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On a Property of Harmonic Measure on Simply Connected Domains

Part of: Riemann surfaces Geometric function theory 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: [email protected]
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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
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Copyright
© Canadian Mathematical Society 2019

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