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NOTE ON THE CONVOLUTION OF HARMONIC MAPPINGS

Part of: Geometric function theory

Published online by Cambridge University Press:  13 February 2019

LIULAN LI  and
SAMINATHAN PONNUSAMY Open the ORCID record for SAMINATHAN PONNUSAMY [Opens in a new window]
LIULAN LI
Affiliation:
College of Mathematics and Statistics, Hunan Provincial Key Laboratory of Intelligent Information Processing and Application, Hengyang Normal University, Hengyang, Hunan 421002, PR China email lanlimail2012@sina.cn
SAMINATHAN PONNUSAMY*
Affiliation:
Department of Mathematics, Indian Institute of Technology Madras, Chennai-600 036, India email samy@iitm.ac.in
*
✉samy@iitm.ac.in
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Abstract

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Dorff et al. [‘Convolutions of harmonic convex mappings’, Complex Var. Elliptic Equ. 57(5) (2012), 489–503] formulated a question concerning the convolution of two right half-plane mappings, where the normalisation of the functions was considered incorrectly. In this paper, we reformulate the problem correctly and provide a solution to it in a more general form. We also obtain two new theorems which correct and improve related results.

Keywords

harmonicunivalentslanted half-plane mappingconvex mappingconvex in a directionconvolution

MSC classification

Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Secondary: 30C20: Conformal mappings of special domains 30C62: Quasiconformal mappings in the plane
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 3 , June 2019 , pp. 421 - 431
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

Footnotes

The work of the first author is supported by NSF of China (No. 11571216), Application-Oriented Characterized Disciplines, Double First-Class University Project of Hunan Province (Xiangjiaotong [2018] 469), the Science and Technology Plan Project of Hunan Province (No. 2016TP1020) and the Science and Technology Plan Project of Hengyang City (2017KJ183). The work of the second author is supported by the Mathematical Research Impact Centric Support of the Department of Science and Technology, India (MTR/2017/000367).

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