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.
|Number of pages||14|
|Journal||Bulletin of the Malaysian Mathematical Sciences Society|
|State||Published - 15 May 2019|
- Gauss–Lucas Theorem
- Harmonic convolution
- Harmonic half-plane mappings
- Harmonic vertical strip mappings