Approximation of analytic functions with prescribed boundary conditions by circle-packing maps (Q2365325)

Suppose that \(\Omega\) is a simply connected domain and that \(F:\Omega\to C\) is analytic in \(\Omega\) with a finite set of critical points, and that \(|F|:\overline\Omega\to[0,\infty)\) is continuous on \(\overline\Omega\) with \(|F|\equiv\lambda\) on \(\partial\Omega\). The problem here is to approximate \(F\) by a sequence of circle-packing maps. The author shows that this can be done by a sequence of circle-packing maps which converge uniformly on compacta of \(\Omega\) to \(F\). In addition, the corresponding sequence of ``ratio functions'', which measure the distortion of the circle packings, converge uniformly on compacta of \(\Omega\) to \(|F|\).