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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" />,
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<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>
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{{TEX|done}}
  
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
+
The [[Harmonic measure|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 $,
  
<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>
+
$$ \tag{1 }
 +
\omega ( z; \alpha _ {z} , D _ {z} )  \leq  \
 +
\omega ( w  ( z); \partial  E, D _ {w}  ^ {*} ),
 +
$$
  
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" />.
+
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
  
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" />.
+
$$
 +
\omega ( z;  \alpha _ {z} , D _ {z} )  \equiv \
 +
\omega ( w  ( z);  \alpha _ {w} , D _ {w} ).
 +
$$
 +
 
 +
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  $  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====
 
====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>
 
<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>

Latest revision as of 19:43, 5 June 2020


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