Level lines
of a Green function
The point sets
![]() |
where is the Green function for the domain
in the complex plane with pole at the point
. If
is simply connected, then the structure of this set is easily determined by conformally mapping
onto the disc
, taking the point
to
. The Green function is invariant under this transformation, while the level lines of the Green function for the disc
with pole at
, i.e.
, are the circles
. So, in the case of a simply-connected domain, the level line
is a simple closed curve, coinciding for
with the boundary of
and tending to
as
. If the domain
is
-connected and its boundary consists of Jordan curves
,
, then: if
is sufficiently large, the level line is a Jordan curve; for
the corresponding level line tends to the point
, while for decreasing
it moves away from
; if
, then for certain values of
the level line has self-intersection, and decomposes into non-intersecting simple closed curves; for sufficiently small
the level line consists of
Jordan curves and for
each of these curves tends to one of the boundary curves of
.
In questions of the approximation of functions by polynomials on a closed bounded set with a simply-connected complement, an important role is played by estimates for the distance between boundary points of
and level lines of the complement of
(cf. [4], [5]).
For univalent conformal mappings of the disc by functions of the class
(cf. Univalent function), the behaviour of the level line
(the image of the circle
) intuitively gives the degree of distortion. Any function of class
maps the disc
,
, onto a convex domain, while the disc
,
, is mapped onto a star-like domain. The level line
,
,
, belongs to the annulus
![]() |
and bounds a simply-connected domain comprising the coordinate origin.
For the curvature of the level line
in the class
one has the following sharp estimate:
![]() |
and equality holds only for the function at the point
. The exact upper bound for
in the class
is at present (1984) not known. The exact upper bound for
in the subclass of star-like functions in
(cf. Star-like function) has the form
![]() |
and equality holds only for the function at
.
For mappings of the disc by functions of the class
the number of points of inflection of the level line
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
runs over the circle
in a given direction) may change non-monotonically for increasing
, i.e. if
, one can show that the level line
may have more points of inflection and more points violating the star-likeness condition than
.
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 |
Level lines. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Level_lines&oldid=47617