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, 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

$\log |f(z)|\leq {\frac {1+|z|}{1-|z|}}(7+\max(0,\log |f(0)|))$.

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