Let
E
E
be a continuum in the closed unit disk
|
z
|
≤
1
|z|\le 1
of the complex
z
z
-plane which divides the open disk
|
z
|
>
1
|z| > 1
into
n
≥
2
n\ge 2
pairwise nonintersecting simply connected domains
D
k
D_k
such that each of the domains
D
k
D_k
contains some point
a
k
a_k
on a prescribed circle
|
z
|
=
ρ
|z| = \rho
,
0
>
ρ
>
1
,
k
=
1
,
…
,
n
.
0 >\rho >1\, , \, k=1,\dots ,n\,.
It is shown that for some increasing function
Ψ
,
\Psi \,,
independent of
E
E
and the choice of the points
a
k
,
a_k,
the mean value of the harmonic measures
\[
Ψ
−
1
[
1
n
∑
k
=
1
k
Ψ
(
ω
(
a
k
,
E
,
D
k
)
)
]
\Psi ^{-1}\left [ \frac {1}{n} \sum _{k=1}^{k} \Psi ( \omega (a_k,E, D_k))\right ]
\]
is greater than or equal to the harmonic measure
ω
(
ρ
,
E
∗
,
D
∗
)
,
\omega (\rho , E^* , D^*)\,,
where
E
∗
=
{
z
:
z
n
∈
[
−
1
,
0
]
}
E^* = \{ z: z^n \in [-1,0] \}
and
D
∗
=
{
z
:
|
z
|
>
1
,
|
arg
z
|
>
π
/
n
}
.
D^* =\{ z: |z|>1, |\textrm {arg}\, z| > \pi /n\} \,.
This implies, for instance, a solution to a problem of R. W. Barnard, L. Cole, and A. Yu. Solynin concerning a lower estimate of the quantity
inf
E
max
k
=
1
,
…
,
n
ω
(
a
k
,
E
,
D
k
)
\inf _{E} \max _{k=1,\dots ,n} \omega (a_k,E, D_k)\,
for arbitrary points of the circle
|
z
|
=
ρ
.
|z| = \rho \,.
These authors stated this hypothesis in the particular case when the points are equally distributed on the circle
|
z
|
=
ρ
.
|z| = \rho \,.