The image of the Ahlfors function
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- by David Minda PDF
- Proc. Amer. Math. Soc. 83 (1981), 751-756 Request permission
Abstract:
Let $\Omega$ denote a maximal region on the Riemann sphere for bounded holomorphic functions and $p \in \Omega$. We present a class of examples to show that the complement in the unit disk of the image of the Ahlfors function for $\Omega$ and $p$ can be a fairly general discrete subset of the unit disk.References
- Lars V. Ahlfors, Bounded analytic functions, Duke Math. J. 14 (1947), 1–11. MR 21108
- Stephen D. Fisher, On Schwarz’s lemma and inner functions, Trans. Amer. Math. Soc. 138 (1969), 229–240. MR 240302, DOI 10.1090/S0002-9947-1969-0240302-2
- Stephen Fisher, The moduli of extremal functions, Michigan Math. J. 19 (1972), 179–183. MR 306506
- T. W. Gamelin, Cluster values of bounded analytic functions, Trans. Amer. Math. Soc. 225 (1977), 295–306. MR 438133, DOI 10.1090/S0002-9947-1977-0438133-8 S. Ya. Havinson, Analytic capacity of sets, joint nontriviality of various classes of analytic functions and the Schwarz lemma in arbitrary domains, Amer. Math. Soc. Transl. (2) 43 (1964), 215-266.
- Maurice Heins, On the Lindelöf principle, Ann. of Math. (2) 61 (1955), 440–473. MR 69275, DOI 10.2307/1969809
- A. Marden, I. Richards, and B. Rodin, Analytic self-mappings of Riemann surfaces, J. Analyse Math. 18 (1967), 197–225. MR 212182, DOI 10.1007/BF02798045
- Christian Pommerenke, Über die Kapazität ebener Kontinuen, Math. Ann. 139 (1959), 64–75 (1959) (German). MR 114922, DOI 10.1007/BF01459823
- Eckehard Röding, Über die Wertannahme der Ahlfors Funktion in beliebigen Gebieten, Manuscripta Math. 20 (1977), no. 2, 133–140. MR 437740, DOI 10.1007/BF01170721
- H. L. Royden, Real analysis, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1963. MR 0151555
- Walter Rudin, Some theorems on bounded analytic functions, Trans. Amer. Math. Soc. 78 (1955), 333–342. MR 67192, DOI 10.1090/S0002-9947-1955-0067192-5
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 83 (1981), 751-756
- MSC: Primary 30D50; Secondary 30C55
- DOI: https://doi.org/10.1090/S0002-9939-1981-0630049-3
- MathSciNet review: 630049