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Some remarks on Schottky's theorem

Published online by Cambridge University Press:  24 October 2008

W. K. Hayman
W. K. Hayman
Affiliation:
St John's College, Cambridge
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Extract

1. We denote by the class of all functions regular in |z| < 1, and never taking the values 0 or 1 in |z| < 1. Then Schottky's(1) theorem states:

Theorem A. Let f(z) = a0 + a1z + … belong to . Then

and hence

where

and the constantsK(a0), K(a0,ρ) depend only on the quantities indicated.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 43 , Issue 4 , October 1947 , pp. 442 - 454
Copyright
Copyright © Cambridge Philosophical Society 1947

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References

(1)Schottky, F. S.B.Preuss. Akad. Wiss. (1904), Math.-Phys. K1 pp. 1244–1262.Google Scholar
(2)Bohr, H. and Landau, E.Nachr. Ges. Wiss. Göttingen (1910), Math.-Phys. K1. pp. 303–30.Google Scholar
(3)Valiron, G.Bull. Sci. Math. 51 (1927), 167–183.Google Scholar
(4)Ostrowsky, A.Studien über den Schottkyschen Salz (Basel, 1931).Google Scholar
(5)PflÜger, A. P.Commentarii Mathematici Helvetici, 7 (1935), 159–70.Google Scholar
(6)For an introduction to these expressions see R., Nevanlinna, Eindeutige Analytische Functionen (Berlin, 1936), chapters I–III. The ideas followed in this paper are very largely due to Nevanlinna.Google Scholar
(7)See, for example, Bieberbach, L., Lehrbuch der Functionentheorie, Vol. 2, chap. I.Google Scholar
(8)See, for example, Littlewood, J. E., Lectures on the Theory of Functions (Oxford, 1944), pp. 163et seq., where the theory of subordination is discussed.Google Scholar
(9)Ahlfors, L. V., Trans. Amer. Math. Soc. 43 (1938), 359–364. The method used is extremely simple and elegant and does not depend on the modular function.Google Scholar