Univalency of convolutions of harmonic mappings
Applied Mathematics and Computation, 2014•Elsevier
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.
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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