Courant theorem

on conformal mapping of domains with variable boundaries

Let $\{D_n\}$ be a sequence of nested simply-connected domains in the complex $z$-plane, $\overline{D_{n+1}}\subset D_n$, which converges to its kernel $D_{z_0}$ relative to some point $z_0$; the set $D_{z_0}$ is assumed to be bounded by a Jordan curve. Then the sequence of functions $\{w=f_n(z)\}$ which map $D_n$ conformally onto the disc $\Delta=\{w:|w|<1\}$, $f_n(z_0)=0$, $f'_n(z_0)>0$, is uniformly convergent in the closed domain $\overline{D_{z_0}}$ to the function $w=f(z)$ which maps $D_{z_0}$ conformally onto $\Delta$, moreover $f(z_0)=0$, $f'(z_0)>0$.

This theorem, due to R. Courant [1], is an extension of the Carathéodory theorem.

References

[1a]	R. Courant, Gott. Nachr. (1914) pp. 101–109
[1b]	R. Courant, Gott. Nachr. (1922) pp. 69–70
[2]	A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) (Translated from Russian)

Comments

Cf. Carathéodory theorem for the definition of "kernel of a sequence of domains" .