# 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)
How to Cite This Entry:
Harmonic measure, principle of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonic_measure,_principle_of&oldid=47183
This article was adapted from an original article by P.M. Tamrazov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article