Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-23T22:57:44.339Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometric Proofs of some Classical Results on Boundary Values for Analytic Functions

Published online by Cambridge University Press:  20 November 2018

Enrique Villamor
Enrique Villamor*
Affiliation:
Department of Mathematics, Florida International University Miami, Florida 33199 U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

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.

In this note we are going to give a geometric proof, using the method of the extremal metric, of the following result of Beurling. For any analytic function f(z) in the unit disc Δ of the plane with a bounded Dirichlet integral, the set E on the boundary of the unit disc where the nontangential limits of f(z) do not exist has logarithmic capacity zero. Also, using an unpublished result of Beurling, we will prove different results on boundary values for different classes of functions.

Keywords

30C45Areally mean p-valentextremal metriclogarithmic capacity
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 37 , Issue 2 , 01 June 1994 , pp. 263 - 269
Copyright
Copyright © Canadian Mathematical Society 1994

References

[BEU] Beurling, A., Ensembles exceptionnels, Acta Math. 72(1940), 1–13.Google Scholar
[CAR] Carleson, L., Sets of uniqueness for functions analytic in the unit disc, Acta Math. 87(1952), 325–345.Google Scholar
[CL] Colling, E. F. wood and Loh water, A. J., Theory of Cluster Sets, University Press, 1966.Google Scholar
[JO] Jenkins, J. A. and Oikawa, K., On the growth of slowly increasing unbounded harmonic functions, Acta Math. 124(1970), 37–63.Google Scholar
[KOS] Koskela, P., A radial uniqueness theorem for Sobolev functions, preprint.Google Scholar
[LV] Lento, O. and Virtanen, K. I., Boundary behavior and normal meromorphic functions, Acta Math. 97 (1957), 47–65.Google Scholar
[MAR] Marshall, D. E., A new proof of a sharp inequality concerning the Dirichlet integral, Ark. Mat. 27 (1989), 131–137.Google Scholar
[PF] Pflüger, A., Extremallängen und kapazität, Comment. Math. Helv. 29(1955), 120–131.Google Scholar
[PO] Pommerenke, Ch., On normal and automorphic functions, Michigan Math. J. 21(1974), 193–202.Google Scholar
[TSU] Tsuji, M., Beurling's theorem on exceptional sets, Tôhoku Math. J. 2(1950), 113–125.Google Scholar
[VII] Villamor, E., On slowly increasing unbounded harmonic functions, Kodai Math. J. 14(1991), 222–238.Google Scholar
[VI2] Villamor, E., Boundary values for some classes of functions, J. London Math. Soc. 45(1992), 508–518.Google Scholar
[VI3] Villamor, E., On uniqueness sets for areally mean p-valent functions, Proc. Amer. Math. Soc, to appear.Google Scholar