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