Schottky's theorem
In mathematical complex analysis, Schottky's theorem, introduced by Schottky (1904) is a quantitative version of Picard's theorem. It states that for a holomorphic function f in the open unit disk that does not take the values 0 or 1, the value of |f(z)| can be bounded in terms of z and f(0). Schottky's original theorem did not give an explicit bound for f. Ostrowski (1931, 1933) gave some weak explicit bounds. (Ahlfors 1938) gave a strong explicit bound, showing that if f is holomorphic in the open unit disk and does not take the values 0 or 1, then
- [math]\displaystyle{ \log |f(z)| \le \frac{1+|z|}{1-|z|}(7+\max(0,\log |f(0)|)) }[/math].
Several authors, such as (Jenkins 1955), have given variations of Ahlfors's bound with better constants: in particular (Hempel 1980) gave some bounds whose constants are in some sense the best possible.
References
- Ahlfors, Lars V. (1938), "An Extension of Schwarz's Lemma", Transactions of the American Mathematical Society 43 (3): 359–364, doi:10.2307/1990065, ISSN 0002-9947
- Hempel, Joachim A. (1980), "Precise bounds in the theorems of Schottky and Picard", Journal of the London Mathematical Society 21 (2): 279–286, doi:10.1112/jlms/s2-21.2.279, ISSN 0024-6107
- Jenkins, J. A. (1955), "On explicit bounds in Schottky's theorem", Canadian Journal of Mathematics 7: 76–82, doi:10.4153/CJM-1955-010-4, ISSN 0008-414X
- Ostrowski, A. M. (1931), Studien über den schottkyschen satz, Basel, B. Wepf & cie., https://books.google.com/books?id=uzwgAAAAIAAJ
- Ostrowski, Alexander (1933), "Asymptotische Abschätzung des absoluten Betrages einer Funktion, die die Werte 0 und 1 nicht annimmt", Commentarii Mathematici Helvetici 5: 55–87, doi:10.1007/bf01297506, ISSN 0010-2571
- Schottky, F. (1904), "Über den Picardschen Satz und die Borelschen Ungleichungen", Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin: 1244–1263
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