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Holomorphic Functions with Spiral asymptotic Paths

Published online by Cambridge University Press:  22 January 2016

W. Seidel
W. Seidel*
Affiliation:
University of Notre Dame
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Let f(z) be a holomorphic and unbounded function in | z | < 1, with the property that it remains bounded on some spiral S in | z | < 1 which approaches | z | = 1 asymptotically. The existence of such functions was first established by G. Valiron. Accordingly, we shall refer to such functions as functions of class (V) relative to S. More recently, F. Bagemihl and W. Seidel obtained examples of functions holomorphic and unbounded in | z | < 1 which approach prescribed finite or infinite values as | z | → 1 on any given enumerable set of disjunct spirals which approach | z | =1 asymptotically, as well as on certain sets of such spirals having the power of the continuum.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 14 , February 1959 , pp. 159 - 171
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1959

References

1) Valiron, G.. 1. Sur certaines singularités des fonctions holomorphes dans un cercle, Comptes Rendus de l’Académie des Sciences de Paris, vol. 198 (1934), pp. 2065–2067 Google Scholar, and 2. Sur les singularités de certaines fonctions holomorphes et de leurs inverses, Journal de Mathématiques pures et appliquées (9), vol. 15 (1936), pp. 423-435.

2) Bagemihl, F. and Seidel, W.. Spiral and other asymptotic paths, and paths of complete indetermination, of analytic and meromorphic functions, Proceedings of the National Academy of Sciences, vol 39 (1953), pp. 1251–1258 CrossRefGoogle ScholarPubMed.

3) Bagemihl, F. and Seidel, W.. Some boundary properties of analytic functions, Mathematische Zeitschrift, vol. 61 (1954), pp. 186–199 Google Scholar.

4) In the sequel, we shall abbreviate the expression “non-Euclidean” to n—E. For the facts concerning n—E geometry which we shall employ in this paper, see, for example, Carathéodory, C.. Conformal Representation, second edition, Cambridge, University Press, 1952, Chapter II.Google Scholar

5) See, for example, Collingwood, E. F. and Cartwright, M. L.. Boundary theorems for a function meromorphic in the unit circle, Acta Mathematica, vol. 87 (1952), pp. 83–146.Google Scholar

6) Valiron 2., loc. cit., pp. 433-435.

7) Bagemihl, F., Erdös, P., Seidel, W.. Sur quelques propriétés frontières des fonctions holomorphes définies par certains produits dans le cercle-unité, Annales de l’École Normale Supérieure (3), vol. 70 (1953), pp. 135–147; in particular, pp. 136–141 Google Scholar.

8) The extreme cases must be interpreted to mean that, in the first case, conclusion 2 holds for every Stolz angle Δτ, 3 while, in the second case, conclusion 1 holds for every Stolz angle Δτ, 3.

9) A point τ, with |τ| = 1, is called a Fatou point of a function f(z), meromorphic in |Z|<1, if there exists a number c, finite or infinite, such that f(z)→c as z→τ uniformly in every Stolz angle Δτ, a. Cf. Collingwood and Cartwright, loc. cit., p. 95.

10) It clearly suffices to assume here that f(z) assumes two distinct finite values at most a finite number of times. An analogous extension of Corollary 4 is also evidently true.

11) Gross, W.. Über die Singularitäten analytischer Filnktionen, Monatshefte für Mathematik und Physik, vol. 29 (1918), pp. 3–47; particularly, p. 26 CrossRefGoogle Scholar.

12) Plessner, A.. Über das Verhalten analytischer Funktionen am Rande ihres Definitionsbereiches, Journal fÜr die reine und angewandte Mathematik, vol. 158 (1927), pp. 219–227 Google Scholar.

13) For the definition of this term, see e.g. Collingwood and Cartwright, loc. cit., p. 139.

14) It is to be noted that both Theorems 6 and 7 are valid for the same function.