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on conformal mapping of domains with variable boundaries
One of the main results in the theory of conformal mapping of domains with variable boundaries; obtained by C. Carathéodory [1].
Let ,
be a sequence of simply-connected domains of the
-plane containing a fixed point
,
. If there exists a disc
,
, belonging to all
, then the kernel of the sequence
,
with respect to
is the largest domain
containing
and such that for each compact set
belonging to
there is an
such that
belongs to
for all
. A largest domain is one which contains any other domain having the same property. If there is no such a disc, then by the kernel
of the sequence
,
one means the point
(in this case one says that the sequence
,
has a degenerate kernel). A sequence of domains
,
converges to a kernel
if any subsequence of
has
as its kernel.
Carathéodory's theorem. Suppose that one is given a sequence of functions ,
,
,
that are regular and univalent in the disc
and that map this disc onto the domains
, respectively. Then in order that the sequence
,
converges in the disc
to a finite function
, it is necessary and sufficient that the sequence
,
converges to a kernel
which is either the point
or a domain containing more than one boundary point. Moreover, the convergence is uniform on compact sets in the interior of the disc
. If the limit function
, then it maps the disc
univalently onto
, and the inverse functions
,
are uniformly convergent on compact sets in the interior of
to the inverse function
of
.
The question of the convergence of univalent functions in multiply-connected domains is considered analogously. One such theorem is given below for unbounded domains. Let ,
be a sequence of domains in the
-plane containing some fixed neighbourhood of
. Then the kernel of the sequence
,
with respect to
is the largest domain
containing
and such that any closed subdomain of it is a subset of all
from some
onwards. Convergence of the sequence
,
to the kernel
is defined as before. The following theorem holds [2]. Let
,
be a sequence of domains in the
-plane containing
and converging to a kernel
, and suppose that the functions
,
map them univalently onto corresponding domains
containing
;
,
,
. Then in order that the sequence
,
converges uniformly on compact sets in the interior of
to a univalent function
, it is necessary and sufficient that the sequence
,
possesses a kernel
and converges to it. In this case
maps
univalently onto
.
It is possible to give other theorems on the convergence of a sequence of univalent functions, in dependence of the method of normalizing them (see [2]).
References
[1] | C. Carathéodory, "Untersuchungen über die konformen Abbildungen von festen und veränderlichen Gebieten" Math. Ann. , 72 (1912) pp. 107–144 |
[2] | G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian) |
Comments
References
[a1] | P.L. Duren, "Univalent functions" , Springer (1983) pp. Chapt. 3 |
Carathéodory theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carath%C3%A9odory_theorem&oldid=17662