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A method which is a synthesis of Goluzin's method of variations (cf. [[Internal variations, method of|Internal variations, method of]]) and Loewner's [[Parametric representation method|parametric representation method]] for the important subclass of univalent functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v0961801.png" /> mapping the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v0961802.png" /> onto domains obtained from the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v0961803.png" /> by cutting along piecewise-continuous arcs. This synthesis is obtained by a special variation which, in the simplest case of one Jordan cut, is determined by the following theorem. Let the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v0961804.png" /> map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v0961805.png" /> onto the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v0961806.png" /> obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v0961807.png" /> by performing the cut
+
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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/v/v096/v096180/v0961808.png" /></td> </tr></table>
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 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v0961809.png" /> is continuous, while the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618011.png" />, is simply connected. A parametrization of the cut <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618012.png" /> may be considered such that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618014.png" />, associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618015.png" />, which univalently and conformally maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618016.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618017.png" />, is normalized by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618018.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618019.png" /> denote the function inverse to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618020.png" /> for a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618021.png" />. Then, for all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618022.png" />, (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618023.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618024.png" />) and for all constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618025.png" />, there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618026.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618027.png" /> which may be represented in the form
+
A method which is a synthesis of Goluzin's method of variations (cf. [[Internal variations, method of|Internal variations, method of]]) and Loewner's [[Parametric representation method|parametric representation method]] for the important subclass of univalent functions of class  $  S $
 +
mapping the disc  $  E = \{ {z } : {| z | < 1 } \} $
 +
onto domains obtained from the plane  $  \mathbf C _ {w} $
 +
by cutting along piecewise-continuous arcs. This synthesis is obtained by a special variation which, in the simplest case of one Jordan cut, is determined by the following theorem. Let the function $  w = f ( z) \in S $
 +
map  $  E $
 +
onto the domain  $  B( 0) $
 +
obtained from  $  {\overline{\mathbf C}\; } _ {w} $
 +
by performing the cut
  
<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/v/v096/v096180/v09618028.png" /></td> </tr></table>
+
$$
 +
= \{ {w } : {w = \phi ( t), 0 \leq  t \leq  \infty } \}
 +
,\ \
 +
\phi ( \infty ) = \infty ,
 +
$$
  
<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/v/v096/v096180/v09618029.png" /></td> </tr></table>
+
where  $  \phi ( t) $
 +
is continuous, while the domain  $  B( \tau ) = {\overline{\mathbf C}\; } _ {w} \setminus  L ( \tau ) $,
 +
where  $  L( \tau ) = \{ {w } : {w = \phi ( t),  0 \leq  \tau \leq  t \leq  \infty } \} $,
 +
is simply connected. A parametrization of the cut  $  L $
 +
may be considered such that the function  $  z = F( w, \tau ) $,
 +
$  F( 0, \tau ) = 0 $,
 +
associated to  $  f( z) $,
 +
which univalently and conformally maps  $  B( \tau ) $
 +
onto  $  E $,
 +
is normalized by the condition  $  F _ {w} ^ { \prime } ( 0, \tau ) = e ^ {- \tau } $.  
 +
Let  $  \Psi ( z, \tau ) $
 +
denote the function inverse to  $  F( w, \tau ) $
 +
for a fixed  $  \tau $.  
 +
Then, for all points  $  z _ {k} \in E $,
 +
( $  k = 1 \dots n $;
 +
$  n = 1, 2 ,\dots $)
 +
and for all constants  $  A _ {k} $,
 +
there exists a function  $  f _ {*} ( z) $
 +
in  $  S $
 +
which may be represented in 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/v/v096/v096180/v09618030.png" /></td> </tr></table>
+
$$
 +
f _ {*} ( z)  = f ( z) +
 +
$$
 +
 
 +
$$
 +
+
 +
\lambda \sum _ {k = 1 } ^ { n }  \left [ 2A _ {k} H  ^ {2} ( z _ {k} ,
 +
\tau )
 +
\frac{f  ^ {2} ( z) }{f ( z) - \Psi ( z _ {k} , \tau ) }
 +
\right . +
 +
$$
 +
 
 +
$$
 +
+ \left .
 +
A _ {k} K ( z, \tau , z _ {k} ) + A
 +
bar _ {k} K \left ( z, \tau ,
 +
\frac{1}{\overline{z}\; _ {k} }
 +
\right ) \right ] + \gamma ( \lambda , E).
 +
$$
  
 
Here
 
Here
  
<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/v/v096/v096180/v09618031.png" /></td> </tr></table>
+
$$
 +
K ( z, \tau , \zeta )  = \
 +
 
 +
\frac{F ( f  ( z), \tau ) }{F _ {w} ^ { \prime } ( f ( z), \tau ) }
 +
\cdot
  
<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/v/v096/v096180/v09618032.png" /></td> </tr></table>
+
\frac{\zeta + F ( f ( z), \tau ) }{\zeta - F ( f ( z), \tau ) }
 +
- f ( z),
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618033.png" /> is a holomorphic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618034.png" />, the limit of which with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618035.png" /> uniformly tends to zero inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618036.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618037.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618038.png" />).
+
$$
 +
H ( z, \tau )  =
 +
\frac{z \Psi _ {z}  ^  \prime  ( z, \tau ) }{\Psi ( z, \tau ) }
 +
,
 +
$$
  
If, in the course of study of extremal problems in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618039.png" />, use is made of the special variation mentioned above and of Loewner's equation
+
and  $  \gamma ( \lambda , E) $
 +
is a holomorphic function in $  E $,
 +
the limit of which with respect to  $  \lambda $
 +
uniformly tends to zero inside  $  E $
 +
as  $  \lambda \rightarrow 0 $(
 +
$  \lambda > 0 $).
  
<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/v/v096/v096180/v09618040.png" /></td> </tr></table>
+
If, in the course of study of extremal problems in  $  S $,
 +
use is made of the special variation mentioned above and of Loewner's equation
  
<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/v/v096/v096180/v09618041.png" /></td> </tr></table>
+
$$
  
which is satisfied by a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618042.png" /> subject to the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618043.png" />, two equations are usually obtained for the function which was associated to the extremal function. Irrespective of the constants contained in the equations, which can be expressed as values of the extremal function, further study of these equations frequently yields complete solutions of the problems under consideration, in particular in the problem of the domain of values of a functional which depends analytically on the function, its derivative and their conjugate values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618044.png" />. The method was proposed by P.P. Kufarev [[#References|[1]]]; for subsequent development and applications of the method see [[#References|[2]]]–[[#References|[5]]].
+
\frac{d \zeta }{d \tau }
 +
  = \
 +
- \zeta
 +
\frac{\mu ( \tau ) + \zeta }{\mu ( \tau ) - \zeta }
 +
,
 +
$$
 +
 
 +
$$
 +
\mu ( \tau )  =  \Psi ( \phi ( \tau ), \tau ),
 +
$$
 +
 
 +
which is satisfied by a function $  F( w, \tau ) $
 +
subject to the condition $  F( f( z), 0) = z $,  
 +
two equations are usually obtained for the function which was associated to the extremal function. Irrespective of the constants contained in the equations, which can be expressed as values of the extremal function, further study of these equations frequently yields complete solutions of the problems under consideration, in particular in the problem of the domain of values of a functional which depends analytically on the function, its derivative and their conjugate values in $  S $.  
 +
The method was proposed by P.P. Kufarev [[#References|[1]]]; for subsequent development and applications of the method see [[#References|[2]]]–[[#References|[5]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.P. Kufarev,  "On one property of extremal domains of coefficient problems"  ''Dokl. Akad. Nauk SSSR'' , '''97''' :  3  (1954)  pp. 391–393  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.A. Aleksandrov,  ''Uchen. Zap. Tomsk. Univ.'' , '''32'''  (1958)  pp. 41–57</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.A. Aleksandrov,  "Boundary values of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618045.png" /> on the class of holomorphic univalent functions in the disc"  ''Sibirsk. Mat. Zh.'' , '''4''' :  1  (1963)  pp. 17–31  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.I. Red'kov,  "The domains of values of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618046.png" /> for certain classes of bounded univalent functions"  ''Soviet Math. Dokl.'' , '''1''' :  2  (1960)  pp. 848–851  ''Dokl. Akad. Nauk SSSR'' , '''133''' :  2  (1960)  pp. 284–287</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.I. Red'kov,  "The domain of values of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618047.png" /> on the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618048.png" />"  ''Izv. Vyssh. Uchebn. Zaved. Mat.'' , '''4(29)'''  (1962)  pp. 134–142  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.P. Kufarev,  "On one property of extremal domains of coefficient problems"  ''Dokl. Akad. Nauk SSSR'' , '''97''' :  3  (1954)  pp. 391–393  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.A. Aleksandrov,  ''Uchen. Zap. Tomsk. Univ.'' , '''32'''  (1958)  pp. 41–57</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.A. Aleksandrov,  "Boundary values of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618045.png" /> on the class of holomorphic univalent functions in the disc"  ''Sibirsk. Mat. Zh.'' , '''4''' :  1  (1963)  pp. 17–31  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.I. Red'kov,  "The domains of values of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618046.png" /> for certain classes of bounded univalent functions"  ''Soviet Math. Dokl.'' , '''1''' :  2  (1960)  pp. 848–851  ''Dokl. Akad. Nauk SSSR'' , '''133''' :  2  (1960)  pp. 284–287</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.I. Red'kov,  "The domain of values of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618047.png" /> on the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096180/v09618048.png" />"  ''Izv. Vyssh. Uchebn. Zaved. Mat.'' , '''4(29)'''  (1962)  pp. 134–142  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:27, 6 June 2020


A method which is a synthesis of Goluzin's method of variations (cf. Internal variations, method of) and Loewner's parametric representation method for the important subclass of univalent functions of class $ S $ mapping the disc $ E = \{ {z } : {| z | < 1 } \} $ onto domains obtained from the plane $ \mathbf C _ {w} $ by cutting along piecewise-continuous arcs. This synthesis is obtained by a special variation which, in the simplest case of one Jordan cut, is determined by the following theorem. Let the function $ w = f ( z) \in S $ map $ E $ onto the domain $ B( 0) $ obtained from $ {\overline{\mathbf C}\; } _ {w} $ by performing the cut

$$ L = \{ {w } : {w = \phi ( t), 0 \leq t \leq \infty } \} ,\ \ \phi ( \infty ) = \infty , $$

where $ \phi ( t) $ is continuous, while the domain $ B( \tau ) = {\overline{\mathbf C}\; } _ {w} \setminus L ( \tau ) $, where $ L( \tau ) = \{ {w } : {w = \phi ( t), 0 \leq \tau \leq t \leq \infty } \} $, is simply connected. A parametrization of the cut $ L $ may be considered such that the function $ z = F( w, \tau ) $, $ F( 0, \tau ) = 0 $, associated to $ f( z) $, which univalently and conformally maps $ B( \tau ) $ onto $ E $, is normalized by the condition $ F _ {w} ^ { \prime } ( 0, \tau ) = e ^ {- \tau } $. Let $ \Psi ( z, \tau ) $ denote the function inverse to $ F( w, \tau ) $ for a fixed $ \tau $. Then, for all points $ z _ {k} \in E $, ( $ k = 1 \dots n $; $ n = 1, 2 ,\dots $) and for all constants $ A _ {k} $, there exists a function $ f _ {*} ( z) $ in $ S $ which may be represented in the form

$$ f _ {*} ( z) = f ( z) + $$

$$ + \lambda \sum _ {k = 1 } ^ { n } \left [ 2A _ {k} H ^ {2} ( z _ {k} , \tau ) \frac{f ^ {2} ( z) }{f ( z) - \Psi ( z _ {k} , \tau ) } \right . + $$

$$ + \left . A _ {k} K ( z, \tau , z _ {k} ) + A bar _ {k} K \left ( z, \tau , \frac{1}{\overline{z}\; _ {k} } \right ) \right ] + \gamma ( \lambda , E). $$

Here

$$ K ( z, \tau , \zeta ) = \ \frac{F ( f ( z), \tau ) }{F _ {w} ^ { \prime } ( f ( z), \tau ) } \cdot \frac{\zeta + F ( f ( z), \tau ) }{\zeta - F ( f ( z), \tau ) } - f ( z), $$

$$ H ( z, \tau ) = \frac{z \Psi _ {z} ^ \prime ( z, \tau ) }{\Psi ( z, \tau ) } , $$

and $ \gamma ( \lambda , E) $ is a holomorphic function in $ E $, the limit of which with respect to $ \lambda $ uniformly tends to zero inside $ E $ as $ \lambda \rightarrow 0 $( $ \lambda > 0 $).

If, in the course of study of extremal problems in $ S $, use is made of the special variation mentioned above and of Loewner's equation

$$ \frac{d \zeta }{d \tau } = \ - \zeta \frac{\mu ( \tau ) + \zeta }{\mu ( \tau ) - \zeta } , $$

$$ \mu ( \tau ) = \Psi ( \phi ( \tau ), \tau ), $$

which is satisfied by a function $ F( w, \tau ) $ subject to the condition $ F( f( z), 0) = z $, two equations are usually obtained for the function which was associated to the extremal function. Irrespective of the constants contained in the equations, which can be expressed as values of the extremal function, further study of these equations frequently yields complete solutions of the problems under consideration, in particular in the problem of the domain of values of a functional which depends analytically on the function, its derivative and their conjugate values in $ S $. The method was proposed by P.P. Kufarev [1]; for subsequent development and applications of the method see [2][5].

References

[1] P.P. Kufarev, "On one property of extremal domains of coefficient problems" Dokl. Akad. Nauk SSSR , 97 : 3 (1954) pp. 391–393 (In Russian)
[2] I.A. Aleksandrov, Uchen. Zap. Tomsk. Univ. , 32 (1958) pp. 41–57
[3] I.A. Aleksandrov, "Boundary values of the functional on the class of holomorphic univalent functions in the disc" Sibirsk. Mat. Zh. , 4 : 1 (1963) pp. 17–31 (In Russian)
[4] M.I. Red'kov, "The domains of values of the functional for certain classes of bounded univalent functions" Soviet Math. Dokl. , 1 : 2 (1960) pp. 848–851 Dokl. Akad. Nauk SSSR , 133 : 2 (1960) pp. 284–287
[5] M.I. Red'kov, "The domain of values of the functional on the class " Izv. Vyssh. Uchebn. Zaved. Mat. , 4(29) (1962) pp. 134–142 (In Russian)

Comments

The idea to combine variational methods and Loewner theory seems to go back to M. Schiffer, cf. [a2], Chapts. 10, 11; see [a2], also for further references.

References

[a1] I.A. Aleksandrov, "Parametric extensions in the theory of univalent functions" , Moscow (1976) (In Russian)
[a2] P.L. Duren, "Univalent functions" , Springer (1983) pp. Sect. 10.11
How to Cite This Entry:
Variation-parametric method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Variation-parametric_method&oldid=19274
This article was adapted from an original article by I.A. Aleksandrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article