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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t0944901.png" /> be a finitely-connected Jordan domain in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t0944902.png" />-plane and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t0944903.png" /> be a regular analytic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t0944904.png" /> satisfying the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t0944905.png" />, as well as the relation
+
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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/t/t094/t094490/t0944906.png" /></td> </tr></table>
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{{TEX|done}}
  
on some arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t0944907.png" /> of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t0944908.png" />. Then, at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t0944909.png" /> of the set
+
Let  $  D $
 +
be a finitely-connected Jordan domain in the $  z $-
 +
plane and let  $  w ( z) $
 +
be a regular analytic function in  $  D $
 +
satisfying the inequality  $  | w ( z) | \leq  M $,  
 +
as well as the relation
  
<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/t/t094/t094490/t09449010.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {z \rightarrow \zeta }  \sup  | w ( z) |  \leq  \
 +
< M ,\  z \in D ,\  \zeta \in \alpha ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449011.png" /> is the [[Harmonic measure|harmonic measure]] of the arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449012.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449013.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449014.png" />, the inequality
+
on some arc  $  \alpha $
 +
of the boundary  $  \partial  D $.  
 +
Then, at each point  $  z $
 +
of the set
  
<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/t/t094/t094490/t09449015.png" /></td> </tr></table>
+
$$
 +
\{ {z \in D } : {0 < \lambda \leq  \omega ( z ; \alpha , D ) < 1 } \}
 +
,
 +
$$
  
is satisfied. If for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449016.png" /> (satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449017.png" />) equality is attained, equality will hold for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449018.png" /> and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449020.png" />, while the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449021.png" /> in this case has the form
+
where  $  \omega ( z ;  \alpha , D ) $
 +
is the [[Harmonic measure|harmonic measure]] of the arc  $  \alpha $
 +
with respect to  $  D $
 +
at  $  z $,  
 +
the inequality
  
<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/t/t094/t094490/t09449022.png" /></td> </tr></table>
+
$$
 +
| w ( z) |  \leq  m  ^  \lambda  \cdot M ^ {1- \lambda }
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449023.png" /> is a real number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449024.png" /> is an analytic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449025.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449026.png" /> [[#References|[1]]], [[#References|[2]]].
+
is satisfied. If for some  $  z $(
 +
satisfying the condition  $  \omega ( z ;  \alpha , D ) = \lambda $)
 +
equality is attained, equality will hold for all  $  z \in D $
 +
and for all  $  \lambda $,
 +
0 \leq  \lambda \leq  1 $,
 +
while the function  $  w ( z) $
 +
in this case has the form
 +
 
 +
$$
 +
w ( z)  = e ^ {ia } m ^ {\phi ( z) } M ^ {1 - \phi ( z) } ,
 +
$$
 +
 
 +
where  $  a $
 +
is a real number and $  \phi ( z) $
 +
is an analytic function in $  D $
 +
for which $  \mathop{\rm Re}  \phi ( z) = \omega ( z ;  \alpha , D ) $[[#References|[1]]], [[#References|[2]]].
  
 
The two-constants theorem gives a quantitative expression of the unique determination of analytic functions by their boundary values and has important applications in the theory of functions [[#References|[3]]]. Hadamard's three-circles theorem (cf. [[Hadamard theorem|Hadamard theorem]]) is obtained from it as a special case. For possible analogues of the two-constants theorem for harmonic functions in space see [[#References|[4]]], [[#References|[5]]].
 
The two-constants theorem gives a quantitative expression of the unique determination of analytic functions by their boundary values and has important applications in the theory of functions [[#References|[3]]]. Hadamard's three-circles theorem (cf. [[Hadamard theorem|Hadamard theorem]]) is obtained from it as a special case. For possible analogues of the two-constants theorem for harmonic functions in space see [[#References|[4]]], [[#References|[5]]].
Line 21: Line 64:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Nevanlinna,  R. Nevanlinna,  "Über die Eigenschaften einer analytischen Funktion in der Umgebung einer singulären Stelle oder Linie"  ''Acta Soc. Sci. Fennica'' , '''5''' :  5  (1922)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Ostrowski,  "Über allgemeine Konvergenzsätze der komplexen Funktionentheorie"  ''Jahresber. Deutsch. Math.-Ver.'' , '''32''' :  9–12  (1923)  pp. 185–194</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Nevanilinna,  "Analytic functions" , Springer  (1970)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.N. Mergelyan,  "Harmonic approximation and approximate solution of the Cauchy problem for the Laplace equation"  ''Uspekhi Mat. Nauk'' , '''11''' :  5  (1956)  pp. 3–26  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.D. Solomentsev,  "Three-spheres theorem for harmonic functions"  ''Dokl. Akad. Nauk ArmSSR'' , '''42''' :  5  (1966)  pp. 274–278  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Nevanlinna,  R. Nevanlinna,  "Über die Eigenschaften einer analytischen Funktion in der Umgebung einer singulären Stelle oder Linie"  ''Acta Soc. Sci. Fennica'' , '''5''' :  5  (1922)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Ostrowski,  "Über allgemeine Konvergenzsätze der komplexen Funktionentheorie"  ''Jahresber. Deutsch. Math.-Ver.'' , '''32''' :  9–12  (1923)  pp. 185–194</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Nevanilinna,  "Analytic functions" , Springer  (1970)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.N. Mergelyan,  "Harmonic approximation and approximate solution of the Cauchy problem for the Laplace equation"  ''Uspekhi Mat. Nauk'' , '''11''' :  5  (1956)  pp. 3–26  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.D. Solomentsev,  "Three-spheres theorem for harmonic functions"  ''Dokl. Akad. Nauk ArmSSR'' , '''42''' :  5  (1966)  pp. 274–278  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
There is a more general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449028.png" />-constants theorem, [[#References|[a2]]]: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449029.png" /> be holomorphic in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449030.png" /> whose boundary is the union of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449031.png" /> distinct rectifiable arcs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449032.png" />; suppose that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449033.png" /> there is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449034.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449035.png" /> approaches any point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449036.png" />, then the limits of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449037.png" /> do not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449038.png" /> in absolute value. Then for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094490/t09449039.png" />,
+
There is a more general $  n $-
 +
constants theorem, [[#References|[a2]]]: Let $  f( z) $
 +
be holomorphic in a domain $  D $
 +
whose boundary is the union of $  n $
 +
distinct rectifiable arcs $  \alpha _ {1} \dots \alpha _ {n} $;  
 +
suppose that for each $  j $
 +
there is a constant $  M _ {j} $
 +
such that if $  z $
 +
approaches any point of $  \alpha _ {j} $,  
 +
then the limits of $  f ( z) $
 +
do not exceed $  M _ {j} $
 +
in absolute value. Then for each $  z \in D $,
  
<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/t/t094/t094490/t09449040.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm log}  | f( z) |  \leq  \sum_{j=1} ^ { m }
 +
\omega ( z, \alpha _ {j} ; D)  \mathop{\rm log}  M _ {j} .
 +
$$
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''2''' , Chelsea  (1977)  pp. 210–214  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Hille,  "Analytic function theory" , '''2''' , Chelsea, reprint  (1987)  pp. 409–410</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''2''' , Chelsea  (1977)  pp. 210–214  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Hille,  "Analytic function theory" , '''2''' , Chelsea, reprint  (1987)  pp. 409–410</TD></TR>
 +
</table>

Latest revision as of 19:15, 11 January 2024


Let $ D $ be a finitely-connected Jordan domain in the $ z $- plane and let $ w ( z) $ be a regular analytic function in $ D $ satisfying the inequality $ | w ( z) | \leq M $, as well as the relation

$$ \lim\limits _ {z \rightarrow \zeta } \sup | w ( z) | \leq \ m < M ,\ z \in D ,\ \zeta \in \alpha , $$

on some arc $ \alpha $ of the boundary $ \partial D $. Then, at each point $ z $ of the set

$$ \{ {z \in D } : {0 < \lambda \leq \omega ( z ; \alpha , D ) < 1 } \} , $$

where $ \omega ( z ; \alpha , D ) $ is the harmonic measure of the arc $ \alpha $ with respect to $ D $ at $ z $, the inequality

$$ | w ( z) | \leq m ^ \lambda \cdot M ^ {1- \lambda } $$

is satisfied. If for some $ z $( satisfying the condition $ \omega ( z ; \alpha , D ) = \lambda $) equality is attained, equality will hold for all $ z \in D $ and for all $ \lambda $, $ 0 \leq \lambda \leq 1 $, while the function $ w ( z) $ in this case has the form

$$ w ( z) = e ^ {ia } m ^ {\phi ( z) } M ^ {1 - \phi ( z) } , $$

where $ a $ is a real number and $ \phi ( z) $ is an analytic function in $ D $ for which $ \mathop{\rm Re} \phi ( z) = \omega ( z ; \alpha , D ) $[1], [2].

The two-constants theorem gives a quantitative expression of the unique determination of analytic functions by their boundary values and has important applications in the theory of functions [3]. Hadamard's three-circles theorem (cf. Hadamard theorem) is obtained from it as a special case. For possible analogues of the two-constants theorem for harmonic functions in space see [4], [5].

References

[1] F. Nevanlinna, R. Nevanlinna, "Über die Eigenschaften einer analytischen Funktion in der Umgebung einer singulären Stelle oder Linie" Acta Soc. Sci. Fennica , 5 : 5 (1922)
[2] A. Ostrowski, "Über allgemeine Konvergenzsätze der komplexen Funktionentheorie" Jahresber. Deutsch. Math.-Ver. , 32 : 9–12 (1923) pp. 185–194
[3] R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)
[4] S.N. Mergelyan, "Harmonic approximation and approximate solution of the Cauchy problem for the Laplace equation" Uspekhi Mat. Nauk , 11 : 5 (1956) pp. 3–26 (In Russian)
[5] E.D. Solomentsev, "Three-spheres theorem for harmonic functions" Dokl. Akad. Nauk ArmSSR , 42 : 5 (1966) pp. 274–278 (In Russian)

Comments

There is a more general $ n $- constants theorem, [a2]: Let $ f( z) $ be holomorphic in a domain $ D $ whose boundary is the union of $ n $ distinct rectifiable arcs $ \alpha _ {1} \dots \alpha _ {n} $; suppose that for each $ j $ there is a constant $ M _ {j} $ such that if $ z $ approaches any point of $ \alpha _ {j} $, then the limits of $ f ( z) $ do not exceed $ M _ {j} $ in absolute value. Then for each $ z \in D $,

$$ \mathop{\rm log} | f( z) | \leq \sum_{j=1} ^ { m } \omega ( z, \alpha _ {j} ; D) \mathop{\rm log} M _ {j} . $$

References

[a1] A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) pp. 210–214 (Translated from Russian)
[a2] E. Hille, "Analytic function theory" , 2 , Chelsea, reprint (1987) pp. 409–410
How to Cite This Entry:
Two-constants theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Two-constants_theorem&oldid=12855
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article