Univalency of convolutions of harmonic mappings

摘要

We consider the convolution or Hadamard product of planar harmonic mappings that are the vertical shears of the canonical half-plane mapping φ(z)=z/(1-z) with respective dilatations -xz and -yz, where |x|=|y|=1. We prove that any such convolution is univalent. Furthermore, in the case that x=y=-1, we show the resulting convolution is convex.