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''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|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. [[#References|[4]]], [[#References|[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|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

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

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Gaier,   "Vorlesungen über Approximation im Komplexen" , Birkhäuser  (1980)  pp. Chapt. 1, §6</TD></TR></table>