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, theorem B) 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
log | f ( z ) | ≤ 1 + | z | 1 − | z | ( 7 + max ( 0 , log | f ( 0 ) | ) ) {\displaystyle \log |f(z)|\leq {\frac {1+|z|}{1-|z|}}(7+\max(0,\log |f(0)|))} . 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, JSTOR 1990065 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, MR 0575385 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, MR 0066460 Ostrowski, A. M. (1931), Studien über den schottkyschen satz, Basel, B. Wepf & cie. 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, S2CID 119852055 Schottky, F. (1904), "Über den Picardschen Satz und die Borelschen Ungleichungen", Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin: 1244–1263