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Conformal radius of a domain

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A characteristic of a conformal mapping of a simply-connected domain, defined as follows: Let be a simply-connected domain with more than one boundary point in the -plane. Let be a point of . If , then there exists a unique function , holomorphic in , normalized by the conditions , , that maps univalently onto the disc . The radius of this disc is called the conformal radius of relative to . If , then there exists a unique function , holomorphic in except at , that, in a neighbourhood of , has a Laurent expansion of the form

and that maps univalently onto a domain . In this case the quantity is called the conformal radius of relative to infinity. The conformal radius of , , relative to infinity is equal to the transfinite diameter of the boundary of and to the capacity of the set .

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

The quantity is called the interior radius of relative to . If is a simply-connected domain whose boundary contains at least two points, then the interior radius of relative to is equal to the conformal radius of relative to . The interior radius of a domain is non-decreasing as the domain increases: If the domains , have Green functions , , respectively, if and if , then the following inequality holds for their interior radii , at :

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

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 in the -plane is defined as the conformal radius of its complement relative to infinity (as defined above). If is contained in a disc of radius and has diameter , then

where 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
How to Cite This Entry:
Conformal radius of a domain. G.V. Kuz'mina (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Conformal_radius_of_a_domain&oldid=18740
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098