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Koebe's covering theorem: There exist an absolute constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k0556601.png" /> (the Koebe constant) such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k0556602.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k0556603.png" /> is the class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k0556604.png" /> that are regular and univalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k0556605.png" />), then the set of values of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k0556606.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k0556607.png" /> fills out the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k0556608.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k0556609.png" /> is the largest number for which this holds. L. Bieberbach (1916) proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566010.png" /> and that on the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566011.png" /> there exists points not belonging to the image of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566012.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566013.png" /> only when
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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/k/k055/k055660/k05566014.png" /></td> </tr></table>
+
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 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566015.png" /> is a real number. Koebe's covering theorem is sometimes stated as follows: If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566017.png" />, is regular and univalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566018.png" /> and maps the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566019.png" /> onto a domain not containing the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566021.png" />.
+
Koebe's covering theorem: There exist an absolute constant  $  K > 0 $(
 +
the Koebe constant) such that if  $  f \in S $(
 +
where $  S $
 +
is the class of functions  $  f ( z) = z + \dots $
 +
that are regular and univalent in  $  | z | < 1 $),
 +
then the set of values of the function  $  w = f ( z) $
 +
for  $  | z | < 1 $
 +
fills out the disc  $  | w | < K $,
 +
where  $  K $
 +
is the largest number for which this holds. L. Bieberbach (1916) proved that  $  K = 1 / 4 $
 +
and that on the circle  $  | w | = 1 / 4 $
 +
there exists points not belonging to the image of the disc  $  | z | < 1 $
 +
under  $  w = f ( z) $
 +
only when
 +
 
 +
$$
 +
f ( z)  = \
 +
 
 +
\frac{z}{( 1 + e ^ {i \alpha } z )  ^ {2} }
 +
,
 +
$$
 +
 
 +
where  $  \alpha $
 +
is a real number. Koebe's covering theorem is sometimes stated as follows: If a function $  w = f ( z) $,
 +
$  f ( 0) = 0 $,  
 +
is regular and univalent in $  | z | < 1 $
 +
and maps the disc $  | z | < 1 $
 +
onto a domain not containing the point $  c $,  
 +
then $  | f ^ { \prime } ( 0) | \leq  4 c $.
  
 
Koebe's distortion theorems.
 
Koebe's distortion theorems.
  
a) There exist positive numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566023.png" />, depending only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566024.png" />, such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566026.png" />,
+
a) There exist positive numbers $  m _ {1} ( r) $,  
 +
$  M _ {1} ( r) $,  
 +
depending only on $  r $,  
 +
such that for any $  f \in S $,
 +
$  | z | = r $,
 +
 
 +
$$
 +
m _ {1} ( r)  \leq  | f ( z) |  \leq  M _ {1} ( r) .
 +
$$
 +
 
 +
b) There exists a number  $  M ( r) $,
 +
depending only on  $  r $,
 +
such that for  $  f \in S $,
 +
$  | z _ {1} | , | z _ {2} | \leq  r $,
 +
 
 +
$$
  
<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/k/k055/k055660/k05566027.png" /></td> </tr></table>
+
\frac{1}{M ( r) }
 +
  \leq  \
 +
\left |
  
b) There exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566028.png" />, depending only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566029.png" />, such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566031.png" />,
+
\frac{f ^ { \prime } ( z _ {1} ) }{f ^ { \prime } ( z _ {2} ) }
  
<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/k/k055/k055660/k05566032.png" /></td> </tr></table>
+
\right |
 +
\leq  M ( r) .
 +
$$
  
This theorem can also be stated as follows: There exist positive numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566034.png" />, depending only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566035.png" />, such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566037.png" />,
+
This theorem can also be stated as follows: There exist positive numbers $  m _ {2} ( r) $,  
 +
$  M _ {2} ( r) $,  
 +
depending only on $  r $,  
 +
such that for any $  f \in S $,  
 +
$  | z | \leq  r $,
  
<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/k/k055/k055660/k05566038.png" /></td> </tr></table>
+
$$
 +
m _ {2} ( r)  \leq  | f ^ { \prime } ( z) |  \leq  M _ {2} ( r) .
 +
$$
  
 
Bieberbach proved that the best possible bounds in Koebe's distortion theorems are:
 
Bieberbach proved that the best possible bounds in Koebe's distortion theorems are:
  
<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/k/k055/k055660/k05566039.png" /></td> </tr></table>
+
$$
 +
m _ {1} ( r)  = \
 +
 
 +
\frac{r}{( 1 + r )  ^ {2} }
 +
,\ \
 +
M _ {1} ( r)  = \
  
<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/k/k055/k055660/k05566040.png" /></td> </tr></table>
+
\frac{r}{( 1 - r )  ^ {2} }
 +
,
 +
$$
 +
 
 +
$$
 +
m _ {2} ( r)  =
 +
\frac{1 - r }{( 1 + r )  ^ {3} }
 +
,\ \
 +
M _ {2} ( r)  =
 +
\frac{1 + r }{( 1 - r )  ^ {3} }
 +
.
 +
$$
  
 
Koebe's theorems on mapping finitely-connected domains onto canonical domains.
 
Koebe's theorems on mapping finitely-connected domains onto canonical domains.
  
a) Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566041.png" />-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566042.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566043.png" />-plane can be univalently mapped onto a circular domain (that is, onto a domain bounded by a finite number of complete non-intersecting circles; here some of the circles may degenerate to a point) of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566044.png" />-plane. There exists just one normalized mapping among these mappings taking a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566045.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566046.png" /> and such that the expansion of the mapping function in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566047.png" /> has the form
+
a) Every $  n $-
 +
connected domain $  B $
 +
of the $  z $-
 +
plane can be univalently mapped onto a circular domain (that is, onto a domain bounded by a finite number of complete non-intersecting circles; here some of the circles may degenerate to a point) of the $  \zeta $-
 +
plane. There exists just one normalized mapping among these mappings taking a given point $  z = a \in B $
 +
to $  \zeta = \infty $
 +
and such that the expansion of the mapping function in a neighbourhood of $  z = a $
 +
has the form
  
<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/k/k055/k055660/k05566048.png" /></td> </tr></table>
+
$$
  
according as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566049.png" /> is finite or not.
+
\frac{1}{z - a }
  
b) Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566050.png" />-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566051.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566052.png" />-plane with boundary continua <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566053.png" /> can be univalently mapped onto the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566054.png" />-plane with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566055.png" /> slits along arcs of logarithmic spirals with respective inclinations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566058.png" />, to the radial directions, and, moreover, such that the continuum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566060.png" />, is taken to the arc with inclination <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566061.png" />, the given points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566062.png" /> are taken to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566064.png" />, and the expansion of the mapping function in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566065.png" /> has the form
+
+ \alpha _ {1} ( z - a )
 +
+ \dots \  \textrm{ or } \ \
 +
z +
  
<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/k/k055/k055660/k05566066.png" /></td> </tr></table>
+
\frac{\alpha _ {1} }{z}
 +
+ \dots ,
 +
$$
  
according as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566067.png" /> is finite or not. The mapping is unique.
+
according as $  a $
 +
is finite or not.
 +
 
 +
b) Every  $  n $-
 +
connected domain  $  B $
 +
of the  $  z $-
 +
plane with boundary continua  $  K _ {1} \dots K _ {n} $
 +
can be univalently mapped onto the  $  \zeta $-
 +
plane with  $  n $
 +
slits along arcs of logarithmic spirals with respective inclinations  $  \theta _ {1} \dots \theta _ {n} $,
 +
$  0 \leq  \theta _  \nu  \leq  \pi / 2 $,
 +
$  \nu = 1 \dots n $,
 +
to the radial directions, and, moreover, such that the continuum  $  K _  \nu  $,
 +
$  \nu = 1 \dots n $,
 +
is taken to the arc with inclination  $  \theta _  \nu  $,
 +
the given points  $  a , b \in B $
 +
are taken to  $  0 $
 +
and  $  \infty $,
 +
and the expansion of the mapping function in a neighbourhood of  $  z = b $
 +
has the form
 +
 
 +
$$
 +
 
 +
\frac{1}{z - b }
 +
 
 +
+ \alpha _ {0} + \alpha _ {1} ( z - b ) + \dots \ \
 +
\textrm{ or } \ \
 +
z + \alpha _ {0} +
 +
\frac{\alpha _ {1} }{z}
 +
+ \dots ,
 +
$$
 +
 
 +
according as  $  b $
 +
is finite or not. The mapping is unique.
  
 
Theorems 1)–3) were established by P. Koebe (see –[[#References|[4]]]).
 
Theorems 1)–3) were established by P. Koebe (see –[[#References|[4]]]).
Line 43: Line 164:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  P. Koebe,  "Ueber die Uniformisierung beliebiger analytischer Kurven"  ''Nachr. K. Ges. Wissenschaft. Göttingen Math. Phys. Kl.'' , '''2'''  (1907)  pp. 191–210</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  P. Koebe,  "Ueber die Uniformisierung der algebraischen Kurven durch automorphe Funktionen mit imaginären Substitutionsgruppe"  ''Nachr. K. Ges. Wissenschaft. Göttingen Math. Phys. Kl.'' , '''4'''  (1908)  pp. 68–76</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P. Koebe,  "Ueber die Uniformisierung der algebraischen Kurven II"  ''Math. Ann.'' , '''69'''  (1910)  pp. 1–81</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P. Koebe,  "Abhandlung zur Theorie der konformen Abbildung IV"  ''Acta Math.'' , '''41'''  (1918)  pp. 305–344</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P. Koebe,  "Abhandlung zur Theorie der konformen Abbildung V"  ''Math. Z'' , '''2'''  (1918)  pp. 198–236</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G.M. Goluzin,  "Intrinsic problems in the theory of univalent functions"  ''Uspekhi Mat. Nauk'' , '''6'''  (1939)  pp. 26–89  (In Russian)</TD></TR><TR><TD valign="top">[6]</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">[7]</TD> <TD valign="top">  J.A. Jenkins,  "Univalent functions and conformal mapping" , Springer  (1958)</TD></TR></table>
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  P. Koebe,  "Ueber die Uniformisierung beliebiger analytischer Kurven"  ''Nachr. K. Ges. Wissenschaft. Göttingen Math. Phys. Kl.'' , '''2'''  (1907)  pp. 191–210</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  P. Koebe,  "Ueber die Uniformisierung der algebraischen Kurven durch automorphe Funktionen mit imaginären Substitutionsgruppe"  ''Nachr. K. Ges. Wissenschaft. Göttingen Math. Phys. Kl.'' , '''4'''  (1908)  pp. 68–76</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P. Koebe,  "Ueber die Uniformisierung der algebraischen Kurven II"  ''Math. Ann.'' , '''69'''  (1910)  pp. 1–81</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P. Koebe,  "Abhandlung zur Theorie der konformen Abbildung IV"  ''Acta Math.'' , '''41'''  (1918)  pp. 305–344</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P. Koebe,  "Abhandlung zur Theorie der konformen Abbildung V"  ''Math. Z'' , '''2'''  (1918)  pp. 198–236</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G.M. Goluzin,  "Intrinsic problems in the theory of univalent functions"  ''Uspekhi Mat. Nauk'' , '''6'''  (1939)  pp. 26–89  (In Russian)</TD></TR><TR><TD valign="top">[6]</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">[7]</TD> <TD valign="top">  J.A. Jenkins,  "Univalent functions and conformal mapping" , Springer  (1958)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Koebe's covering theorem is related to Bloch's theorem: There exists an absolute constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566068.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566069.png" /> is analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566070.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566071.png" /> contains a disc of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566072.png" /> which is the one-to-one image of a subdomain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566073.png" />. The best (largest) value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055660/k05566074.png" /> is called Bloch's constant. It is known that
+
Koebe's covering theorem is related to Bloch's theorem: There exists an absolute constant $  B $
 +
such that if $  f ( z) = z + a _ {2} z  ^ {2} + \dots $
 +
is analytic in $  D = \{ {z } : {| z | < 1 } \} $,  
 +
then $  f ( D) $
 +
contains a disc of radius $  B $
 +
which is the one-to-one image of a subdomain of $  D $.  
 +
The best (largest) value of $  B $
 +
is called Bloch's constant. It is known that
  
<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/k/k055/k055660/k05566075.png" /></td> </tr></table>
+
$$
 +
B  \leq 
 +
\frac{\Gamma ( 1/3) \Gamma ( 11/12) }{\sqrt {1 + \sqrt 3 } \Gamma ( 1/4) }
 +
,
 +
$$
  
 
and equality has been conjectured. For an up-to-date discussion of these matters, see [[#References|[a1]]].
 
and equality has been conjectured. For an up-to-date discussion of these matters, see [[#References|[a1]]].

Latest revision as of 22:14, 5 June 2020


Koebe's covering theorem: There exist an absolute constant $ K > 0 $( the Koebe constant) such that if $ f \in S $( where $ S $ is the class of functions $ f ( z) = z + \dots $ that are regular and univalent in $ | z | < 1 $), then the set of values of the function $ w = f ( z) $ for $ | z | < 1 $ fills out the disc $ | w | < K $, where $ K $ is the largest number for which this holds. L. Bieberbach (1916) proved that $ K = 1 / 4 $ and that on the circle $ | w | = 1 / 4 $ there exists points not belonging to the image of the disc $ | z | < 1 $ under $ w = f ( z) $ only when

$$ f ( z) = \ \frac{z}{( 1 + e ^ {i \alpha } z ) ^ {2} } , $$

where $ \alpha $ is a real number. Koebe's covering theorem is sometimes stated as follows: If a function $ w = f ( z) $, $ f ( 0) = 0 $, is regular and univalent in $ | z | < 1 $ and maps the disc $ | z | < 1 $ onto a domain not containing the point $ c $, then $ | f ^ { \prime } ( 0) | \leq 4 c $.

Koebe's distortion theorems.

a) There exist positive numbers $ m _ {1} ( r) $, $ M _ {1} ( r) $, depending only on $ r $, such that for any $ f \in S $, $ | z | = r $,

$$ m _ {1} ( r) \leq | f ( z) | \leq M _ {1} ( r) . $$

b) There exists a number $ M ( r) $, depending only on $ r $, such that for $ f \in S $, $ | z _ {1} | , | z _ {2} | \leq r $,

$$ \frac{1}{M ( r) } \leq \ \left | \frac{f ^ { \prime } ( z _ {1} ) }{f ^ { \prime } ( z _ {2} ) } \right | \leq M ( r) . $$

This theorem can also be stated as follows: There exist positive numbers $ m _ {2} ( r) $, $ M _ {2} ( r) $, depending only on $ r $, such that for any $ f \in S $, $ | z | \leq r $,

$$ m _ {2} ( r) \leq | f ^ { \prime } ( z) | \leq M _ {2} ( r) . $$

Bieberbach proved that the best possible bounds in Koebe's distortion theorems are:

$$ m _ {1} ( r) = \ \frac{r}{( 1 + r ) ^ {2} } ,\ \ M _ {1} ( r) = \ \frac{r}{( 1 - r ) ^ {2} } , $$

$$ m _ {2} ( r) = \frac{1 - r }{( 1 + r ) ^ {3} } ,\ \ M _ {2} ( r) = \frac{1 + r }{( 1 - r ) ^ {3} } . $$

Koebe's theorems on mapping finitely-connected domains onto canonical domains.

a) Every $ n $- connected domain $ B $ of the $ z $- plane can be univalently mapped onto a circular domain (that is, onto a domain bounded by a finite number of complete non-intersecting circles; here some of the circles may degenerate to a point) of the $ \zeta $- plane. There exists just one normalized mapping among these mappings taking a given point $ z = a \in B $ to $ \zeta = \infty $ and such that the expansion of the mapping function in a neighbourhood of $ z = a $ has the form

$$ \frac{1}{z - a } + \alpha _ {1} ( z - a ) + \dots \ \textrm{ or } \ \ z + \frac{\alpha _ {1} }{z} + \dots , $$

according as $ a $ is finite or not.

b) Every $ n $- connected domain $ B $ of the $ z $- plane with boundary continua $ K _ {1} \dots K _ {n} $ can be univalently mapped onto the $ \zeta $- plane with $ n $ slits along arcs of logarithmic spirals with respective inclinations $ \theta _ {1} \dots \theta _ {n} $, $ 0 \leq \theta _ \nu \leq \pi / 2 $, $ \nu = 1 \dots n $, to the radial directions, and, moreover, such that the continuum $ K _ \nu $, $ \nu = 1 \dots n $, is taken to the arc with inclination $ \theta _ \nu $, the given points $ a , b \in B $ are taken to $ 0 $ and $ \infty $, and the expansion of the mapping function in a neighbourhood of $ z = b $ has the form

$$ \frac{1}{z - b } + \alpha _ {0} + \alpha _ {1} ( z - b ) + \dots \ \ \textrm{ or } \ \ z + \alpha _ {0} + \frac{\alpha _ {1} }{z} + \dots , $$

according as $ b $ is finite or not. The mapping is unique.

Theorems 1)–3) were established by P. Koebe (see –[4]).

References

[1a] P. Koebe, "Ueber die Uniformisierung beliebiger analytischer Kurven" Nachr. K. Ges. Wissenschaft. Göttingen Math. Phys. Kl. , 2 (1907) pp. 191–210
[1b] P. Koebe, "Ueber die Uniformisierung der algebraischen Kurven durch automorphe Funktionen mit imaginären Substitutionsgruppe" Nachr. K. Ges. Wissenschaft. Göttingen Math. Phys. Kl. , 4 (1908) pp. 68–76
[2] P. Koebe, "Ueber die Uniformisierung der algebraischen Kurven II" Math. Ann. , 69 (1910) pp. 1–81
[3] P. Koebe, "Abhandlung zur Theorie der konformen Abbildung IV" Acta Math. , 41 (1918) pp. 305–344
[4] P. Koebe, "Abhandlung zur Theorie der konformen Abbildung V" Math. Z , 2 (1918) pp. 198–236
[5] G.M. Goluzin, "Intrinsic problems in the theory of univalent functions" Uspekhi Mat. Nauk , 6 (1939) pp. 26–89 (In Russian)
[6] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[7] J.A. Jenkins, "Univalent functions and conformal mapping" , Springer (1958)

Comments

Koebe's covering theorem is related to Bloch's theorem: There exists an absolute constant $ B $ such that if $ f ( z) = z + a _ {2} z ^ {2} + \dots $ is analytic in $ D = \{ {z } : {| z | < 1 } \} $, then $ f ( D) $ contains a disc of radius $ B $ which is the one-to-one image of a subdomain of $ D $. The best (largest) value of $ B $ is called Bloch's constant. It is known that

$$ B \leq \frac{\Gamma ( 1/3) \Gamma ( 11/12) }{\sqrt {1 + \sqrt 3 } \Gamma ( 1/4) } , $$

and equality has been conjectured. For an up-to-date discussion of these matters, see [a1].

See also Landau theorems.

References

[a1] C.D. Minda, "Bloch constants" J. d'Anal. Math. , 41 (1982) pp. 54–84
[a2] J.B. Conway, "Functions of a complex variable" , Springer (1978)
How to Cite This Entry:
Koebe theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Koebe_theorem&oldid=47511
This article was adapted from an original article by E.G. Goluzina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article