Extension of domain, principle of
Carleman's principle
The harmonic measure of an arc
of the boundary
of a domain
can only increase when
is extended across arcs
,
. More precisely, let the boundary
of a domain
in the complex
-plane consist of a finite number of Jordan curves, let
be a part of
consisting of a finite number of arcs of
, and let
be an extension of the domain
across the complementary arcs
, that is,
and
is a part of the boundary
of
. Then for the harmonic measures one has the inequality
,
, where equality only holds if
. The principle of extension of domain also holds for harmonic measure on domains in the Euclidean space
,
, or
,
.
The principle of extension of domain finds important applications in various problems concerned with estimating harmonic measure. For example, T. Carleman [1] used the principle of extension of domain to obtain the solution of the Carleman–Milloux problem: Let the boundary of a simply-connected domain
consist of a finite number of Jordan arcs, let
be a point on
, or let
, let
be the disc of radius
with centre
, and let
be the part of
in
. It is required to find a lower bound for the harmonic measure
depending only on
and
,
. The solution is given by
![]() | (1) |
where is the sum of the lengths of arcs of the intersection
![]() |
Since , it follows that
![]() | (2) |
There exist generalizations of the Carleman–Milloux problem and refinements of formulas (1), (2) (see [3]). The principle of extension of domain also allows one to prove the Lindelöf theorems (cf. 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 [2], and also [3], [4]).
References
[1] | T. Carleman, "Sur les fonctions inverses des fonctions entières" Ark. Mat. Ast. Fys. , 15 : 10 (1921) |
[2] | H. Milloux, "Le théorème de M. Picard, suites des fonctions holomorphes, fonctions méromorphes et fonctions entières" J. Math. Pures Appl. , 3 (1924) |
[3] | R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German) |
[4] | M.A. Evgrafov, "Analytic functions" , Saunders , Philadelphia (1966) (Translated from Russian) |
Extension of domain, principle of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Extension_of_domain,_principle_of&oldid=15054