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Some Remarks on Boundary Behavior of Analytic and Meromorphic Functions

Published online by Cambridge University Press:  22 January 2016

F. Bagemihl  and
W. Seidel
F. Bagemihl
Affiliation:
The Institute for Advanced Study, University of Notre Dame
W. Seidel
Affiliation:
The Institute for Advanced Study, University of Notre Dame
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This paper is concerned with regular and meromorphic functions in |z| < 1 and their behavior near |z| = 1. Among the results obtained are the following. In section 2 we prove the existence of a non-constant meromorphic function that tends to zero at every point of |z| = 1 along almost all chords of |z| < 1 terminating in that point. Section 3 deals with the impossibility of ex tending this result to regular functions. In section 4 it is shown that a regular function can tend to infinity along every member of a set of spirals approach ing |z| = 1 and exhausting |z| < 1 in a simple manner. Finally, in section 5 we prove that this set of spirals cannot be replaced by an exhaustive set of Jordan arcs terminating in points of |z| = 1; Theorem 3 of this section can be interpreted as a uniqueness theorem for meromorphic functions.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 9 , October 1955 , pp. 79 - 85
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1955

References

1) By this we mean, more pr ecisely, that if φ denotes the angle which a chord terminating in eiθ makes with the radius at eiθ , where — π/2 < φ π/2 and we observe the usual convention regarding the sign of φ, then the set of values of φ for which f(z) → ∞ along the corresponding chord is of Lebesgue measure π.

2) The use of a product of this form was suggested to us by a product which P. Erdös proposed for another purpose.

3) We remark that f(z) also possesses the properties described in Theorems 1 to 4 of Bagemihl, F., Erdös, P., and Seidel, W.: Sur quelques propriétés frontières des fonctions holomorphes définies par certains produits dans le cercle-unité, Ann. Sci. École Norm. Sup. (3) 70 (1953), pp. 135–147.CrossRefGoogle Scholar

4) This result was announced in Bagemihl, F. and Seidel, W.: A general principle involving Baire category, with applications to function theory and other fields, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), pp. 1068–1075 (see pp. 1073–1074).CrossRefGoogle ScholarPubMed

5) Meier, K.: Über die Randwerte meromorpher Funktionen und hinreichende Bedingungen für Regularitãt von Funktionen einer komplexen Variablen, Comm. Math. Helv. 24 (1950), pp. 238–259 (see p. 241).CrossRefGoogle Scholar

6) Cf.Lusin, N. and Priwaloff, J.: Sur l’unicité et la multiplicité des fonctions analytiques, Ann. Sci. École Norm. Sup. (3) 42 (1925), pp. 143–191 (see p. 164).CrossRefGoogle Scholar

7) We regard Q as belonging to J 0 but to no other J 0 (0 < θ < 2 π).

8) By continuous we mean continuous in the extended sense, so that, e.g., a meromorphic function is to be regarded as continuous at a pole.

9) A subset of A is of first category if it is the union of enumerably many nowhere dense subsets of A, and a residual subset of A is the complement (with respect to A) of a subset of A of first category.

10) We take this opportunity to note that, in the aforementioned paper of the present authors’, Theorems 5, 6, and 9, which are stated there for meromorphic functions, are obviously valid if the functions are assumed to be merely continuous.

11) For the notion of harmonic measure, see, e.g., Nevanlinna, R.: Eindeutige analytische Funktionen, 2nd ed., Berlin-Göttingen-Heidelberg, 1953.CrossRefGoogle Scholar

12) Bagemihl, F.: Curvilinear cluster sets of arbitrary functions, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), pp. 379–382, Theorem 2.Google Scholar

13) Cf. R. Nevanlinna: loc. cit. 11) p. 209.