Hostname: page-component-669899f699-chc8l Total loading time: 0 Render date: 2025-04-28T09:27:43.007Z Has data issue: false hasContentIssue false
  • English
  • Français

On a Property of Harmonic Measure on Simply Connected Domains

Part of: Geometric function theory Riemann surfaces Two-dimensional theory

Published online by Cambridge University Press:  22 November 2019

Christina Karafyllia
Christina Karafyllia*
Affiliation:
Department of Mathematics, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece Email: karafyllc@math.auth.gr
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $D\subset \mathbb{C}$ be a domain with $0\in D$. For $R>0$, let $\widehat{\unicode[STIX]{x1D714}}_{D}(R)$ denote the harmonic measure of $D\cap \{|z|=R\}$ at $0$ with respect to the domain $D\cap \{|z|<R\}$ and let $\unicode[STIX]{x1D714}_{D}(R)$ denote the harmonic measure of $\unicode[STIX]{x2202}D\cap \{|z|\geqslant R\}$ at $0$ with respect to $D$. The behavior of the functions $\unicode[STIX]{x1D714}_{D}$ and $\widehat{\unicode[STIX]{x1D714}}_{D}$ near $\infty$ determines (in some sense) how large $D$ is. However, it is not known whether the functions $\unicode[STIX]{x1D714}_{D}$ and $\widehat{\unicode[STIX]{x1D714}}_{D}$ always have the same behavior when $R$ tends to $\infty$. Obviously, $\unicode[STIX]{x1D714}_{D}(R)\leqslant \widehat{\unicode[STIX]{x1D714}}_{D}(R)$ for every $R>0$. Thus, the arising question, first posed by Betsakos, is the following: Does there exist a positive constant $C$ such that for all simply connected domains $D$ with $0\in D$ and all $R>0$,

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{D}(R)\geqslant C\widehat{\unicode[STIX]{x1D714}}_{D}(R)?\end{eqnarray}$$
In general, we prove that the answer is negative by means of two different counter-examples. However, under additional assumptions involving the geometry of $D$, we prove that the answer is positive. We also find the value of the optimal constant for starlike domains.

Keywords

Harmonic measureconformal mappinghyperbolic distance

MSC classification

Primary: 30C85: Capacity and harmonic measure in the complex plane
Secondary: 30F45: Conformal metrics (hyperbolic, Poincaré, distance functions) 30C35: General theory of conformal mappings 31A15: Potentials and capacity, harmonic measure, extremal length
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 2 , April 2021 , pp. 297 - 317
Check for updates
Copyright
© Canadian Mathematical Society 2019

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

Ahlfors, L. V., Conformal invariants: topics in geometric function theory. McGraw-Hill, New York, 1973.Google Scholar
Baernstein, A., The size of the set where a univalent function is large. J. Anal. Math. 70(1996), 157–173. https://doi.org/10.1007/BF02820443CrossRefGoogle Scholar
Balogh, Z. and Bonk, M., Lengths of radii under conformal maps of the unit disk. Proc. Amer. Math. Soc. 127(1999), 801–804. https://doi.org/10.1090/S0002-9939-99-04565-7CrossRefGoogle Scholar
Beardon, A. F. and Minda, D., The hyperbolic metric and geometric function theory. In: Quasiconformal mappings and their applications, Narosa, New Delhi, 2007, pp. 9–56.Google Scholar
Betsakos, D., Harmonic measure on simply connected domains of fixed inradius. Ark. Mat. 36(1998), 275–306. https://doi.org/10.1007/BF02384770CrossRefGoogle Scholar
Betsakos, D., Geometric theorems and problems for harmonic measure. Rocky Mountain J. Math. 31(2001), 773–795. https://doi.org/10.1216/rmjm/1020171668CrossRefGoogle Scholar
Beurling, A., The collected works of Arne Beurling. Vol. 1, Complex analysis. Birkhäuser, Boston, 1989.Google Scholar
Essén, M., On analytic functions which are in H p for some positive p. Ark. Mat. 19(1981), 43–51. https://doi.org/10.1007/BF02384468CrossRefGoogle Scholar
Essén, M., Harmonic majorization and thinness. In: Proceedings of the 14th Winter School on abstract analysis. Rend. Circ. Mat. Palermo (2) Suppl. No. 14, 1987, pp. 295–304.Google Scholar
Essén, M., Haliste, K., Lewis, J. L., and Shea, D. F., Harmonic majorization and classical analysis. J. London Math. Soc. 32(1985), 506–520. https://doi.org/10.1112/jlms/s2-32.3.506CrossRefGoogle Scholar
Garnett, J. B. and Marshall, D. E., Harmonic measure. Cambridge University Press, Cambridge, 2005. https://doi.org/10.1017/CBO9780511546617CrossRefGoogle Scholar
Hayman, W. K., Subharmonic functions. Vol. 2, Academic Press, London, 1989, pp. 285–875.Google Scholar
Hayman, W. K. and Weitsman, A., On the coefficients and means of functions omitting values. Math. Proc. Cambridge Philos. Soc. 77(1975), 119–137. https://doi.org/10.1017/S030500410004946XCrossRefGoogle Scholar
Jørgensen, V., On an inequality for the hyperbolic measure and its applications in the theory of functions. Math. Scand. 4(1956), 113–124. https://doi.org/10.7146/math.scand.a-10460CrossRefGoogle Scholar
Karafyllia, C., On a relation between harmonic measure and hyperbolic distance on planar domains. Indiana Univ. Math. J., to appear. arxiv:1908.11830.Google Scholar
Karafyllia, C., On the Hardy number of a domain in terms of harmonic measure and hyperbolic distance. Ark. Mat., to appear. arxiv:1908.11845.Google Scholar
Kim, Y. C. and Sugawa, T., Hardy spaces and unbounded quasidisks. Ann. Acad. Sci. Fenn. 36(2011), 291–300. https://doi.org/10.5186/aasfm.2011.3618CrossRefGoogle Scholar
Minda, D., Inequalities for the hyperbolic metric and applications to geometric function theory. Lecture Notes in Math., 1275, Springer, Berlin, 1987, pp. 235–252.. https://doi.org/10.1007/BFb0078356Google Scholar
Poggi-Corradini, P., Geometric models, iteration and composition operators. Ph.D. Thesis, University of Washington, 1996.Google Scholar
Poggi-Corradini, P., The Hardy class of geometric models and the essential spectral radius of composition operators. J. Funct. Anal. 143(1997), 129–156. https://doi.org/10.1006/jfan.1996.2978CrossRefGoogle Scholar
Poggi-Corradini, P., The Hardy class of Kœnigs maps. Michigan Math. J. 44(1997), 495–507. https://doi.org/10.1307/mmj/1029005784Google Scholar
Pommerenke, C., Boundary behaviour of conformal maps. Grundlehren der Mathematischen Wissenschaften, 299, Springer-Verlag, Berlin, 1992. https://doi.org/10.1007/978-3-662-02770-7CrossRefGoogle Scholar
Port, S. C. and Stone, C. J., Brownian motion and classical potential theory. Probability and Mathematical Statistics. Academic Press, New York–London, 1978.Google Scholar
Ransford, T., Potential theory in the complex plane. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
Sakai, M., Isoperimetric inequalities for the least harmonic majorant of |x|p. Trans. Amer. Math. Soc. 299(1987), 431–472. https://doi.org/10.2307/2000507Google Scholar
Solynin, A. Yu., The boundary distortion and extremal problems in certain classes of univalent functions. J. Math. Sci. 79(1996), 1341–1358. https://doi.org/10.1007/BF02366464Google Scholar
Solynin, A. Yu., Functional inequalities via polarization. St. Petersburg Math. J. 8(1997), 1015–1038.Google Scholar
Tsuji, M., A theorem on the majoration of harmonic measure and its applications. Tohoku Math. J. (2) 3(1951), 13–23. https://doi.org/10.2748/tmj/1178245553CrossRefGoogle Scholar
Tsuji, M., Potential theory in modern function theory. Maruzen, Tokyo, 1959.Google Scholar