Abstract
LetD⊂C be a simply connected domain that contains 0 and does not contain any disk of radius larger than 1. ForR>0, letω D (R) denote the harmonic measure at 0 of the set {z:|z|≽R}⋔∂D. Then it is shown thatthere exist β>0and C>0such that for each such D,ω D (R)≤Ce −βR,for every R>0. Thus a natural question is: What is the supremum of all β′s , call it β0, for which the above inequality holds for every suchD? Another formulation of the problem involves hyperbolic metric instead of harmonic measure. Using this formulation a lower bound for β0 is found. Upper bounds for β0 can be obtained by constructing examples of domainsD. It is shown that a certain domain whose boundary consists of an infinite number of vertical half-lines, i.e. a comb domain, gives a good upper bound. This bound disproves a conjecture of C. Bishop which asserted that the strips of width 2 are extremal domains. Harmonic measures on comb domains are also studied.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
[A]Akhiezer, N. I.,Elements of the Theory of Elliptic Functions, Amer. Math. Soc., Providence, R. I., 1990.
[B1]Baernstein, A. II, Analytic functions of bounded mean oscillation, inAspects of Contemporary Complex Analysis (Brannan, D. A. and Clunie, J., eds.), pp. 3–36, Academic Press, New York, 1980.
[B2]Baernstein, A. II. Unpublished manuscript, 1981.
[B3]Baernstein, A. II, The size of the set where a univalent function is large, Preprint, 1996.
[BC]Bañuelos, R. andCarroll, T., Brownian motion and the fundamental frequency of a drum,Duke Math. J. 75 (1994), 575–602.
[Be]Beurling, A.,The Collected Works of Arne Beurling, Vol. 1, Complex Analysis, Birkhäuser, Boston, Mass., 1989.
[Bi]Bishop, C. J., How geodesics approach the boundary in a simply connected domain,J. Anal. Math.,64 (1994), 291–325.
[D]Duren, P. L.,Theory of H p Spaces, Academic Press, New York, 1970.
[FM]Feng, J. andMacGregor, T. H., Estimates on the integral means of the derivatives of univalent functions,J. Anal. Math.,29 (1976), 203–231.
[Go]Goodman, R. E., On the Bloch-Landau constant for schlicht functions,Bull. Amer. Math. Soc. 51 (1945), 234–239.
[Gw]Gwilliam, A. E., On Lipschitz conditions,Proc. London Math. Soc. 40 (1936), 353–364.
[HK]Hayman, W. K. andKennedy, P. B.,Subharmonic Functions, Vol. 1, Academic Press, London, 1976.
[HP]Hayman, W. K. andPommerenke, C., On analytic functions of bounded mean oscillation,Bull. London Math. Soc.,10 (1978), 219–224.
[K]Koosis, P.,The Logarithmic Integral II, Cambridge Univ. Press, Cambridge, 1992.
[N]Nevanlinna, R.,Eindeutige analytische Funktionen, 2nd ed., Springer-Verlag, Berlin, 1953. English transl.:Analytic Functions, Springer-Verlag, New York, 1970.
[O]Ohtsuka, M.,Dirichlet Problem, Extremal Length and Prime Ends, Van Nostrand, New York, 1970.
[P]Pommerenke, C.,Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin-Heidelberg, 1992.
[Z]Zhang, S. Y., On the schlicht Bloch constant,Beijing Daxue Xuebao 25 (1989), 537–540.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Betsakos, D. Harmonic measure on simply connected domains of fixed inradius. Ark. Mat. 36, 275–306 (1998). https://doi.org/10.1007/BF02384770
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02384770