Harmonic measure, principle of
The harmonic measure does not decrease under mappings realized by single-valued analytic functions. If is the harmonic measure of a boundary set
with respect to a domain
in the complex
-plane, one specific formulation of the principle of harmonic measure is as follows. In a domain
with boundary
consisting of a finite number of Jordan arcs let there be given a single-valued analytic function
which satisfies the following conditions: the values
,
, form part of the domain
with boundary
consisting of a finite number of Jordan arcs; the function
can be continuously extended onto some set
consisting of a finite number of arcs; and the values of
on
form part of a set
with boundary
consisting of a finite number of Jordan arcs. Under these conditions one has, at any point
at which
,
![]() | (1) |
where denotes the subdomain of
such that
and
. If (1) becomes an equality at any point
, then equality will be valid everywhere in
. In particular, for a one-to-one conformal mapping from
onto
one has the identity
![]() |
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 that is holomorphic in a domain
, the maximum value of
on the level line
is a convex function of the parameter
.
The principle of harmonic measure has been generalized to holomorphic functions ,
, of several complex variables,
.
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) |
Harmonic measure, principle of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonic_measure,_principle_of&oldid=47183