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On Explicit Bounds In Schottky's Theorem

  • J. A. Jenkins (a1)
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1. Introduction. To Schottky is due the theorem which states that a function F(Z), regular and not taking the values 0 and 1 in |Z| < 1 and for which F(0) = a0, is bounded in absolute value in |Z| ≤ r, 0 ≤ r < 1, by a number depending only on a0 and r. Let K(a0 r) denote the best possible bound in this result. Various authors have dealt with the problem of giving an explicit estimate for this bound.

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References
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1. Ahlfors, L. V., An extension of Schwarz's Lemma, Trans. Amer. Math. Soc, 43 (1938), 359364.
2. Bohr, H. and Landau, E., Über das Verhalten von ζ(s) und ζK(s) in der Nähe der Geraden σ = 1, Nach. Akad. Wiss. Göttingen, Math.-Phys. Kl., 1910, 303330.
3. Hayman, W. K., Some remarks on Schottky's Theorem, Proc. Cambridge Phil. Soc, 43 (1947), 442454.
4. Ostrowski, A., Asymptotische Abschätzung des absoluten Betrages einer Funktion, die die Werte 0 und 1 nicht annimmt, Comm. Math. Helv., 5 (1933), 5587.
5. Pfluger, A., Über numerische Schranken im Schottky'schen Satz, Comm. Math. Helv., 7 (1934-35), 159170.
6. Robinson, R. M., On numerical bounds in Schottky's Theorem, Bull. Amer. Math. Soc, 45 (1939), 907910.
7. Valiron, G., Complements au théorème de Picard-Julia, Bull. Se Math., 51 (1927), 167183.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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