Abstract
New conditions are given for continuously differentiable mappings of some plane domains to be injective. In the case of a circular plane domain, these criteria coincide with well-known conditions. The method of locally homeomorphic extension is used. Bibliography: 18 titles.
Similar content being viewed by others
REFERENCES
F. G. Avkhadiev, “Sufficient conditions for univalence of quasiconformal mapping,” Mat.Zametki, 18, 793–802 (1975).
L. V. Ahlfors and G. Weil, “A uniqueness theorem for Beltrami equations,” Proc.Am.Math.Soc., 13, 975–978 (1962).
F. G. Avkhadiev, Conformal Mappings and Boundary-Value Problems, Kazan Foundation “Matematika,” Kazan (1996).
J. Becker, “Lüwnershe Differentialgleichung und quasikonform fortsetsbare schlichte Functionen,” J.Reine Angew.Math., 255, 23–43 (1972).
J. Becker, “Lüwnershe Differentialgleichung und schlichtheitscriterien,” Math.Ann., 202, 321–325 (1973).
L. A. Aksentiev, “Conditions of univalence for solutions of the basic inverse boundary-value problems,” Usp.Mat.Nauk, 15, 119–124 (1960).
F. G. Avkhadiev and L. A. Aksentiev, “Basic results on sufficient conditions for univalence of analytic functions,” Usp.Mat.Nauk, 30, 3–60 (1975).
F. G. Avkhadiev and L. A. Aksentiev, “Basic results on sufficient conditions under which an analytic function is finite-sheeted,” Izv.VUZ.Mat., 3–13 (1986).
F. G. Avkhadiev, L. A. Aksentiev, and A. M. Elizarov, “Sufficient conditions under which an analytic function is finite-sheeted and their applications,” in: Itogi Nauki Tekh., Mat.Analiz [in Russian], 25, VINITI, Moscow (1987), pp. 3–121.
G. M. Golusin, Geometric Theory of Functions of Complex Variables [in Russian], Moscow (1966).
S. Stoilov, Lectures on Topological Principles in the Theory of Analytic Functions [in Russian], Moscow (1964).
F. G. Avkhadiev, “On the method of locally homeomorphic extension in the theory of sufficient conditions for univalence,” in: Tr.Semin.Kraev.Zad. [in Russian], 20, Kazan (1983), pp. 3–10.
Maria Fait, J. Krzyz, and J. Zygmunt, “Explicit quasiconformal extensions for some classes of univalent functions,” Comm.Math.Helv., 51, 279–285 (1976).
D. A. Brannau and W. E. Kirwan, “On some classes of bounded univalent functions,” J.London Math.Soc., 1, 431–443 (1969).
L. A. Aksentiev and P. L. Shabalin, “Conditions of univalence with quasiconformal extension and their application,” Izv.VUZ.Mat., 2(249), 6–14 (1983).
L. A. Aksentiev and P. L. Shabalin, “Univalence conditions in star-shaped and convex domains,” in: Tr.Semin.Kraev.Zad. [in Russian], 20, Kazan (1983), pp. 35–42.
F. F. Maier and M. A. Sevodin, “Univalence conditions in domains with convex complement,” in: Tr.Semin.Kraev.Zad. [in Russian], 22, Kazan (1985), pp. 151–160.
M. A. Sevodin, “Univalence conditions in spiral domains,” in: Tr.Semin.Kraev.Zad. [in Russian], 23, Kazan (1986), pp. 193–200.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Patsevich, E.L. Injectivity Conditions for Some Classes of Domains. I. Journal of Mathematical Sciences 107, 4038–4053 (2001). https://doi.org/10.1023/A:1012488616627
Issue Date:
DOI: https://doi.org/10.1023/A:1012488616627