On the crowding of parameters associated with Schwarz-Christoffel transformations
Let \(z_ 1,z_ 2,...,z_ n\) be n distinct points on \(\partial {\mathbb{D}}\) in counter clockwise order. Then the authors' problem is: For which linear map L of \({\mathbb{D}}\) onto itself are the images \(Z_ k=L(z_ k)\) least crowded, i.e. when is \(\min_{k}| Z_{k+1}-Z_ k|\) maximal \((Z_{n+1}=Z_ 1)?\) It is shown that the extremal L is uniquely determined, up to rotation, and that the \(Z_ k\) are least crowded if and only if for every semicircle \(S\subset \partial {\mathbb{D}}\) there is a pair \(Z_ k\), \(Z_{k+1}\) such that \[ | Z_{k+1}-Z_ k| =M:=\min_{k}| Z_{k+1}-Z_ k|, \] and at least one half of the arc from \(Z_ k\) to \(Z_{k+1}\) lies outside S. The problem is relevant in the use of the Schwarz-Christoffel formula; renormalization of the mapping can reduce the crowding phenomenon of the inverse images first noticed by the reviewer in Numer. Math. 19, 179-194 (1972; Zbl 0221.65053).
scientific article; zbMATH DE number 4081897