Convolutions of Harmonic Right Half-Plane Mappings with Harmonic Strip Mappings

  • Zhi-Hong Liu
  • Zhi-Gang Wang
  • Antti RasilaEmail author
  • Yue-Ping Jiang


We prove that convolutions of harmonic right half-plane mappings with harmonic vertical strip mappings are univalent and convex in the horizontal direction. The proofs of these results make use the Gauss–Lucas Theorem. Our results show that two recent conjectures, the one by Kumar, Gupta, Singh and Dorff, and the one of Liu, Jiang and Li, are true. Moreover, examples of univalent harmonic mappings related to the above-mentioned results are presented, suggesting that the bounds given by our results may be sharp.


Harmonic convolution Harmonic half-plane mappings Harmonic vertical strip mappings Gauss–Lucas Theorem 

Mathematics Subject Classification

Primary 58E20 Secondary 30C55 



The research of the first author was supported by the Foundation of Guilin University of Technology under Grant No. GUTQD-JJ2018080, the second author was supported by the Natural Science Foundation of Hunan Province under Grant no. 2016JJ2036. The research is financially supported by Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (Changsha University of Science & Technology). The authors would like to thank the anonymous referee and Mr. Jonathan Leake for his helpful discussions and suggestions in the preparation of this paper.


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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  1. 1.College of ScienceGuilin University of TechnologyGuilinPeople’s Republic of China
  2. 2.School of Mathematics and Computing ScienceHunan First Normal UniversityChangshaPeople’s Republic of China
  3. 3.Guangdong Technion – Israel Institute of TechnologyShantouPeople’s Republic of China
  4. 4.School of Mathematics and EconometricsHunan UniversityChangshaPeople’s Republic of China

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