Conformal radius of a domain
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 |
Conformal radius of a domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conformal_radius_of_a_domain&oldid=46458