View source for Harmonic measure, principle of

The [[Harmonic measure|harmonic measure]] does not decrease under mappings realized by single-valued analytic functions. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h0465101.png" /> is the harmonic measure of a boundary set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h0465102.png" /> with respect to a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h0465103.png" /> in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h0465104.png" />-plane, one specific formulation of the principle of harmonic measure is as follows. In a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h0465105.png" /> with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h0465106.png" /> consisting of a finite number of Jordan arcs let there be given a single-valued analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h0465107.png" /> which satisfies the following conditions: the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h0465108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h0465109.png" />, form part of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651010.png" /> with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651011.png" /> consisting of a finite number of Jordan arcs; the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651012.png" /> can be continuously extended onto some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651013.png" /> consisting of a finite number of arcs; and the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651014.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651015.png" /> form part of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651016.png" /> with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651017.png" /> consisting of a finite number of Jordan arcs. Under these conditions one has, at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651018.png" /> at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651019.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651021.png" /> denotes the subdomain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651022.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651024.png" />. If (1) becomes an equality at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651025.png" />, then equality will be valid everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651026.png" />. In particular, for a one-to-one conformal mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651027.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651028.png" /> one has the identity

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651029.png" /></td> </tr></table>

The principle of harmonic measure, including its numerous applications [[#References|[1]]], [[#References|[2]]], was established by R. Nevanlinna. In particular, a corollary of the principle is the [[Two-constants theorem|two-constants theorem]], which implies, in turn, that for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651030.png" /> that is holomorphic in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651031.png" />, the maximum value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651032.png" /> on the level line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651033.png" /> is a convex function of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651034.png" />.

The principle of harmonic measure has been generalized to holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651036.png" />, of several complex variables, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046510/h04651037.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Nevanilinna,   "Analytic functions" , Springer  (1970)  (Translated from German)</TD></TR></table>