It is well known that complex harmonic polynomials of degree n may have more than n zeros. In this paper, we examine a one-parameter family of harmonic trinomials and determine how the number of zeros depends on the parameter. Our proof heavily utilizes the Argument Principle for Harmonic Functions and involves finding the winding numbers about the origin for a family of hypocycloids.

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A JS surface is a minimal graph over a polygonal domain that becomes infinite in magnitude at the domain boundary. Jenkins and Serrin characterized the existence of these minimal graphs in terms of the signs of the boundary values and the side-lengths of the polygon. For a convex polygon, there can be essentially only one JS surface, but a non-convex domain may admit several distinct JS surfaces. We consider two families of JS surfaces corresponding to different boundary values, namely JS 0 and JS 1 , over domains in the form of regular stars. We give parameterizations for these surfaces as lifts of harmonic maps, and observe that all previously constructed JS surfaces have been of type JS 0 . We give an example of a JS 1 surface that is a new complete embedded minimal surface generalizing Scherk's doubly periodic surface, and show also that the JS 0 surface over a regular convex 2n-gon is the limit of JS 1 surfaces over non-convex stars. Finally we consider the construction of other JS surfaces over stars that belong neither to JS 0 nor to JS 1 .

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A general version of the Radó-Kneser-Choquet theorem implies that a piecewise constant sensepreserving mapping of the unit circle onto the vertices of a convex polygon extends to a univalent harmonic mapping of the unit disk onto the polygonal domain. This paper discusses similarly generated harmonic mappings of the disk onto nonconvex polygonal regions in the shape of regular stars. Calculation of the Blaschke product dilatation allows a determination of the exact range of parameters that produce univalent mappings.  2004 Elsevier Inc. All rights reserved.