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On Explicit Bounds in Landau's Theorem. II

Published online by Cambridge University Press:  20 November 2018

James A. Jenkins
James A. Jenkins*
Affiliation:
Washington University, St. Louis, Missouri
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Quite some years ago a number of mathematicians were interested in obtaining explicit expressions for the bounds in Schottky's and Landau's theorems, specifically numerically évaluable bounds of a particular form. The best bounds of this type in Schottky's theorem were given by the author [3]. For Landau's theorem the chosen form is as follows. Let F(Z) be regular in |Z| < 1, omit the values 0 and 1 and have Taylor expansion about Z = 0

Then

Using the same method employed for Schottky's theorem the author showed that one can take K = 5.94. By a slight modification of the author's method Lai [6] gave the further value K = 4.76.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 3 , 01 June 1981 , pp. 559 - 562
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Ahlfors, Lars V., Conformai invariants (McGraw-Hill, 1973).Google Scholar
2. Jenkins, James A. and Morse, Marston, Curve families F* locally the level curves of a pseudoharmonic function, Acta Mathematica 91 (1954), 1–42.Google Scholar
3. Jenkins, James A., On explicit bounds in Schottkys theorem, Can. J. Math. 7 (1955), 76–82.Google Scholar
4. Jenkins, James A., On explicit bounds in Landau's theorem, Can. J. Math. 8 (1956), 423–425.Google Scholar
5. Jenkins, James A., A topological Three Pole Theorem, Indiana University Mathematics Journal 21 (1972), 1013–1018.Google Scholar
6. Lai, W., Ûber den Satz von Landau, Science Record J+ (1960), 339–342.Google Scholar
7. Review of [3], Mathematical Reviews 16 (1955), 579.Google Scholar