Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-25T13:29:52.368Z Has data issue: false hasContentIssue false

On the Boundary Behavior of Holomorphic Functions in the Unit Disk

Published online by Cambridge University Press:  22 January 2016

Claude Marie Faust C. C. V. I.*
Affiliation:
University of Notre Dame, Notre Dame, Indiana and Incarnate Word College, San Antonio, Texas
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let f(z) be a holomorphic function defined in the unit disk |z|<1, which we shall denote by D. Let Σ be a subset of D, whose closure has at least one point in common with C, the circumference of the unit disk. The set of all values a such that the equation f(z) = a has infinitely many solutions in Σ is called the range of f(z) in Σ, and is denoted by R(f, Σ). Let τ be a point of C, and let {zn) be a sequence of points in D with the properties: . The non-Euclidean (hyperbolic) distance ρ(zn, zn+1) between two points zn and zn+1 of the sequence is defined to be equal to

(cf.[3], Ch. II).

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1962

References

[1] Bagemihl, F. and Seidel, W., Sequential and continuous limits of meromorphic functions, Annales Academiae Scientiarum Fennicae, Series A I, No. 280 (1960), pp. 117.Google Scholar
[2] Blaschke, W., Eine Erweiterung des Satzes von Vitali über Folgen analytischer Funktionen, Leipziger Berichte vol. 67 (1915), pp. 194200.Google Scholar
[3] Carathéodory, C., Conformal Representation, second edition, Cambridge, University Press, 1952, Chapter II.Google Scholar
[4] Gattegno, C. and Ostrowski, A., Representation Conforme à la Frontière. Domaines Généraux, Mémorial des Sciences Mathématiques, No. 109, (1949).Google Scholar
[5] Julia, G., Leçons sur les fonctions uniformes à point singulier essentiel isolé, Paris, 1924, p. 102 ff.Google Scholar
[6] Montel, Paul, Leçons sur les familles normales de fonctions analytiques et leurs applications, Paris: Gauthiers-Villars, 1927.Google Scholar
[7] Noshiro, Kiyoshi, Cluster Sets, Berlin-Göttingen-Heidelberg, 1960.Google Scholar
[8] Piranian, G., The boundary of a simply connected domain, Bulletin of the American Mathematical Society, 64, (1958), pp. 4555.Google Scholar
[9] Seidel, W., Holomorphic functions with spiral asymptotic paths, Nagoya Mathematical Journal, vol. 14 (1959), pp. 159171.Google Scholar