Univalency of convolutions of harmonic mappings

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

doi:10.1016/j.amc.2014.01.162

Univalency of convolutions of harmonic mappings

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

Department of Mathematical Sciences

Brigham Young University
Maria Curie-Skłodowska University in Lublin

Research output: Contribution to journal › Article › peer-review

14 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 language	English
Pages (from-to)	326-332
Number of pages	7
Journal	Applied Mathematics and Computation
Volume	234
DOIs	https://doi.org/10.1016/j.amc.2014.01.162
State	Published - 15 May 2014

Keywords

Convolutions
Harmonic mappings
Univalence

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10.1016/j.amc.2014.01.162

Cite this

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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.",

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AU - Boyd, Z.

AU - Dorff, M.

AU - Nowak, M.

AU - Romney, M.

AU - Wołoszkiewicz, M.

PY - 2014/5/15

Y1 - 2014/5/15

N2 - 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.

AB - 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.

KW - Convolutions

KW - Harmonic mappings

KW - Univalence

UR - https://www.scopus.com/pages/publications/84896529728

UR - https://www.scopus.com/pages/publications/84896529728#tab=citedBy

U2 - 10.1016/j.amc.2014.01.162

DO - 10.1016/j.amc.2014.01.162

M3 - Article

AN - SCOPUS:84896529728

SN - 0096-3003

VL - 234

SP - 326

EP - 332

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

ER -

Univalency of convolutions of harmonic mappings

Abstract

Keywords

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