Conformal mapping of long quadrilaterals and thick doubly connected domains
Conformal maps \(f\) from a region \(R\) to a region \(G\) are in a sense localized. They do not change very much in a part \(R_ 1\) of \(R\) if the boundary of \(G\) is changed far away from the image \(f(R_ 1)\). The author gives several quantitative versions of this principle. The first result concerns the mapping \(f\) from the annulus \(\{r_ 2< | z|< r_ 1\}\) with \(r_ 2< 1<r_ 1\) to a doubly connected region \(D\). The region \(D\) and \(f\) are symmetric with respect to the real axis. Assume that the inner component \(K_ 2\) of the complement of \(D\) contains 0. Let \(f_ 1\) be the mapping of the disc \(\{| z|< r_ 1\}\) to \(D\cup K_ 2\) which satisfies \(f_ 1(0) =0\), \(f_ 1 (r_ 1)= f(r_ 1)\). The function \(f_ 1\) is assumed to be Hölder continuous of order \(\alpha\) with \(0<\alpha \leq 1\). Then \(|\log f(z)- \log f_ 1(z)|= O(r_ *^{\min (1,2 \alpha)})\) for \(1\leq| z|< r_ 1\) with \(r_ *= \max (r_ 2, 1/r_ 1)\). It is shown by an example that the order of approximation is best possible. There is also an estimate for the difference of the inverse maps. These results are used to obtain similar bounds for the approximation of the mapping from a rectangle to a long quadrilateral. The approximating function is the mapping from a semi-infinite strip to the quadrilateral with one of the sides moved to infinity. This extends results of Papamichael and Stylianopoulos. The dependence on the Hölder exponent \(\alpha\) is stronger than conjectured.
scientific article; zbMATH DE number 700845
scientific article