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''Löwner's method of parametric representation of univalent functions, Löwner's parametric method''
 
''Löwner's method of parametric representation of univalent functions, Löwner's parametric method''
  
A method in the theory of univalent functions that consists in using the [[Löwner equation|Löwner equation]] to solve extremal problems. The method was proposed by K. Löwner [[#References|[1]]]. It is based on the fact that the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l0609601.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l0609602.png" />, that are regular and univalent in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l0609603.png" /> and that map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l0609604.png" /> onto domains of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l0609606.png" /> (cf. [[Smirnov domain|Smirnov domain]]), which are obtained from the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l0609607.png" /> by making a slit along a part of a Jordan arc starting from a point on the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l0609608.png" /> and not passing through the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l0609609.png" />, is complete (in the topology of uniform convergence of functions inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096010.png" />) in the whole family of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096012.png" />, that are regular and univalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096013.png" /> and are such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096014.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096015.png" />. Associating the length of the arc that has been removed with a parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096016.png" />, it has been established that a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096018.png" />, that maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096019.png" /> univalently onto a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096020.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096021.png" /> is a solution of the differential equation (see [[Löwner equation|Löwner equation]])
+
A method in the theory of univalent functions that consists in using the [[Löwner equation|Löwner equation]] to solve extremal problems. The method was proposed by K. Löwner [[#References|[1]]]. It is based on the fact that the set of functions $  f ( z) $,
 +
$  f ( 0) = 0 $,  
 +
that are regular and univalent in the disc $  E = \{ {z } : {| z | < 1 } \} $
 +
and that map $  E $
 +
onto domains of type $  ( s) $(
 +
cf. [[Smirnov domain|Smirnov domain]]), which are obtained from the disc $  | w | < 1 $
 +
by making a slit along a part of a Jordan arc starting from a point on the circle $  | w | = 1 $
 +
and not passing through the point $  w = 0 $,  
 +
is complete (in the topology of uniform convergence of functions inside $  E $)  
 +
in the whole family of functions $  f ( z) $,
 +
$  f ( 0) = 0 $,  
 +
that are regular and univalent in $  E $
 +
and are such that $  | f ( z) | < 1 $
 +
in $  E $.  
 +
Associating the length of the arc that has been removed with a parameter $  t $,  
 +
it has been established that a function $  w = f ( z) $,
 +
$  f ( 0) = 0 $,  
 +
that maps $  E $
 +
univalently onto a domain $  D $
 +
of type $  ( s) $
 +
is a solution of the differential equation (see [[Löwner equation|Löwner equation]])
 +
 
 +
$$ \tag{* }
  
<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/l/l060/l060960/l06096022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
\frac{\partial  f ( z , t ) }{\partial  t }
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096023.png" />, satisfying the initial condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096024.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096026.png" /> is a continuous complex-valued function on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096027.png" /> corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096028.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096029.png" />. Löwner used this method to obtain sharp estimates of the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096032.png" /> in the expansions
+
= - f ( z , t )
  
<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/l/l060/l060960/l06096033.png" /></td> </tr></table>
+
\frac{1 + k ( t) f ( z , t ) }{1 - k ( t) f ( z , t ) }
 +
,
 +
$$
 +
 
 +
$  f ( z , t _ {0} ) = f ( z) $,
 +
satisfying the initial condition  $  f ( z , 0 ) = z $.
 +
Here  $  t \in [ 0 , t _ {0} ] $
 +
and  $  k ( t) $
 +
is a continuous complex-valued function on the interval  $  [ 0 , t _ {0} ] $
 +
corresponding to  $  D $
 +
with  $  | k ( t) | = 1 $.
 +
Löwner used this method to obtain sharp estimates of the coefficients  $  c _ {3} $
 +
and  $  b _ {n} $,
 +
$  n = 2 , 3 \dots $
 +
in the expansions
 +
 
 +
$$
 +
= f ( z)  = z +
 +
\sum _ { n= } 2 ^  \infty 
 +
c _ {n} z  ^ {n}
 +
$$
  
 
and
 
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/l/l060/l060960/l06096034.png" /></td> </tr></table>
+
$$
 +
= f ^ { - 1 } ( w)  = w +
 +
\sum _ { n= } 2 ^  \infty 
 +
b _ {n} w  ^ {n}
 +
$$
  
in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096035.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096038.png" />, that are regular and univalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096039.png" />.
+
in the class $  S $
 +
of functions $  w = f ( z) $,
 +
$  f ( 0) = 0 $,  
 +
$  f ^ { \prime } ( 0) = 1 $,  
 +
that are regular and univalent in $  E $.
  
The Löwner method has been used (see [[#References|[3]]]) to obtain fundamental results in the theory of univalent functions (distortion theorems, reciprocal growth theorems, rotation theorems). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096040.png" /> be the subclass of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096041.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096042.png" /> that have in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096043.png" /> the representation
+
The Löwner method has been used (see [[#References|[3]]]) to obtain fundamental results in the theory of univalent functions (distortion theorems, reciprocal growth theorems, rotation theorems). Let $  S ^ { \prime } $
 +
be the subclass of functions $  f ( z) $
 +
in $  S $
 +
that have in $  E $
 +
the representation
  
<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/l/l060/l060960/l06096044.png" /></td> </tr></table>
+
$$
 +
f ( z)  = \lim\limits _ {t \rightarrow \infty } \
 +
e  ^ {t} f ( z , t ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096045.png" />, as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096046.png" />, is regular and univalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096048.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096051.png" />, and as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096053.png" />, is a solution of the differential equation (*) satisfying the initial condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096054.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096055.png" /> in (*) is any complex-valued function that is piecewise continuous and has modulus 1 on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096056.png" />. To estimate any quantity on the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096057.png" /> it is sufficient to estimate it on the subclass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096058.png" />, since any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096059.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096060.png" /> can be approximated by functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096063.png" />, each of which maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096064.png" /> univalently onto the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096065.png" />-plane with a slit along a Jordan arc starting at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096066.png" /> and not passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096067.png" />, and hence by functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096068.png" />. Under this approximation the quantities to be estimated for the approximating functions converge to the same quantity as for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096069.png" />.
+
where $  f ( z , t ) $,  
 +
as a function of $  z $,  
 +
is regular and univalent in $  E $,
 +
$  | f ( z , t) | < 1 $
 +
in $  E $,
 +
$  f ( 0 ,t ) = 0 $,  
 +
$  f _ {z} ^ { \prime } ( 0 , t ) > 0 $,  
 +
and as a function of $  t $,
 +
$  0 < t < \infty $,  
 +
is a solution of the differential equation (*) satisfying the initial condition $  f ( z , 0 ) = z $;  
 +
$  k ( t) $
 +
in (*) is any complex-valued function that is piecewise continuous and has modulus 1 on the interval $  [ 0 , \infty ) $.  
 +
To estimate any quantity on the class $  S $
 +
it is sufficient to estimate it on the subclass $  S ^ { \prime } $,  
 +
since any function $  f ( z) $
 +
of class $  S $
 +
can be approximated by functions $  f _ {n} ( z) $,
 +
$  f _ {n} ( 0) = 0 $,  
 +
$  f _ {n} ^ { \prime } ( 0) > 0 $,  
 +
each of which maps $  E $
 +
univalently onto the $  w $-
 +
plane with a slit along a Jordan arc starting at $  \infty $
 +
and not passing through $  w = 0 $,  
 +
and hence by functions $  f _ {n} ( z) / f _ {n} ^ { \prime } ( 0) \in S ^ { \prime } $.  
 +
Under this approximation the quantities to be estimated for the approximating functions converge to the same quantity as for the function $  f ( z) $.
  
 
Löwner's method has been used in work on the theory of univalent functions (see [[#References|[3]]]); it often leads to success in obtaining explicit estimates, but as a rule it does not ensure the classification of all extremal functions and does not give complete information about their uniqueness. For a complete solution of extremal problems Löwner's method is usually combined with a variational method (see [[#References|[3]]] and [[Variation-parametric method|Variation-parametric method]]). Löwner's method has been extended to doubly-connected domains. A generalized equation of the type of Löwner's equation has been obtained for multiply-connected domains and for automorphic functions (see [[#References|[4]]]).
 
Löwner's method has been used in work on the theory of univalent functions (see [[#References|[3]]]); it often leads to success in obtaining explicit estimates, but as a rule it does not ensure the classification of all extremal functions and does not give complete information about their uniqueness. For a complete solution of extremal problems Löwner's method is usually combined with a variational method (see [[#References|[3]]] and [[Variation-parametric method|Variation-parametric method]]). Löwner's method has been extended to doubly-connected domains. A generalized equation of the type of Löwner's equation has been obtained for multiply-connected domains and for automorphic functions (see [[#References|[4]]]).
Line 25: Line 118:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Löwner,  "Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I"  ''Math. Ann.'' , '''89'''  (1923)  pp. 103–121</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Peschl,  "Zur Theorie der schlichten Funktionen"  ''J. Reine Angew. Math.'' , '''176'''  (1936)  pp. 61–94</TD></TR><TR><TD valign="top">[3]</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">[4]</TD> <TD valign="top">  I.A. Aleksandrov,  "Parametric extensions in the theory of univalent functions" , Moscow  (1976)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Löwner,  "Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I"  ''Math. Ann.'' , '''89'''  (1923)  pp. 103–121</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Peschl,  "Zur Theorie der schlichten Funktionen"  ''J. Reine Angew. Math.'' , '''176'''  (1936)  pp. 61–94</TD></TR><TR><TD valign="top">[3]</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">[4]</TD> <TD valign="top">  I.A. Aleksandrov,  "Parametric extensions in the theory of univalent functions" , Moscow  (1976)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 04:11, 6 June 2020


Löwner's method of parametric representation of univalent functions, Löwner's parametric method

A method in the theory of univalent functions that consists in using the Löwner equation to solve extremal problems. The method was proposed by K. Löwner [1]. It is based on the fact that the set of functions $ f ( z) $, $ f ( 0) = 0 $, that are regular and univalent in the disc $ E = \{ {z } : {| z | < 1 } \} $ and that map $ E $ onto domains of type $ ( s) $( cf. Smirnov domain), which are obtained from the disc $ | w | < 1 $ by making a slit along a part of a Jordan arc starting from a point on the circle $ | w | = 1 $ and not passing through the point $ w = 0 $, is complete (in the topology of uniform convergence of functions inside $ E $) in the whole family of functions $ f ( z) $, $ f ( 0) = 0 $, that are regular and univalent in $ E $ and are such that $ | f ( z) | < 1 $ in $ E $. Associating the length of the arc that has been removed with a parameter $ t $, it has been established that a function $ w = f ( z) $, $ f ( 0) = 0 $, that maps $ E $ univalently onto a domain $ D $ of type $ ( s) $ is a solution of the differential equation (see Löwner equation)

$$ \tag{* } \frac{\partial f ( z , t ) }{\partial t } = - f ( z , t ) \frac{1 + k ( t) f ( z , t ) }{1 - k ( t) f ( z , t ) } , $$

$ f ( z , t _ {0} ) = f ( z) $, satisfying the initial condition $ f ( z , 0 ) = z $. Here $ t \in [ 0 , t _ {0} ] $ and $ k ( t) $ is a continuous complex-valued function on the interval $ [ 0 , t _ {0} ] $ corresponding to $ D $ with $ | k ( t) | = 1 $. Löwner used this method to obtain sharp estimates of the coefficients $ c _ {3} $ and $ b _ {n} $, $ n = 2 , 3 \dots $ in the expansions

$$ w = f ( z) = z + \sum _ { n= } 2 ^ \infty c _ {n} z ^ {n} $$

and

$$ z = f ^ { - 1 } ( w) = w + \sum _ { n= } 2 ^ \infty b _ {n} w ^ {n} $$

in the class $ S $ of functions $ w = f ( z) $, $ f ( 0) = 0 $, $ f ^ { \prime } ( 0) = 1 $, that are regular and univalent in $ E $.

The Löwner method has been used (see [3]) to obtain fundamental results in the theory of univalent functions (distortion theorems, reciprocal growth theorems, rotation theorems). Let $ S ^ { \prime } $ be the subclass of functions $ f ( z) $ in $ S $ that have in $ E $ the representation

$$ f ( z) = \lim\limits _ {t \rightarrow \infty } \ e ^ {t} f ( z , t ) , $$

where $ f ( z , t ) $, as a function of $ z $, is regular and univalent in $ E $, $ | f ( z , t) | < 1 $ in $ E $, $ f ( 0 ,t ) = 0 $, $ f _ {z} ^ { \prime } ( 0 , t ) > 0 $, and as a function of $ t $, $ 0 < t < \infty $, is a solution of the differential equation (*) satisfying the initial condition $ f ( z , 0 ) = z $; $ k ( t) $ in (*) is any complex-valued function that is piecewise continuous and has modulus 1 on the interval $ [ 0 , \infty ) $. To estimate any quantity on the class $ S $ it is sufficient to estimate it on the subclass $ S ^ { \prime } $, since any function $ f ( z) $ of class $ S $ can be approximated by functions $ f _ {n} ( z) $, $ f _ {n} ( 0) = 0 $, $ f _ {n} ^ { \prime } ( 0) > 0 $, each of which maps $ E $ univalently onto the $ w $- plane with a slit along a Jordan arc starting at $ \infty $ and not passing through $ w = 0 $, and hence by functions $ f _ {n} ( z) / f _ {n} ^ { \prime } ( 0) \in S ^ { \prime } $. Under this approximation the quantities to be estimated for the approximating functions converge to the same quantity as for the function $ f ( z) $.

Löwner's method has been used in work on the theory of univalent functions (see [3]); it often leads to success in obtaining explicit estimates, but as a rule it does not ensure the classification of all extremal functions and does not give complete information about their uniqueness. For a complete solution of extremal problems Löwner's method is usually combined with a variational method (see [3] and Variation-parametric method). Löwner's method has been extended to doubly-connected domains. A generalized equation of the type of Löwner's equation has been obtained for multiply-connected domains and for automorphic functions (see [4]).

References

[1] K. Löwner, "Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I" Math. Ann. , 89 (1923) pp. 103–121
[2] E. Peschl, "Zur Theorie der schlichten Funktionen" J. Reine Angew. Math. , 176 (1936) pp. 61–94
[3] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[4] I.A. Aleksandrov, "Parametric extensions in the theory of univalent functions" , Moscow (1976) (In Russian)

Comments

The Löwner equation has been used to solve the Bieberbach conjecture, [a1]; cf. [a2]. Further references on the method include [a3][a6].

References

[a1] L. de Branges, "A proof of the Bieberbach conjecture" Acta. Math. , 154 (1985) pp. 137–152
[a2] C.H. FitzGerald, C. Pommerenke, "The de Branges theorem on univalent functions" Trans. Amer. Math. Soc. , 290 (1985) pp. 683–690
[a3] W.K. Hayman, "Multivalent functions" , Cambridge Univ. Press (1958)
[a4] C. Pommerenke, "Univalent functions" , Vandenhoeck & Ruprecht (1975)
[a5] P.L. Duren, "Univalent functions" , Springer (1983) pp. 258
[a6] D.A. Brannan, "The Löwner differential equation" D.A. Brannan (ed.) J.G. Clunie (ed.) , Aspects of Contemporary Complex Analysis , Acad. Press (1980) pp. 79–95
How to Cite This Entry:
Löwner method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L%C3%B6wner_method&oldid=23408
This article was adapted from an original article by E.G. Goluzina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article