Univalency of convolutions of harmonic mappings

Z Boyd, M Dorff, M Nowak, M Romney… - Applied Mathematics …, 2014 - Elsevier
Z Boyd, M Dorff, M Nowak, M Romney, M Wołoszkiewicz
Applied Mathematics and Computation, 2014Elsevier
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.
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.
Elsevier
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