Harmonic measure, principle of

The harmonic measure does not decrease under mappings realized by single-valued analytic functions. If $ \omega ( z; \alpha , D) $ is the harmonic measure of a boundary set $ \alpha $ with respect to a domain $ D $ in the complex $ z $- plane, one specific formulation of the principle of harmonic measure is as follows. In a domain $ D _ {z} $ with boundary $ \Gamma _ {z} $ consisting of a finite number of Jordan arcs let there be given a single-valued analytic function $ w = w( z) $ which satisfies the following conditions: the values $ w = w( z) $, $ z \in D _ {z} $, form part of the domain $ D _ {w} $ with boundary $ \Gamma _ {w} $ consisting of a finite number of Jordan arcs; the function $ w( z) $ can be continuously extended onto some set $ \alpha _ {z} \subset \Gamma _ {z} $ consisting of a finite number of arcs; and the values of $ w( z) $ on $ \alpha _ {z} $ form part of a set $ E \subset \overline{D}\; _ {w} $ with boundary $ \partial E $ consisting of a finite number of Jordan arcs. Under these conditions one has, at any point $ z \in D _ {z} $ at which $ w( z) \notin E $,

$$ \tag{1 } \omega ( z; \alpha _ {z} , D _ {z} ) \leq \ \omega ( w ( z); \partial E, D _ {w} ^ {*} ), $$

where $ D _ {w} ^ {*} $ denotes the subdomain of $ D _ {w} $ such that $ w( z) \in D _ {w} ^ {*} $ and $ \partial D _ {w} ^ {*} \subset \Gamma _ {w} \cup \partial E $. If (1) becomes an equality at any point $ z $, then equality will be valid everywhere in $ D _ {z} $. In particular, for a one-to-one conformal mapping from $ D _ {z} $ onto $ D _ {w} $ one has the identity

$$ \omega ( z; \alpha _ {z} , D _ {z} ) \equiv \ \omega ( w ( z); \alpha _ {w} , D _ {w} ). $$

The principle of harmonic measure, including its numerous applications [1], [2], was established by R. Nevanlinna. In particular, a corollary of the principle is the two-constants theorem, which implies, in turn, that for a function $ w( z) $ that is holomorphic in a domain $ D _ {z} $, the maximum value of $ \mathop{\rm ln} w( z) $ on the level line $ \{ {z } : {\omega ( z ; \alpha _ {z} , D _ {z} ) = t } \} $ is a convex function of the parameter $ t \in ( 0, 1) $.

The principle of harmonic measure has been generalized to holomorphic functions $ w = w( z) $, $ z = ( z _ {1} \dots z _ {n} ) $, of several complex variables, $ n \geq 1 $.

References

[1]	F. Nevanlinna, R. Nevanlinna, "Ueber die Eigenschaften einer analytischen Funktion in der Umgebung einer singulären Stelle oder Linie" Acta Soc. Sci. Fennica , 50 : 5 (1922) pp. 1–46
[2]	R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)