Level lines

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