Univalency of convolutions of harmonic mappings

Z. Boyd, M. Dorff, M. Nowak, M. Romney, M. Wołoszkiewicz

Research output: Contribution to journalArticlepeer-review

13 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)326-332
Number of pages7
JournalApplied Mathematics and Computation
Volume234
DOIs
StatePublished - 15 May 2014

Keywords

  • Convolutions
  • Harmonic mappings
  • Univalence

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