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''Carleman's principle''

The [[Harmonic measure|harmonic measure]]  $  \omega ( z, \alpha , D) $
of an arc  $  \alpha $
of the boundary  $  \Gamma $
of a domain  $  D $
can only increase when  $  D $
is extended across arcs  $  \beta \subset  \Gamma $, 
$  \alpha \cup \beta = \Gamma $. 
More precisely, let the boundary  $  \Gamma $
of a domain  $  D $
in the complex  $  z $-
plane consist of a finite number of Jordan curves, let  $  \alpha $
be a part of  $  \Gamma $
consisting of a finite number of arcs of  $  \Gamma $, 
and let  $  D  ^  \prime  $
be an extension of the domain  $  D $
across the complementary arcs  $  \beta = \Gamma \setminus  \alpha $, 
that is,  $  D \subset  D  ^  \prime  $
and  $  \alpha $
is a part of the boundary  $  \Gamma  ^  \prime  $
of  $  D  ^  \prime  $. 
Then for the harmonic measures one has the inequality  $  \omega ( z, \alpha , D) \leq  \omega ( z, \alpha  , D  ^  \prime  ) $, 
$  z \in D $, 
where equality only holds if  $  D  ^  \prime  = D $. 
The principle of extension of domain also holds for harmonic measure on domains in the Euclidean space  $  \mathbf R  ^ {n} $, 
$  n \geq  2 $, 
or  $  \mathbf C  ^ {n} $, 
$  n \geq  1 $.

The principle of extension of domain finds important applications in various problems concerned with estimating harmonic measure. For example, T. Carleman [[#References|[1]]] used the principle of extension of domain to obtain the solution of the Carleman–Milloux problem: Let the boundary  $  \Gamma $
of a simply-connected domain  $  D $
consist of a finite number of Jordan arcs, let  $  \zeta $
be a point on  $  \Gamma $, 
or let  $  \zeta \notin \overline{D}\; $, 
let  $  \Delta = \{ {z } : {| z - \zeta | < R } \} $
be the disc of radius  $  R $
with centre  $  \zeta $, 
and let  $  \alpha $
be the part of  $  \Gamma $
in  $  \Delta _ {R} = \Delta \cap D $. 
It is required to find a lower bound for the harmonic measure  $  \omega ( z, \alpha , \Delta _ {R} ) $
depending only on  $  R $
and  $  | z - \zeta | $, 
$  z \in \Delta _ {R} $. 
The solution is given by

$$ \tag{1 }
\omega ( z, \alpha , \Delta _ {R} )  \geq  \ 
{
\frac{2} \pi 
 }   \mathop{\rm arc}   \mathop{\rm tan} \ 
\left (
{
\frac{2}{\theta ( R) }
 }   \mathop{\rm ln} \ 
{
\frac{R}{| z - \zeta | }
 }
\right ) ,
$$

where  $  R \theta ( R) $
is the sum of the lengths of arcs of the intersection

$$ 
\{ {z } : {| z - \zeta | = R } \}
\cap D.
$$

Since  $  \theta ( R) \leq  2 \pi $, 
it follows that

$$ \tag{2 }
\omega ( z, \alpha , \Delta _ {R} )  \geq  \ 
{
\frac{2} \pi 
 }   \mathop{\rm arc}   \mathop{\rm tan} \ 
\left ( {
\frac{1} \pi 
 }   \mathop{\rm ln} \ 
{
\frac{R}{| z - \zeta | }
 }
\right ) ,\ \ 
z \in \Delta _ {R} .
$$

There exist generalizations of the Carleman–Milloux problem and refinements of formulas (1), (2) (see [[#References|[3]]]). The principle of extension of domain also allows one to prove the Lindelöf theorems (cf. [[Lindelöf theorem|Lindelöf theorem]]). Various applications of the principle of extension of domain and of formulas of the type (1), (2) were given by H. Milloux (see [[#References|[2]]], and also [[#References|[3]]], [[#References|[4]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T. Carleman,   "Sur les fonctions inverses des fonctions entières"  ''Ark. Mat. Ast. Fys.'' , '''15''' :  10  (1921)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Milloux,   "Le théorème de M. Picard, suites des fonctions holomorphes, fonctions méromorphes et fonctions entières"  ''J. Math. Pures Appl.'' , '''3'''  (1924)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Nevanilinna,   "Analytic functions" , Springer  (1970)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.A. Evgrafov,   "Analytic functions" , Saunders , Philadelphia  (1966)  (Translated from Russian)</TD></TR></table>