Valancy of harmonic mappings onto bounded convex domains (Q1851477)

The authors show that the Sheil-Small conjecture is false. This conjecture is: if \(f\) is a harmonic mapping of \(\mathbb{D}\) onto a convex Jordan domain \(K\) that extends continuously to \(n\)-valent sense preserving local homeomorphism between \(\partial\mathbb{D}\) and \(\partial K\) and assumes every point in \(\partial K\) exactly \(n\) times, then the valency of \(f\) is at most \(n^2\).