Carathéodory theorem

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 $B_n$, $n=1,2,\dots,$ be a sequence of simply-connected domains of the $z$-plane containing a fixed point $z_0$, $z_0\neq\infty$. If there exists a disc $|z-z_0|<\rho$, $\rho>0$, belonging to all $B_n$, then the kernel of the sequence $B_n$, $n=1,2,\dots,$ with respect to $z_0$ is the largest domain $B$ containing $z_0$ and such that for each compact set $E$ belonging to $B$ there is an $N$ such that $E$ belongs to $B_n$ for all $n\geq N$. 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 $B$ of the sequence $B_n$, $n=1,2,\dots,$ one means the point $z_0$ (in this case one says that the sequence $B_n$, $n=1,2,\dots,$ has a degenerate kernel). A sequence of domains $B_n$, $n=1,2,\dots,$ converges to a kernel $B$ if any subsequence of $B_n$ has $B$ as its kernel.

Carathéodory's theorem. Suppose that one is given a sequence of functions $z=f_n(\zeta)$, $f_n(\zeta_0)=z_0$, $f'_n(\zeta_0)>0$, $n=1,2,\dots,$ that are regular and univalent in the disc $|\zeta-\zeta_0|<1$ and that map this disc onto the domains $B_n$, respectively. Then in order that the sequence $f_n(\zeta)$, $n=1,2,\dots,$ converges in the disc $|\zeta-\zeta_0|<1$ to a finite function $f(\zeta)$, it is necessary and sufficient that the sequence $B_n$, $n=1,2,\dots,$ converges to a kernel $B$ which is either the point $z_0$ or a domain containing more than one boundary point. Moreover, the convergence is uniform on compact sets in the interior of the disc $|\zeta-\zeta_0|<1$. If the limit function $f(\zeta)\not\equiv\mathrm{const}$, then it maps the disc $|\zeta-\zeta_0|<1$ univalently onto $B$, and the inverse functions $\phi_n(z)$, $n=1,2,\dots,$ are uniformly convergent on compact sets in the interior of $B$ to the inverse function $\phi(z)$ of $f(\zeta)$.

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 $B_n$, $n=1,2,\dots,$ be a sequence of domains in the $z$-plane containing some fixed neighbourhood of $z=\infty$. Then the kernel of the sequence $B_n$, $n=1,2,\dots,$ with respect to $z=\infty$ is the largest domain $B$ containing $z=\infty$ and such that any closed subdomain of it is a subset of all $B_n$ from some $n$ onwards. Convergence of the sequence $B_n$, $n=1,2,\dots,$ to the kernel $B$ is defined as before. The following theorem holds [2]. Let $A_n$, $n=1,2,\dots,$ be a sequence of domains in the $z$-plane containing $z=\infty$ and converging to a kernel $A$, and suppose that the functions $\zeta=f_n(z)$, $n=1,2,\dots,$ map them univalently onto corresponding domains $B_n$ containing $\zeta=\infty$; $f_n(\infty)=\infty$, $f'_n(\infty)=1$, $n=1,2,\dots$. Then in order that the sequence $f_n(z)$, $n=1,2,\dots,$ converges uniformly on compact sets in the interior of $A$ to a univalent function $f(z)$, it is necessary and sufficient that the sequence $B_n$, $n=1,2,\dots,$ possesses a kernel $B$ and converges to it. In this case $\zeta=f(z)$ maps $A$ univalently onto $B$.

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)

References

[a1]	P.L. Duren, "Univalent functions" , Springer (1983) pp. Chapt. 3