Hostname: page-component-6bb9c88b65-vmslq Total loading time: 0 Render date: 2025-07-21T05:12:47.869Z Has data issue: false hasContentIssue false
  • English
  • Français

An estimate of harmonic measure in Ṟd, d ≧ 2

Published online by Cambridge University Press:  14 February 2012

Matts Essén
Matts Essén
Affiliation:
Department of Mathematics, Royal Institute of Technology, Stockholm
Get access Rights & Permissions [Opens in a new window]

Synopsis

Let D be an open connected subset of the open unit ball in Ṟd, d ≧ 2. We give an estimate of the harmonic measure of ∂D∩{|x| = 1} with respect to D. This estimate depends in a simple way on the geometry of D. An essential tool is a rearrangement theorem for differential inequalities. When d = 2, examples are given which illustrate the precision of the results.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 78 , Issue 1-2 , 1977 , pp. 129 - 138
Copyright
Copyright © Royal Society of Edinburgh 1977

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1Baernstein, A. II and Taylor, B. A.Spherical rearrangements, subharmonic functions and functions in n-space. Duke Math. J. 43 (1976), 245–268.CrossRefGoogle Scholar
2Beurling, A.Études sur un problème de majoration (Uppsala Univ. Thèse, 1933).Google Scholar
3Carleman, T.Sur une inégalité différentielle dans la théorie des fonctions analytiques. C.R. Acad. Sci. Paris 196 (1933), 995–997.Google Scholar
4Essén, M.The cos πλ Theorem. Lecture Notes in Mathematics 467 (Berlin: Springer, 1975).Google Scholar
5Essén, M. A generalization of Beurling's estimate of harmonic measure. Proc. Summer Course in Complex Analysis (Trieste: Internat. Centre Theoret. Phys., 1975).Google Scholar
6Essén, M. and Everitt, W. N.A singular integral inequality associated with a problem in estimating harmonic measure, to appear.Google Scholar
7Friedland, S. and Hayman, W. K.Eigenvalue inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions. Comment. Math. Helv. 51 (1976), 133–161.CrossRefGoogle Scholar
8Hayman, W. K. On the domains where a harmonic or subharmonic function is positive. Lecture Notes in Mathematics 505, 24–55 (Berlin: Springer, 1976).Google Scholar
9Hayman, W. K. and Weitsman, A.On the coefficients and means of functions omitting values. Math. Proc. Cambridge Philos. Soc. 77 (1975), 119–137.CrossRefGoogle Scholar
10Heins, M.On a notion of convexity connected with a method of Carleman. J. Analyse Math. 7 (1960), 53–77.CrossRefGoogle Scholar
11Huber, A.Über Wachstumseigenschaften gewisser Klassen von subharmonischen Funktionen. Comment. Math. Helv. 26 (1952), 81–116.CrossRefGoogle Scholar
12Tsuji, M.Potential theory in modern function theory (Tokyo: Maruzen, 1959).Google Scholar
13Sperner, E.Zur Symmetrisierung von Funktionen auf Sphären. Math. Z. 134 (1973), 317–327.CrossRefGoogle Scholar