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''of a Green function''
 
''of a Green function''
  
 
The point sets
 
The point sets
  
<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/l058/l058210/l0582101.png" /></td> </tr></table>
+
$$
 +
L _  \lambda  = \{ {z \in D } : {
 +
G ( z , z _ {0} ) = \lambda = \textrm{ const } } \}
 +
,\
 +
0 \leq  \lambda < \infty ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l0582102.png" /> is the [[Green function|Green function]] for the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l0582103.png" /> in the complex plane with pole at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l0582104.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l0582105.png" /> is simply connected, then the structure of this set is easily determined by conformally mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l0582106.png" /> onto the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l0582107.png" />, taking the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l0582108.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l0582109.png" />. The Green function is invariant under this transformation, while the level lines of the Green function for the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821010.png" /> with pole at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821011.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821012.png" />, are the circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821013.png" />. So, in the case of a simply-connected domain, the level line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821014.png" /> is a simple closed curve, coinciding for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821015.png" /> with the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821016.png" /> and tending to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821017.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821018.png" />. If the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821019.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821020.png" />-connected and its boundary consists of Jordan curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821022.png" />, then: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821023.png" /> is sufficiently large, the level line is a Jordan curve; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821024.png" /> the corresponding level line tends to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821025.png" />, while for decreasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821026.png" /> it moves away from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821027.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821028.png" />, then for certain values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821029.png" /> the level line has self-intersection, and decomposes into non-intersecting simple closed curves; for sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821030.png" /> the level line consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821031.png" /> Jordan curves and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821032.png" /> each of these curves tends to one of the boundary curves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821033.png" />.
+
where $  G ( z , z _ {0} ) $
 +
is the [[Green function|Green function]] for the domain $  D $
 +
in the complex plane with pole at the point $  z _ {0} \in D $.  
 +
If $  D $
 +
is simply connected, then the structure of this set is easily determined by conformally mapping $  D $
 +
onto the disc $  | \zeta | < 1 $,  
 +
taking the point $  z _ {0} $
 +
to $  \zeta = 0 $.  
 +
The Green function is invariant under this transformation, while the level lines of the Green function for the disc $  | \zeta | < 1 $
 +
with pole at $  \zeta = 0 $,  
 +
i.e. $  -  \mathop{\rm log}  | \zeta | $,  
 +
are the circles $  | \zeta | = \textrm{ const } $.  
 +
So, in the case of a simply-connected domain, the level line $  G ( z , z _ {0} ) = \lambda $
 +
is a simple closed curve, coinciding for $  \lambda = 0 $
 +
with the boundary of $  D $
 +
and tending to $  z _ {0} $
 +
as $  \lambda \rightarrow + \infty $.  
 +
If the domain $  D $
 +
is $  m $-
 +
connected and its boundary consists of Jordan curves $  C _  \nu  $,  
 +
$  \nu = 1 \dots m $,  
 +
then: if $  \lambda > 0 $
 +
is sufficiently large, the level line is a Jordan curve; for $  \lambda \rightarrow + \infty $
 +
the corresponding level line tends to the point $  z _ {0} $,  
 +
while for decreasing $  \lambda $
 +
it moves away from $  z _ {0} $;  
 +
if $  m > 1 $,  
 +
then for certain values of $  \lambda $
 +
the level line has self-intersection, and decomposes into non-intersecting simple closed curves; for sufficiently small $  \lambda $
 +
the level line consists of $  m $
 +
Jordan curves and for $  \lambda \rightarrow 0 $
 +
each of these curves tends to one of the boundary curves of $  D $.
  
In questions of the approximation of functions by polynomials on a closed bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821034.png" /> with a simply-connected complement, an important role is played by estimates for the distance between boundary points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821035.png" /> and level lines of the complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821036.png" /> (cf. [[#References|[4]]], [[#References|[5]]]).
+
In questions of the approximation of functions by polynomials on a closed bounded set $  B $
 +
with a simply-connected complement, an important role is played by estimates for the distance between boundary points of $  B $
 +
and level lines of the complement of $  B $(
 +
cf. [[#References|[4]]], [[#References|[5]]]).
  
For univalent conformal mappings of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821037.png" /> by functions of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821038.png" /> (cf. [[Univalent function|Univalent function]]), the behaviour of the level line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821039.png" /> (the image of the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821040.png" />) intuitively gives the degree of distortion. Any function of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821041.png" /> maps the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821043.png" />, onto a [[Convex domain|convex domain]], while the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821045.png" />, is mapped onto a [[Star-like domain|star-like domain]]. The level line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821048.png" />, belongs to the annulus
+
For univalent conformal mappings of the disc $  | z | < 1 $
 +
by functions of the class $  S = \{ {f } : {f ( z) = z + \dots,  f  \textrm{ regular  and  univalent  in  }  | z | < 1 } \} $(
 +
cf. [[Univalent function|Univalent function]]), the behaviour of the level line $  L ( f , r ) $(
 +
the image of the circle $  | z | = r < 1 $)  
 +
intuitively gives the degree of distortion. Any function of class $  S $
 +
maps the disc $  | z | < r $,
 +
0 < r < 2 - \sqrt 3 $,  
 +
onto a [[Convex domain|convex domain]], while the disc $  | z | < r $,
 +
0 < r <  \mathop{\rm tanh}  \pi / 4 $,  
 +
is mapped onto a [[Star-like domain|star-like domain]]. The level line $  L ( f , r ) $,
 +
$  f \in S $,
 +
0 < r < 1 $,  
 +
belongs to the annulus
  
<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/l058/l058210/l05821049.png" /></td> </tr></table>
+
$$
 +
K _ {r}  = \{ {w } : {r ( 1 + r )  ^ {- 2} \leq  | w | \leq  r ( 1 - r )  ^ {- 2} } \}
 +
$$
  
 
and bounds a simply-connected domain comprising the coordinate origin.
 
and bounds a simply-connected domain comprising the coordinate origin.
  
For the curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821050.png" /> of the level line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821051.png" /> in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821052.png" /> one has the following sharp estimate:
+
For the curvature $  K ( f , r ) $
 +
of the level line $  L ( f , r ) $
 +
in the class  $  S $
 +
one has the following sharp estimate:
 +
 
 +
$$
 +
K ( f , r )  \geq  \
 +
 
 +
\frac{1 - 4 r + r  ^ {2} }{r}
 +
 
 +
\left ( 1+
 +
\frac{r}{1-r} \right )  ^ {2} ,
 +
$$
 +
 
 +
and equality holds only for the function  $  f ( z) = z / ( 1 + z )  ^ {2} $
 +
at the point  $  z = r $.  
 +
The exact upper bound for  $  K ( f , r ) $
 +
in the class $  S $
 +
is at present (1984) not known. The exact upper bound for  $  K ( f , r ) $
 +
in the subclass of star-like functions in  $  S $(
 +
cf. [[Star-like function|Star-like function]]) 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/l/l058/l058210/l05821053.png" /></td> </tr></table>
+
$$
 +
K ( f , r )  \leq  \
  
and equality holds only for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821054.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821055.png" />. The exact upper bound for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821056.png" /> in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821057.png" /> is at present (1984) not known. The exact upper bound for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821058.png" /> in the subclass of star-like functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821059.png" /> (cf. [[Star-like function|Star-like function]]) has the form
+
\frac{1 + 4 r + r  ^ {2} }{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/l/l058/l058210/l05821060.png" /></td> </tr></table>
+
\left ( 1-  
 +
\frac{r}{1+r} \right )  ^ {2} ,
 +
$$
  
and equality holds only for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821061.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821062.png" />.
+
and equality holds only for the function $  f ( z) = z / ( 1 - z )  ^ {2} $
 +
at $  z = r $.
  
For mappings of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821063.png" /> by functions of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821064.png" /> the number of points of inflection of the level line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821065.png" /> and the number of points violating the star-likeness condition (i.e. points of the level line at which the direction of rotation of the radius vector changes when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821066.png" /> runs over the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821067.png" /> in a given direction) may change non-monotonically for increasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821068.png" />, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821069.png" />, one can show that the level line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821070.png" /> may have more points of inflection and more points violating the star-likeness condition than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058210/l05821071.png" />.
+
For mappings of the disc $  | z | < 1 $
 +
by functions of the class $  S $
 +
the number of points of inflection of the level line $  L ( f , r ) $
 +
and the number of points violating the star-likeness condition (i.e. points of the level line at which the direction of rotation of the radius vector changes when $  z $
 +
runs over the circle $  | z | = r $
 +
in a given direction) may change non-monotonically for increasing $  r $,  
 +
i.e. if $  r _ {1} < r _ {2} $,  
 +
one can show that the level line $  L ( f , r _ {1} ) $
 +
may have more points of inflection and more points violating the star-likeness condition than $  L ( f , r _ {2} ) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Stoilov,  "The theory of functions of a complex variable" , '''1''' , Moscow  (1962)  (In Russian; translated from Rumanian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  pp. Appendix  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.A. Aleksandrov,  "Parametric extensions in the theory of univalent functions" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.K. Dzyadyk,  "On a problem of S.M. Nikol'skii in a complex region"  ''Izv. Akad. Nauk SSSR Mat.'' , '''23''' :  5  (1959)  pp. 697–763  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.A. Lebedev,  N.A. Shirokov,  "The uniform approximation of functions on closed sets with a finite number of angular points with non-zero exterior angles"  ''Izv. Akad. Nauk Armen. SSR Ser. Mat.'' , '''6''' :  4  (1971)  pp. 311–341  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Stoilov,  "The theory of functions of a complex variable" , '''1''' , Moscow  (1962)  (In Russian; translated from Rumanian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  pp. Appendix  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.A. Aleksandrov,  "Parametric extensions in the theory of univalent functions" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.K. Dzyadyk,  "On a problem of S.M. Nikol'skii in a complex region"  ''Izv. Akad. Nauk SSSR Mat.'' , '''23''' :  5  (1959)  pp. 697–763  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.A. Lebedev,  N.A. Shirokov,  "The uniform approximation of functions on closed sets with a finite number of angular points with non-zero exterior angles"  ''Izv. Akad. Nauk Armen. SSR Ser. Mat.'' , '''6''' :  4  (1971)  pp. 311–341  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 14:03, 18 May 2023


of a Green function

The point sets

$$ L _ \lambda = \{ {z \in D } : { G ( z , z _ {0} ) = \lambda = \textrm{ const } } \} ,\ 0 \leq \lambda < \infty , $$

where $ G ( z , z _ {0} ) $ is the Green function for the domain $ D $ in the complex plane with pole at the point $ z _ {0} \in D $. If $ D $ is simply connected, then the structure of this set is easily determined by conformally mapping $ D $ onto the disc $ | \zeta | < 1 $, taking the point $ z _ {0} $ to $ \zeta = 0 $. The Green function is invariant under this transformation, while the level lines of the Green function for the disc $ | \zeta | < 1 $ with pole at $ \zeta = 0 $, i.e. $ - \mathop{\rm log} | \zeta | $, are the circles $ | \zeta | = \textrm{ const } $. So, in the case of a simply-connected domain, the level line $ G ( z , z _ {0} ) = \lambda $ is a simple closed curve, coinciding for $ \lambda = 0 $ with the boundary of $ D $ and tending to $ z _ {0} $ as $ \lambda \rightarrow + \infty $. If the domain $ D $ is $ m $- connected and its boundary consists of Jordan curves $ C _ \nu $, $ \nu = 1 \dots m $, then: if $ \lambda > 0 $ is sufficiently large, the level line is a Jordan curve; for $ \lambda \rightarrow + \infty $ the corresponding level line tends to the point $ z _ {0} $, while for decreasing $ \lambda $ it moves away from $ z _ {0} $; if $ m > 1 $, then for certain values of $ \lambda $ the level line has self-intersection, and decomposes into non-intersecting simple closed curves; for sufficiently small $ \lambda $ the level line consists of $ m $ Jordan curves and for $ \lambda \rightarrow 0 $ each of these curves tends to one of the boundary curves of $ D $.

In questions of the approximation of functions by polynomials on a closed bounded set $ B $ with a simply-connected complement, an important role is played by estimates for the distance between boundary points of $ B $ and level lines of the complement of $ B $( cf. [4], [5]).

For univalent conformal mappings of the disc $ | z | < 1 $ by functions of the class $ S = \{ {f } : {f ( z) = z + \dots, f \textrm{ regular and univalent in } | z | < 1 } \} $( cf. Univalent function), the behaviour of the level line $ L ( f , r ) $( the image of the circle $ | z | = r < 1 $) intuitively gives the degree of distortion. Any function of class $ S $ maps the disc $ | z | < r $, $ 0 < r < 2 - \sqrt 3 $, onto a convex domain, while the disc $ | z | < r $, $ 0 < r < \mathop{\rm tanh} \pi / 4 $, is mapped onto a star-like domain. The level line $ L ( f , r ) $, $ f \in S $, $ 0 < r < 1 $, belongs to the annulus

$$ K _ {r} = \{ {w } : {r ( 1 + r ) ^ {- 2} \leq | w | \leq r ( 1 - r ) ^ {- 2} } \} $$

and bounds a simply-connected domain comprising the coordinate origin.

For the curvature $ K ( f , r ) $ of the level line $ L ( f , r ) $ in the class $ S $ one has the following sharp estimate:

$$ K ( f , r ) \geq \ \frac{1 - 4 r + r ^ {2} }{r} \left ( 1+ \frac{r}{1-r} \right ) ^ {2} , $$

and equality holds only for the function $ f ( z) = z / ( 1 + z ) ^ {2} $ at the point $ z = r $. The exact upper bound for $ K ( f , r ) $ in the class $ S $ is at present (1984) not known. The exact upper bound for $ K ( f , r ) $ in the subclass of star-like functions in $ S $( cf. Star-like function) has the form

$$ K ( f , r ) \leq \ \frac{1 + 4 r + r ^ {2} }{r} \left ( 1- \frac{r}{1+r} \right ) ^ {2} , $$

and equality holds only for the function $ f ( z) = z / ( 1 - z ) ^ {2} $ at $ z = r $.

For mappings of the disc $ | z | < 1 $ by functions of the class $ S $ the number of points of inflection of the level line $ L ( f , r ) $ and the number of points violating the star-likeness condition (i.e. points of the level line at which the direction of rotation of the radius vector changes when $ z $ runs over the circle $ | z | = r $ in a given direction) may change non-monotonically for increasing $ r $, i.e. if $ r _ {1} < r _ {2} $, one can show that the level line $ L ( f , r _ {1} ) $ may have more points of inflection and more points violating the star-likeness condition than $ L ( f , r _ {2} ) $.

References

[1] S. Stoilov, "The theory of functions of a complex variable" , 1 , Moscow (1962) (In Russian; translated from Rumanian)
[2] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) pp. Appendix (Translated from Russian)
[3] I.A. Aleksandrov, "Parametric extensions in the theory of univalent functions" , Moscow (1976) (In Russian)
[4] V.K. Dzyadyk, "On a problem of S.M. Nikol'skii in a complex region" Izv. Akad. Nauk SSSR Mat. , 23 : 5 (1959) pp. 697–763 (In Russian)
[5] N.A. Lebedev, N.A. Shirokov, "The uniform approximation of functions on closed sets with a finite number of angular points with non-zero exterior angles" Izv. Akad. Nauk Armen. SSR Ser. Mat. , 6 : 4 (1971) pp. 311–341 (In Russian)

Comments

Some non-Soviet references for the approximation questions mentioned are [a1] and [a2], in which other references can be found. See also Approximation of functions of a complex variable.

References

[a1] L. Bijvoets, W. Hogeveen, J. Korevaar, "Inverse approximation theorems of Lebedev and Tamrazov" P.L. Butzer (ed.) , Functional analysis and approximation (Oberwolfach 1980) , Birkhäuser (1981) pp. 265–281
[a2] D. Gaier, "Vorlesungen über Approximation im Komplexen" , Birkhäuser (1980) pp. Chapt. 1, §6
How to Cite This Entry:
Level lines. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Level_lines&oldid=13582
This article was adapted from an original article by E.G. Goluzina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article