Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On a property of harmonic measure on simply connected domains   Let $D \subset \mathbb{C}$ be a domain with $0 \in D$. For $R>0$, let ${{\hat
\omega }_D}\left( {R} \right)$ denote the harmonic measure of $ D \cap \left\{
{\left| z \right| = R} \right\}$ at $0$ with respect to the domain $ D \cap
\left\{ {\left| z \right| < R} \right\} $ and ${\omega_D}\left( {R} \right)$
denote the harmonic measure of $\partial D \cap \left\{ {\left| z \right| \ge
R} \right\}$ at $0$ with respect to $D$. The behavior of the functions
${\omega_D}$ and ${{\hat \omega }_D}$ near $\infty$ determines (in some sense)
how large $D$ is. However, it is not known whether the functions ${\omega_D}$
and ${{\hat \omega }_D}$ always have the same behavior when $R$ tends to
$\infty$. Obviously, ${\omega_D}\left( {R} \right) \le {{\hat \omega }_D}\left(
{R} \right)$ 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 $R>0$,
\[{\omega_D}\left( {R} \right) \ge C{{\hat \omega }_D}\left( {R} \right)? \] 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.

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The categories of this paper are: gr-qc
9
The categories of this paper are: nlin
31
The categories of this paper are: ao-sci
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-ex
33
The categories of this paper are: plasm-ph
29
The categories of this paper are: funct-an
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The categories of this paper are: stat
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-ex
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The categories of this paper are: math-ph
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The categories of this paper are: q-bio
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The categories of this paper are: cmp-lg
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The categories of this paper are: solv-int
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The categories of this paper are: mtrl-th