Hostname: page-component-7dd5485656-2pp2p Total loading time: 0 Render date: 2025-10-31T04:02:31.141Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuous Boundary Behaviour for Functions Defined in the Open Unit Disc

Published online by Cambridge University Press:  22 January 2016

Leon Brown ,
P. M. Gauthier  and
Walter Hengartner
Leon Brown
Affiliation:
Wayne State University, Université de Montréal, Université Laval
P. M. Gauthier
Affiliation:
Wayne State University, Université de Montréal, Université Laval
Walter Hengartner
Affiliation:
Wayne State University, Université de Montréal, Université Laval
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.

This paper deals with cluster sets. While cluster sets can be considered in a more abstract setting, we shall limit ourselves to the study of functions f defined in the open unit disc D of the complex plane and taking their values on the Riemann sphere C̅. For p a point of the unit circle C, we denote by C(f,p) the cluster set of f at p, i.e., the set of all values w ∈ C̅ for which there is a sequence {zn}, zn ∈ D, such that zn → p and f(zn) → w. The point p is called a point of determination for f if C(f, p) is a singleton. In Section 1 we characterize the set of points of determination of a function f defined in D.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 57 , June 1975 , pp. 49 - 58
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1975

References

[1] Allen, H.: Distinct holomorphic functions with idetical boundary values (preprint).Google Scholar
[2] Cima, J. A., Rung, D. C.: Normal functions and a class of associated boundary functions. Israel J. Math. 4 (1966), 119–126. MR 34 #6019.CrossRefGoogle Scholar
[3] Clunie, J.: On a problem of Gauthier. Mathematika 18 (1971), 126–139.Google Scholar
[4] Collingwood, E. F., Lohwater, A. J.: The theory of cluster sets. Cambridge University Press, Cambridge, 1966. MR 38 #325.Google Scholar
[5] Colwell, Peter: Cluster set theorems for uniformly convergent sequences of functions. Proc. Amer. Math. Soc. 18 (1967), 48–52. MR 37 #5398.Google Scholar
[6] Dollinger, M. B.: Some aspects of spectral theory on banach spaces. Doctoral dissertation, University of Illinois, 1968.Google Scholar
[7] Fort, M. K. Jr.: A unified theory of semi-continuity. Duke Math. J. 16 (1949), 237–246. MR 10–716.Google Scholar
[8] Gauthier, P. M., Hengartner, Walter: Local harmonic majorants of functions subharmonic in the unit disc. J. Analyse Math. 26 (1973), 405–412.Google Scholar
[9] Heins, Maurice: The boundary values of meromorphic functions having bounded Nevanlinna characteristic. J. Analysis Math. 18 (1967), 121–131. MR 35 #4401.Google Scholar
[10] Kierst, S., Szpilrajn, E.: Sur certaines singularités des fonctions analytiques uniformes. Fund. Math. 21 (1933), 276–294.Google Scholar
[11] Koo, Shu-Chung: Some properties of functions and cluster sets (unpublished).Google Scholar
[12] Lappan, Peter: Some results on harmonic normal functions. Math. Z. 90 (1965), 155–159. MR 35 #397.Google Scholar
[13] Piranian, George; Titus, C. J. and Young, G. S.: Conformal mappings and Peano curves. Michigan Math. J. 1 (1952), 69–72. MR 14–262.Google Scholar
[14] Rudin, W.: Boundary values of continuous analytic functions, Proc. A.M.S., 7 (1956), 808–811. MR 18–472.Google Scholar
[15] Salem, Raphael and Zygmund, A.: Lacunary power series and Peano curves. Duke Math. J. 12 (1945), 569–578. MR 7–378.Google Scholar