Regions of variability for univalent functions
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- by Peter Duren and Ayşenur Ünal PDF
- Trans. Amer. Math. Soc. 295 (1986), 119-126 Request permission
Abstract:
Let $S$ be the standard class of univalent functions in the unit disk, and let ${S_0}$ be the class of nonvanishing univalent functions $g$ with $g(0) = 1$. It is shown that the regions of variability $\{ g(r):g \in {S_0}\}$ and $\{ (1 - {r^2})f\prime (r):f \in S\}$ are very closely related but are not quite identical.References
- Lars V. Ahlfors, Complex analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable. MR 510197
- I. A. Aleksandrov, Parametricheskie prodolzheniya v teorii odnolistnykh funktsiĭ, Izdat. “Nauka”, Moscow, 1976 (Russian). MR 0480952
- I. A. Aleksandrov and S. A. Kopanev, The range of values of the derivative on the class of holomorphic univalent functions, Ukrain. Mat. Ž. 22 (1970), 660–664 (Russian). MR 0283186 Bateman Manuscript Project (A. Erdélyi, W. Magnus, F. Oberhettinger, F. Tricomi), Higher transcendental functions, Vol. II, McGraw-Hill, New York, 1953.
- Louis Brickman and Stephan Ruscheweyh, On certain support points of the class $S$, Proc. Amer. Math. Soc. 92 (1984), no. 1, 61–63. MR 749891, DOI 10.1090/S0002-9939-1984-0749891-6
- Peter L. Duren, Arcs omitted by support points of univalent functions, Comment. Math. Helv. 56 (1981), no. 3, 352–365. MR 639357, DOI 10.1007/BF02566218
- Peter L. Duren, Univalent functions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
- Peter Duren and Glenn Schober, Nonvanishing univalent functions, Math. Z. 170 (1980), no. 3, 195–216. MR 564200, DOI 10.1007/BF01214860
- P. Duren and G. Schober, Nonvanishing univalent functions. II, Ann. Univ. Mariae Curie-Skłodowska Sect. A 36/37 (1982/83), 33–43 (1985) (English, with Russian and Polish summaries). MR 808431
- Arthur Grad, The region of values of the derivative of a schlicht function, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 198–202. MR 33885, DOI 10.1073/pnas.36.3.198
- David Hamilton, Extremal problems for nonvanishing univalent functions, Aspects of contemporary complex analysis (Proc. NATO Adv. Study Inst., Univ. Durham, Durham, 1979) Academic Press, London-New York, 1980, pp. 415–420. MR 623484 —, Covering theorems for univalent functions, preprint.
- W. K. Hayman, Multivalent functions, Cambridge Tracts in Mathematics and Mathematical Physics, No. 48, Cambridge University Press, Cambridge, 1958. MR 0108586
- James A. Jenkins, On univalent functions omitting two values, Complex Variables Theory Appl. 3 (1984), no. 1-3, 169–172. MR 737478, DOI 10.1080/17476938408814069
- Zdzisław Lewandowski, Richard Libera, and Eligiusz Złotkiewicz, Values assumed by Gel′fer functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 31 (1977), 75–84 (1979) (English, with Russian and Polish summaries). MR 569967 A. Pfluger, The range of $\log f\prime (a)$, informal manuscript, 1982. A. Ünal, Derivative type support points of the class $S$, Ph.D. Thesis, University of Michigan, 1986.
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 295 (1986), 119-126
- MSC: Primary 30C70
- DOI: https://doi.org/10.1090/S0002-9947-1986-0831192-5
- MathSciNet review: 831192