J. G. Krzyz} and \textit{J. Zygmunt}, ``Explicit quasiconformal extensions for some classes of univalent functions'', Comment. Math. Helv. 51, 279--285 (1976; Zbl 0332.30010)]. The reader should be warned that the translation of the paper is in some parts somewhat awkward. This should not, however, deter an interested reader with some background in geometric function theory and a modicum of patience.
scientific article; zbMATH DE number 5975568