Walter Hengartner

Univalent harmonic functions

Several families of complex-valued, univalent, harmonic functions are studied from the point of view of geometric function theory. One class consists of mappings of a simply-connected domain onto an infinite horizontal strip with a normalization at the origin. Extreme points and support points are determined, as well as sharp estimates for Fourier coefficients and distortion theorems. Next, mappings in Izi > 1 are considered that leave infinity fixed. Some coefficient estimates, distortion theorems, and covering properties are obtained. For such mappings with real boundary values, many extremal problems are solved explicitly.

Spirallike logharmonic mappings

Univalent logharmonic mappings from the unit disc onto spirallike domains are represented by means of univalent conformal spirallike mappings. Furthermore the class of logharmonic automorphisms on U is completely characterized.

On the boundary behavior of orientation-preserving harmonic mappings

Nonconstant soluliuns of the partial differential equation where a is analytic in the open unit disk and |a| <1, are orientation-preserving harmonic mappings. We consider the case where ais a finite Blaschke product, so that ‖a‖= 1, and describe the behavior at the boundary. Some applications include conditions under which the image is a polygon and a situation where a univalent solution does not exist that maps onto a prescribed domain with a normalization at three boundary points.

Boundary values versus dilatations of harmonic mappings

AbstractThis article is divided into two parts. In the first part, we consider univalent harmonic mappings from the unit diskU onto a Jordan domain Ω whose dilatation functions

Curvature Estimates for some Minimal Surfaces

a = \bar f_{\bar z} /f_z

Shift Generated Haar Spaces on Unbounded, Closed Domains in the Complex Plane

have modulus one on an interval of the unit circle. The boundary values off depend very strongly on the values ofa(eit). A complete characterization of the inverse imagef-1(q) of a pointq on ∂Ω is given. We then consider the case where the dilatation functiona(z) is a finite Blaschke product of degreeN. It is shown that in this case, Ω can have at mostN+2 points of convexity. Finally, we give a complete characterization of simply connected Jordan domains Ω with the property that there exists a nonparametric minimal surface over Ω such that the image of its Gaussian map is the upper half-sphere covered exactly once.

Minimal Surfaces whose Gauss Map Covers Periodically the Pointed Upper Half-Sphere Exactly Once

A harmonic polynomial of degreen has at mostn2 zeros. It is shown that this bound is exact.

Canonical point mappings in

Let D be a simply-connected domain in the complex plane ℂ, and let f = u + iv be a univalent, orientation-preserving, harmonic mapping from D into ℂ. Then f can be written in the form f = h + \(\bar g\) where h and g belong to the linear space H(D) of analytic functions on D. In addition, f can be viewed as a solution of the elliptic partial differential equation