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.
p Harmonic Measure in Simply Connected Domains