Namespaces
Variants
Actions

Difference between revisions of "Schottky theorem"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
 +
<!--
 +
s0833901.png
 +
$#A+1 = 24 n = 0
 +
$#C+1 = 24 : ~/encyclopedia/old_files/data/S083/S.0803390 Schottky theorem
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
If a function
 
If a function
  
<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/s/s083/s083390/s0833901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
= f( z)  = c _ {0} + c _ {1} z + \dots
 +
$$
  
is regular and analytic in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s0833902.png" /> and does not take certain finite values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s0833903.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s0833904.png" />, then in any disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s0833905.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s0833906.png" />, the modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s0833907.png" /> is bounded by a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s0833908.png" /> that depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s0833909.png" /> (see [[#References|[1]]]). A more complete formulation can be obtained, by combining the generalized Schottky theorem and Landau's theorem, for an arbitrary number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s08339010.png" /> of exceptional values. Suppose that the function (*) does not take some finite values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s08339011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s08339012.png" />. Then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s08339013.png" />, the radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s08339014.png" /> is bounded above by a number that depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s08339015.png" /> (Landau's theorem). Moreover, in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s08339016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s08339017.png" />, the modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s08339018.png" /> is bounded by a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s08339019.png" /> that depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s08339020.png" /> (Schottky's theorem).
+
is regular and analytic in the disc $  D = \{ {z } : {| z | < R } \} $
 +
and does not take certain finite values $  a _ {1} , a _ {2} $
 +
in $  D $,  
 +
then in any disc $  | z | \leq  R _ {1} $,
 +
$  0 < R _ {1} < R $,  
 +
the modulus $  | f( z) | $
 +
is bounded by a number $  M( a _ {1} , a _ {2} , c _ {0} , R _ {1} ) $
 +
that depends only on $  a _ {1} , a _ {2} , c _ {0} , R _ {1} $(
 +
see [[#References|[1]]]). A more complete formulation can be obtained, by combining the generalized Schottky theorem and Landau's theorem, for an arbitrary number $  q \geq  2 $
 +
of exceptional values. Suppose that the function (*) does not take some finite values $  a _ {1} \dots a _ {q} $,  
 +
$  q \geq  2 $.  
 +
Then for $  c _ {1} \neq 0 $,  
 +
the radius $  R $
 +
is bounded above by a number that depends only on $  a _ {1} \dots a _ {q} , c _ {0} , c _ {1} $(
 +
Landau's theorem). Moreover, in the disc $  | z | \leq  R _ {1} $,
 +
$  0 < R _ {1} < R $,  
 +
the modulus $  | f( z) | $
 +
is bounded by a number $  M( a _ {1} \dots a _ {q} , c _ {0} , R _ {1} ) $
 +
that depends only on $  a _ {1} \dots a _ {q} , c _ {0} , R _ {1} $(
 +
Schottky's theorem).
  
From the geometrical point of view Schottky's theorem means that the spherical distance (i.e. the distance on the [[Riemann sphere|Riemann sphere]]) of the image of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s08339021.png" /> from the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s08339022.png" /> is no less than a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s08339023.png" /> that depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083390/s08339024.png" />. Schottky's theorem is one of the classical results in the theory of functions of a complex variable of the type of [[Distortion theorems|distortion theorems]].
+
From the geometrical point of view Schottky's theorem means that the spherical distance (i.e. the distance on the [[Riemann sphere|Riemann sphere]]) of the image of the disc $  | z | \leq  R _ {1} $
 +
from the points $  a _ {1} \dots a _ {q} $
 +
is no less than a number $  d( a _ {1} \dots a _ {q} , c _ {0} , R _ {1} ) $
 +
that depends only on $  a _ {1} \dots a _ {q} , c _ {0} , R _ {1} $.  
 +
Schottky's theorem is one of the classical results in the theory of functions of a complex variable of the type of [[Distortion theorems|distortion theorems]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Schottky,  "Ueber den Picard'schen Satz und die Borel'schen Ungleichungen"  ''Sitzungsber. Preuss. Akad. Wiss.'' , '''2'''  (1904)  pp. 1244–1262</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Stoilov,  "The theory of functions of a complex variable" , '''1–2''' , Moscow  (1962)  (In Russian; translated from Rumanian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Schottky,  "Ueber den Picard'schen Satz und die Borel'schen Ungleichungen"  ''Sitzungsber. Preuss. Akad. Wiss.'' , '''2'''  (1904)  pp. 1244–1262</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Stoilov,  "The theory of functions of a complex variable" , '''1–2''' , Moscow  (1962)  (In Russian; translated from Rumanian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:12, 6 June 2020


If a function

$$ \tag{* } w = f( z) = c _ {0} + c _ {1} z + \dots $$

is regular and analytic in the disc $ D = \{ {z } : {| z | < R } \} $ and does not take certain finite values $ a _ {1} , a _ {2} $ in $ D $, then in any disc $ | z | \leq R _ {1} $, $ 0 < R _ {1} < R $, the modulus $ | f( z) | $ is bounded by a number $ M( a _ {1} , a _ {2} , c _ {0} , R _ {1} ) $ that depends only on $ a _ {1} , a _ {2} , c _ {0} , R _ {1} $( see [1]). A more complete formulation can be obtained, by combining the generalized Schottky theorem and Landau's theorem, for an arbitrary number $ q \geq 2 $ of exceptional values. Suppose that the function (*) does not take some finite values $ a _ {1} \dots a _ {q} $, $ q \geq 2 $. Then for $ c _ {1} \neq 0 $, the radius $ R $ is bounded above by a number that depends only on $ a _ {1} \dots a _ {q} , c _ {0} , c _ {1} $( Landau's theorem). Moreover, in the disc $ | z | \leq R _ {1} $, $ 0 < R _ {1} < R $, the modulus $ | f( z) | $ is bounded by a number $ M( a _ {1} \dots a _ {q} , c _ {0} , R _ {1} ) $ that depends only on $ a _ {1} \dots a _ {q} , c _ {0} , R _ {1} $( Schottky's theorem).

From the geometrical point of view Schottky's theorem means that the spherical distance (i.e. the distance on the Riemann sphere) of the image of the disc $ | z | \leq R _ {1} $ from the points $ a _ {1} \dots a _ {q} $ is no less than a number $ d( a _ {1} \dots a _ {q} , c _ {0} , R _ {1} ) $ that depends only on $ a _ {1} \dots a _ {q} , c _ {0} , R _ {1} $. Schottky's theorem is one of the classical results in the theory of functions of a complex variable of the type of distortion theorems.

References

[1] F. Schottky, "Ueber den Picard'schen Satz und die Borel'schen Ungleichungen" Sitzungsber. Preuss. Akad. Wiss. , 2 (1904) pp. 1244–1262
[2] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[3] S. Stoilov, "The theory of functions of a complex variable" , 1–2 , Moscow (1962) (In Russian; translated from Rumanian)

Comments

The theorems of Landau and Schottky are also related to the Picard theorem.

References

[a1] E. Landau, D. Gaier, "Darstellung und Begrundung einiger neuerer Ergebnisse der Funktionentheorie" , Springer, reprint (1986)
[a2] J.B. Conway, "Functions of one complex variable" , Springer (1978)
How to Cite This Entry:
Schottky theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schottky_theorem&oldid=18792
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article