Boundary values versus dilatations of harmonic mappings (Q1385425)

The authors study univalent harmonic mappings from the unit disc onto a Jordan domain. They show that the boundary values of a function \(f\) depend very strongly on the boundary values of the dilatation function \(a= f_z/f_z\). When the function \(a\) is one on an interval of the unit circle, they give a complete characterization of the inverse image \(f^{-1}(q)\) of a point \(q\) on the boundary of the image domain. The authors also study the case where the dilatation function \(a(z)\) is a finite Blaschke product of degree \(N\). They show under this assumption that the image domain \(f(D)\) can have at most \(N+2\) points of convexity. Finally, a nice application to minimal surfaces is included.