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.

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

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