Many properties are known about analytic functions, however the class of harmonic functions which are the sum of an analytic function and the conjugate of an analytic function is less understood. We wish to find conditions such that linear combinations of univalent harmonic functions are univalent. We focus on functions whose image is convex in one direction i.e. each line segment in that direction between points in the image is contained in the image. M. Dorff proved sufficient conditions such that the linear combination of univalent harmonic functions will be univalent on the unit disk. The conditions are: the mappings must be locally univalent, their images must be convex in the imaginary direction and they must satisfy a normalization which states that the right and left extremes of the image are the image of 1 and -1 respectively. In this paper we generalize this existing theorem. The conditions of this theorem are geometric, and we would like to maintain this feature in the generalization. We show that the image may be convex in any direction and that any points on the boundary of the domain, which no longer must be the unit disk, can be the points that are mapped to the extrema, which now must be in the direction perpendicular to the direction of convexity.
Dr. David R. Martin
"Linear Combinations of Harmonic Univalent Mappings,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 21
, Article 4.
Available at: https://scholar.rose-hulman.edu/rhumj/vol21/iss2/4