Boundary properties of planar harmonic mappings

A planar harmonic mapping of a domain ;For the mappings produced by the shear construction, relationships between the geometry of the mapping and its dilatation are suggested by calculating examples and displaying them graphically with Mathematica. An integral representation for the shear operator shows that the boundary functions of harmonic mappings with ;Mappings onto bounded convex domains can be represented by the Poisson integral of the boundary function. As with the shear construction, choosing simple examples suggests relationships between the boundary function and the dilatation. An examination of the Poisson integral shows that if the boundary function has a jump-discontinuity at a point z, then ;The Poisson integral representation reduces linear extremal problems for self-mappings of the disk to those of maximizing a functional over admissible boundary correspondences. The problems of maximizing a balanced sum of Fourier coefficients and of maximizing...