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''in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b0161601.png" />''
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The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b0161602.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b0161603.png" />, regular in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b0161604.png" />, which have an expansion of the form
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/b/b016/b016160/b0161605.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
''in the disc  $  | z | < 1 $''
 +
 
 +
The class $  R $
 +
of functions  $  f(z) $,
 +
regular in the disc  $  | z | < 1 $,
 +
which have an expansion of the form
 +
 
 +
$$ \tag{1 }
 +
f(z) = c _ {1} z + \dots + c _ {n} z  ^ {n} + \dots
 +
$$
  
 
and which satisfy the condition
 
and which satisfy the condition
  
<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/b/b016/b016160/b0161606.png" /></td> </tr></table>
+
$$
 +
f(z _ {1} ) f (z _ {2} )  \neq  1 ,\ \
 +
| z _ {1} | < 1,\  | z _ {2} | < 1.
 +
$$
 +
 
 +
This class of functions is a natural extension of the class $  B $
 +
of functions  $  f(z) $,
 +
regular in the disc  $  | z | < 1 $,
 +
with an expansion (1) and such that  $  | f(z) | < 1 $
 +
for  $  | z | < 1 $.
 +
The class of univalent functions (cf. [[Univalent function|Univalent function]]) in  $  R $
 +
is denoted by  $  \widetilde{R}  $.  
 +
The functions in  $  R $
 +
were named after L. Bieberbach [[#References|[1]]], who showed that for  $  f(z) \in \widetilde{R}  $
 +
the inequality
  
This class of functions is a natural extension of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b0161607.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b0161608.png" />, regular in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b0161609.png" />, with an expansion (1) and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616010.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616011.png" />. The class of univalent functions (cf. [[Univalent function|Univalent function]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616012.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616013.png" />. The functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616014.png" /> were named after L. Bieberbach [[#References|[1]]], who showed that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616015.png" /> the inequality
+
$$ \tag{2 }
 +
| c _ {1} | \leq  1
 +
$$
  
<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/b/b016/b016160/b01616016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
is valid, while equality holds only for the function  $  f(z) = e ^ {i \theta } z $,
 +
where  $  \theta $
 +
is real, and after S. Eilenberg [[#References|[2]]], who proved that the inequality (2) is valid for the whole class $  R $.
 +
It was shown by W. Rogosinski [[#References|[3]]] that every function in  $  R $
 +
is subordinate (cf. [[Subordination principle|Subordination principle]]) to some function in  $  \widetilde{R}  $.  
 +
Inequality (2) yields the following sharp inequality for  $  f(z) \in R $:
  
is valid, while equality holds only for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616018.png" /> is real, and after S. Eilenberg [[#References|[2]]], who proved that the inequality (2) is valid for the whole class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616019.png" />. It was shown by W. Rogosinski [[#References|[3]]] that every function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616020.png" /> is subordinate (cf. [[Subordination principle|Subordination principle]]) to some function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616021.png" />. Inequality (2) yields the following sharp inequality for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616022.png" />:
+
$$ \tag{3 }
 +
| f ^ { \prime } (z) | \leq  \
  
<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/b/b016/b016160/b01616023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
\frac{| 1 - f  ^ {2} (z) | }{1- | z |  ^ {2} }
 +
,\ \
 +
| z | < 1.
 +
$$
  
The following bound on the modulus of a function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616024.png" /> has been obtained: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616025.png" />, then
+
The following bound on the modulus of a function in $  R $
 +
has been obtained: If $  f(z) \in R $,  
 +
then
  
<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/b/b016/b016160/b01616026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
| f(z) |  \leq 
 +
\frac{r}{(1-r  ^ {2} )  ^ {1/2} }
 +
,\ \
 +
| z | = r ,\  0 < r < 1,
 +
$$
  
and (4) becomes an equality only for the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616028.png" /> is real and
+
and (4) becomes an equality only for the functions $  \pm  f(ze ^ {i \theta } ;  r) $,  
 +
where $  \theta $
 +
is real and
  
<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/b/b016/b016160/b01616029.png" /></td> </tr></table>
+
$$
 +
f (z; r)  = \
  
The method of the extremal metric (cf. [[Extremal metric, method of the|Extremal metric, method of the]]) provided the solution of the problem of the maximum and minimum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616030.png" /> in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616031.png" /> of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616032.png" /> with a fixed value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616034.png" />, in the expansion (1): For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616036.png" />, the following sharp inequalities are valid:
+
\frac{(1 - r  ^ {2} ) ^ {1/2} z }{1 + irz }
 +
.
 +
$$
  
<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/b/b016/b016160/b01616037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
The method of the extremal metric (cf. [[Extremal metric, method of the|Extremal metric, method of the]]) provided the solution of the problem of the maximum and minimum of  $  | f(z) | $
 +
in the class $  \widetilde{R}  (c) $
 +
of functions in  $  \widetilde{R}  $
 +
with a fixed value  $  | c _ {1} | = c $,
 +
0 < c \leq  1 $,
 +
in the expansion (1): For  $  f(z) \in \widetilde{R}  (c) $,
 +
$  0 < c < 1 $,
 +
the following sharp inequalities are valid:
  
Here the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616039.png" /> map the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616040.png" /> onto domains which are symmetric with respect to the imaginary axis of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616041.png" />-plane, and the boundaries of which belong to the union of the closures of certain trajectories or orthogonal trajectories of a [[Quadratic differential|quadratic differential]] in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616042.png" />-plane with a certain symmetry in the distribution of the zeros and poles [[#References|[4]]], [[#References|[5]]]. Certain optimal results for functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616043.png" /> were obtained by the simultaneous use of the method of the extremal metric and the symmetrization method [[#References|[4]]].
+
$$ \tag{5 }
 +
\mathop{\rm Im}  H (ir;  r, c)  \leq  \
 +
| f (re ^ {i \theta } ) | \leq  \
 +
\mathop{\rm Im}  F (ir;  r, c).
 +
$$
  
Many results obtained for the functions in the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616045.png" /> are consequences of corresponding results for systems of functions mapping the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616046.png" /> onto disjoint domains [[#References|[6]]]. The analogue of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616047.png" /> for a finitely-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616048.png" /> without isolated boundary points and not containing the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616049.png" />, is the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616051.png" />, of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616052.png" /> regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616053.png" /> and satisfying the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616055.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616056.png" /> are arbitrary points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616057.png" />. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616058.png" /> extends the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616059.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616060.png" />, regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616061.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616063.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616064.png" />. The following sharp estimate is an extension of the result of Bieberbach–Eilenberg and of inequality (3) to functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616065.png" />: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616066.png" />, then
+
Here the functions $  w = H(z;  r, c) $
 +
and $  w = F(z;  r, c) $
 +
map the disc $  | z | < 1 $
 +
onto domains which are symmetric with respect to the imaginary axis of the  $  w $-
 +
plane, and the boundaries of which belong to the union of the closures of certain trajectories or orthogonal trajectories of a [[Quadratic differential|quadratic differential]] in the $  w $-
 +
plane with a certain symmetry in the distribution of the zeros and poles [[#References|[4]]], [[#References|[5]]]. Certain optimal results for functions in $  \widetilde{R}  (c) $
 +
were obtained by the simultaneous use of the method of the extremal metric and the symmetrization method [[#References|[4]]].
  
<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/b/b016/b016160/b01616067.png" /></td> </tr></table>
+
Many results obtained for the functions in the classes  $  \widetilde{R}  $
 +
and  $  R $
 +
are consequences of corresponding results for systems of functions mapping the disc  $  | z | < 1 $
 +
onto disjoint domains [[#References|[6]]]. The analogue of  $  R $
 +
for a finitely-connected domain  $  G $
 +
without isolated boundary points and not containing the point  $  z = \infty $,
 +
is the class  $  R _ {a} (G) $,
 +
$  a \in G $,
 +
of functions  $  f(z) $
 +
regular in  $  G $
 +
and satisfying the conditions  $  f(a) = 0 $,
 +
$  f(z _ {1} )f(z _ {2} ) \neq 1 $,
 +
where  $  z _ {1} , z _ {2} $
 +
are arbitrary points in  $  G $.
 +
The class  $  R _ {a} (G) $
 +
extends the class  $  B _ {a} (G) $
 +
of functions  $  f(z) $,
 +
regular in  $  G $
 +
and such that  $  f(a) = 0 $,
 +
$  | f(z) | < 1 $
 +
in  $  G $.
 +
The following sharp estimate is an extension of the result of Bieberbach–Eilenberg and of inequality (3) to functions of class  $  R _ {a} (G) $:  
 +
If  $  f(z) \in R _ {a} (G) $,
 +
then
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616068.png" />, is that function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616069.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016160/b01616070.png" /> in this class.
+
$$
 +
| f ^ { \prime } (z) |  \leq  \
 +
| 1 - f ^ { 2 } (z) | \
 +
F ^ { \prime } (z, z),\ \
 +
z \in G.
 +
$$
 +
 
 +
where $  F(z, b), b \in G $,  
 +
is that function in $  B _ {b} (G) $
 +
for which $  F ^ { \prime } (b, b) = \max  | f ^ { \prime } (b) | $
 +
in this class.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Bieberbach,  "Ueber einige Extremalprobleme im Gebiete der konformen Abbildung"  ''Math. Ann.'' , '''77'''  (1916)  pp. 153–172</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Eilenberg,  "Sur quelques propriétés topologiques de la surface de sphère"  ''Fund. Math.'' , '''25'''  (1935)  pp. 267–272</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W. Rogosinski,  "On a theorem of Bieberbach–Eilenberg"  ''J. London Math. Soc. (1)'' , '''14'''  (1939)  pp. 4–11</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.A. Jenkins,  "Univalent functions and conformal mappings" , Springer  (1958)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.A. Jenkins,  "On Bieberbach–Eilenberg functions III"  ''Trans. Amer. Math. Soc.'' , '''119''' :  2  (1965)  pp. 195–215</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  N.A. Lebedev,  "The area principle in the theory of univalent functions" , Moscow  (1975)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Bieberbach,  "Ueber einige Extremalprobleme im Gebiete der konformen Abbildung"  ''Math. Ann.'' , '''77'''  (1916)  pp. 153–172</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Eilenberg,  "Sur quelques propriétés topologiques de la surface de sphère"  ''Fund. Math.'' , '''25'''  (1935)  pp. 267–272</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W. Rogosinski,  "On a theorem of Bieberbach–Eilenberg"  ''J. London Math. Soc. (1)'' , '''14'''  (1939)  pp. 4–11</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.A. Jenkins,  "Univalent functions and conformal mappings" , Springer  (1958)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.A. Jenkins,  "On Bieberbach–Eilenberg functions III"  ''Trans. Amer. Math. Soc.'' , '''119''' :  2  (1965)  pp. 195–215</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  N.A. Lebedev,  "The area principle in the theory of univalent functions" , Moscow  (1975)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Duren,  "Univalent functions" , Springer  (1983)  pp. Chapt. 10</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Duren,  "Univalent functions" , Springer  (1983)  pp. Chapt. 10</TD></TR></table>

Latest revision as of 10:59, 29 May 2020


in the disc $ | z | < 1 $

The class $ R $ of functions $ f(z) $, regular in the disc $ | z | < 1 $, which have an expansion of the form

$$ \tag{1 } f(z) = c _ {1} z + \dots + c _ {n} z ^ {n} + \dots $$

and which satisfy the condition

$$ f(z _ {1} ) f (z _ {2} ) \neq 1 ,\ \ | z _ {1} | < 1,\ | z _ {2} | < 1. $$

This class of functions is a natural extension of the class $ B $ of functions $ f(z) $, regular in the disc $ | z | < 1 $, with an expansion (1) and such that $ | f(z) | < 1 $ for $ | z | < 1 $. The class of univalent functions (cf. Univalent function) in $ R $ is denoted by $ \widetilde{R} $. The functions in $ R $ were named after L. Bieberbach [1], who showed that for $ f(z) \in \widetilde{R} $ the inequality

$$ \tag{2 } | c _ {1} | \leq 1 $$

is valid, while equality holds only for the function $ f(z) = e ^ {i \theta } z $, where $ \theta $ is real, and after S. Eilenberg [2], who proved that the inequality (2) is valid for the whole class $ R $. It was shown by W. Rogosinski [3] that every function in $ R $ is subordinate (cf. Subordination principle) to some function in $ \widetilde{R} $. Inequality (2) yields the following sharp inequality for $ f(z) \in R $:

$$ \tag{3 } | f ^ { \prime } (z) | \leq \ \frac{| 1 - f ^ {2} (z) | }{1- | z | ^ {2} } ,\ \ | z | < 1. $$

The following bound on the modulus of a function in $ R $ has been obtained: If $ f(z) \in R $, then

$$ \tag{4 } | f(z) | \leq \frac{r}{(1-r ^ {2} ) ^ {1/2} } ,\ \ | z | = r ,\ 0 < r < 1, $$

and (4) becomes an equality only for the functions $ \pm f(ze ^ {i \theta } ; r) $, where $ \theta $ is real and

$$ f (z; r) = \ \frac{(1 - r ^ {2} ) ^ {1/2} z }{1 + irz } . $$

The method of the extremal metric (cf. Extremal metric, method of the) provided the solution of the problem of the maximum and minimum of $ | f(z) | $ in the class $ \widetilde{R} (c) $ of functions in $ \widetilde{R} $ with a fixed value $ | c _ {1} | = c $, $ 0 < c \leq 1 $, in the expansion (1): For $ f(z) \in \widetilde{R} (c) $, $ 0 < c < 1 $, the following sharp inequalities are valid:

$$ \tag{5 } \mathop{\rm Im} H (ir; r, c) \leq \ | f (re ^ {i \theta } ) | \leq \ \mathop{\rm Im} F (ir; r, c). $$

Here the functions $ w = H(z; r, c) $ and $ w = F(z; r, c) $ map the disc $ | z | < 1 $ onto domains which are symmetric with respect to the imaginary axis of the $ w $- plane, and the boundaries of which belong to the union of the closures of certain trajectories or orthogonal trajectories of a quadratic differential in the $ w $- plane with a certain symmetry in the distribution of the zeros and poles [4], [5]. Certain optimal results for functions in $ \widetilde{R} (c) $ were obtained by the simultaneous use of the method of the extremal metric and the symmetrization method [4].

Many results obtained for the functions in the classes $ \widetilde{R} $ and $ R $ are consequences of corresponding results for systems of functions mapping the disc $ | z | < 1 $ onto disjoint domains [6]. The analogue of $ R $ for a finitely-connected domain $ G $ without isolated boundary points and not containing the point $ z = \infty $, is the class $ R _ {a} (G) $, $ a \in G $, of functions $ f(z) $ regular in $ G $ and satisfying the conditions $ f(a) = 0 $, $ f(z _ {1} )f(z _ {2} ) \neq 1 $, where $ z _ {1} , z _ {2} $ are arbitrary points in $ G $. The class $ R _ {a} (G) $ extends the class $ B _ {a} (G) $ of functions $ f(z) $, regular in $ G $ and such that $ f(a) = 0 $, $ | f(z) | < 1 $ in $ G $. The following sharp estimate is an extension of the result of Bieberbach–Eilenberg and of inequality (3) to functions of class $ R _ {a} (G) $: If $ f(z) \in R _ {a} (G) $, then

$$ | f ^ { \prime } (z) | \leq \ | 1 - f ^ { 2 } (z) | \ F ^ { \prime } (z, z),\ \ z \in G. $$

where $ F(z, b), b \in G $, is that function in $ B _ {b} (G) $ for which $ F ^ { \prime } (b, b) = \max | f ^ { \prime } (b) | $ in this class.

References

[1] L. Bieberbach, "Ueber einige Extremalprobleme im Gebiete der konformen Abbildung" Math. Ann. , 77 (1916) pp. 153–172
[2] S. Eilenberg, "Sur quelques propriétés topologiques de la surface de sphère" Fund. Math. , 25 (1935) pp. 267–272
[3] W. Rogosinski, "On a theorem of Bieberbach–Eilenberg" J. London Math. Soc. (1) , 14 (1939) pp. 4–11
[4] J.A. Jenkins, "Univalent functions and conformal mappings" , Springer (1958)
[5] J.A. Jenkins, "On Bieberbach–Eilenberg functions III" Trans. Amer. Math. Soc. , 119 : 2 (1965) pp. 195–215
[6] N.A. Lebedev, "The area principle in the theory of univalent functions" , Moscow (1975) (In Russian)

Comments

References

[a1] P.L. Duren, "Univalent functions" , Springer (1983) pp. Chapt. 10
How to Cite This Entry:
Bieberbach-Eilenberg functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bieberbach-Eilenberg_functions&oldid=22121
This article was adapted from an original article by G.V. Kuz'mina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article