Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On a relation between harmonic measure and hyperbolic distance on planar
  domains   Let $\psi $ be a conformal map of $\mathbb{D}$ onto an unbounded domain and,
for $\alpha >0$, let ${F_\alpha }=\left\{ {z \in \mathbb{D}:\left| {\psi \left(
z \right)} \right| = \alpha } \right\}$. If $\omega _\mathbb{D}\left(
{0,{F_\alpha }} \right)$ denotes the harmonic measure at $0$ of $F_\alpha $ and
$d_\mathbb{D} {\left( {0,{F_\alpha }} \right)}$ denotes the hyperbolic distance
between $0$ and $F_\alpha$ in $\mathbb{D}$, then an application of the
Beurling-Nevanlinna projection theorem implies that ${\omega _\mathbb{D}}\left(
{0,{F_\alpha }} \right) \ge \frac{2}{\pi }{e^{ - {d_\mathbb{D}}\left(
{0,{F_\alpha }} \right)}}$. Thus a natural question, first stated by P.
Poggi-Corradini, is the following: Does there exist a positive constant $K$
such that for every $\alpha >0$, ${\omega _\mathbb{D}}\left( {0,{F_\alpha }}
\right) \le K{e^{ - {d_\mathbb{D}}\left( {0,{F_\alpha }} \right)}}$? In
general, we prove that the answer is negative by means of two different
examples. However, under additional assumptions involving the number of
components of $F_\alpha$ and the hyperbolic geometry of the domain $\psi \left(
\mathbb{D} \right)$, we prove that the answer is positive.

8
The categories of this paper are: hep-ex
5
The categories of this paper are: gr-qc
15
The categories of this paper are: nucl-ex
23
The categories of this paper are: solv-int
26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
37
The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
19
The categories of this paper are: eess
20
The categories of this paper are: acc-phys
36
The categories of this paper are: cmp-lg
28
The categories of this paper are: alg-geom
24
The categories of this paper are: dg-ga
34
The categories of this paper are: chem-ph
17
The categories of this paper are: q-fin
25
The categories of this paper are: patt-sol