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)