# 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)