Conformal radius of a domain

A characteristic of a conformal mapping of a simply-connected domain, defined as follows: Let $D$ be a simply-connected domain with more than one boundary point in the $z$- plane. Let $z _ {0}$ be a point of $D$. If $z _ {0} \neq \infty$, then there exists a unique function $w = f ( z)$, holomorphic in $D$, normalized by the conditions $f ( z _ {0} ) = 0$, $f ^ { \prime } ( z _ {0} ) = 1$, that maps $D$ univalently onto the disc $\{ {w } : {| w | < r } \}$. The radius $r = r ( z _ {0} , D )$ of this disc is called the conformal radius of $D$ relative to $z _ {0}$. If $\infty \in D$, then there exists a unique function $w = f ( z)$, holomorphic in $D$ except at $\infty$, that, in a neighbourhood of $\infty$, has a Laurent expansion of the form

$$f ( z) = z + c _ {0} + c _ {1} z ^ {-} 1 + \dots ,$$

and that maps $D$ univalently onto a domain $\{ {w } : {| w | > r } \}$. In this case the quantity $r = r ( \infty , D )$ is called the conformal radius of $D$ relative to infinity. The conformal radius of $D$, $\infty \in D$, relative to infinity is equal to the transfinite diameter of the boundary $C$ of $D$ and to the capacity of the set $C$.

An extension of the notion of the conformal radius of a domain to the case of an arbitrary domain $D$ in the complex $z$- plane is that of the interior radius of $D$ relative to a point $z _ {0} \in D$( in the non-Soviet literature the term "interior radius" is used primarily in the case of a simply-connected domain). Let $D$ be a domain in the complex $z$- plane, let $z _ {0}$ be a point of $D$ and suppose that a Green function $g ( z , z _ {0} )$ for $D$ with pole at $z _ {0}$ exists. Let $\gamma$ be the Robin constant of $D$ with respect to $z _ {0}$, i.e.

$$\gamma = \ \left \{ \begin{array}{lll} \lim\limits _ {z \rightarrow z _ {0} } [ g ( z , z _ {0} ) + \mathop{\rm ln} | z - z _ {0} | ] & \textrm{ for } &z _ {0} \neq \infty , \\ \lim\limits _ {z \rightarrow \infty } [ g ( z , \infty ) - \mathop{\rm ln} | z | ] & \textrm{ for } &z _ {0} = \infty . \\ \end{array} \right.$$

The quantity $r = {e ^ \gamma }$ is called the interior radius of $D$ relative to $z _ {0}$. If $D$ is a simply-connected domain whose boundary contains at least two points, then the interior radius of $D$ relative to $z _ {0} \in D$ is equal to the conformal radius of $D$ relative to $z _ {0}$. The interior radius of a domain is non-decreasing as the domain increases: If the domains $D$, $D _ {1}$ have Green functions $g ( z _ {1} , z _ {0} )$, $g _ {1} ( z , z _ {0} )$, respectively, if $z _ {0} \in D$ and if $D \subset D _ {1}$, then the following inequality holds for their interior radii $r$, $r _ {1}$ at $z _ {0}$:

$$r \leq r _ {1} .$$

The interior radius of an arbitrary domain $D$ relative to a point $z _ {0} \in D$ is defined as the least upper bound of the set of interior radii at $z _ {0}$ of all domains containing $z _ {0}$, contained in $D$ and having a Green function. In accordance with this definition, if $D$ does not have a generalized Green function, then the interior radius $r$ of $D$ at $z _ {0} \in D$ is equal to $\infty$.

References

[1]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[2]	V.I. Smirnov, A.N. Lebedev, "Functions of a complex variable" , M.I.T. (1968) (Translated from Russian)
[3]	W.K. Hayman, "Multivalent functions" , Cambridge Univ. Press (1958)

Comments

In [a2] the conformal radius of a compact connected set $E$ in the $z$- plane is defined as the conformal radius of its complement relative to infinity (as defined above). If $E$ is contained in a disc of radius $r$ and has diameter $d \geq r$, then

$$\rho \leq r \leq 4 \rho ,$$

where $\rho$ is its conformal radius (in the sense of [a2], cf. [a2]).

References

[a1]	M. Tsuji, "Potential theory in modern function theory" , Chelsea, reprint (1975)
[a2]	P.L. Duren, "Univalent functions" , Springer (1983) pp. Sect. 10.11