On Convolution and Convex Combination of Harmonic Mappings
In this paper, the subclass of harmonic univalent functions by shearing construction is studied and this subclass of harmonic mappings needs a necessary and adequate condition to be convex in the horizontal direction. Furthermore, convolutions of two special subclasses of univalent harmonic mappings...
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| Format: | Article |
| Language: | English |
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Wiley
2021-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2021/6553600 |
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| _version_ | 1832566394880786432 |
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| author | Ahmad Sulaiman Ahmad El-Faqeer Zhen Chuan Ng Shamani Supramaniam |
| author_facet | Ahmad Sulaiman Ahmad El-Faqeer Zhen Chuan Ng Shamani Supramaniam |
| author_sort | Ahmad Sulaiman Ahmad El-Faqeer |
| collection | DOAJ |
| description | In this paper, the subclass of harmonic univalent functions by shearing construction is studied and this subclass of harmonic mappings needs a necessary and adequate condition to be convex in the horizontal direction. Furthermore, convolutions of two special subclasses of univalent harmonic mappings are shown to be convex in the horizontal direction. Also, the family of univalent harmonic mappings of the unit disk onto a region convex in the direction of the imaginary axis is introduced. Sufficient conditions for convex combinations of harmonic mappings of this family to be univalently convex in the direction of the imaginary axis are obtained. |
| format | Article |
| id | doaj-art-2bfc0df5c4f543aba11884c0e5e879b2 |
| institution | Kabale University |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-2bfc0df5c4f543aba11884c0e5e879b22025-02-03T01:04:10ZengWileyJournal of Mathematics2314-46292314-47852021-01-01202110.1155/2021/65536006553600On Convolution and Convex Combination of Harmonic MappingsAhmad Sulaiman Ahmad El-Faqeer0Zhen Chuan Ng1Shamani Supramaniam2School of Mathematical Sciences, Universiti Sains Malaysia, Penang 11800, MalaysiaSchool of Mathematical Sciences, Universiti Sains Malaysia, Penang 11800, MalaysiaSchool of Mathematical Sciences, Universiti Sains Malaysia, Penang 11800, MalaysiaIn this paper, the subclass of harmonic univalent functions by shearing construction is studied and this subclass of harmonic mappings needs a necessary and adequate condition to be convex in the horizontal direction. Furthermore, convolutions of two special subclasses of univalent harmonic mappings are shown to be convex in the horizontal direction. Also, the family of univalent harmonic mappings of the unit disk onto a region convex in the direction of the imaginary axis is introduced. Sufficient conditions for convex combinations of harmonic mappings of this family to be univalently convex in the direction of the imaginary axis are obtained.http://dx.doi.org/10.1155/2021/6553600 |
| spellingShingle | Ahmad Sulaiman Ahmad El-Faqeer Zhen Chuan Ng Shamani Supramaniam On Convolution and Convex Combination of Harmonic Mappings Journal of Mathematics |
| title | On Convolution and Convex Combination of Harmonic Mappings |
| title_full | On Convolution and Convex Combination of Harmonic Mappings |
| title_fullStr | On Convolution and Convex Combination of Harmonic Mappings |
| title_full_unstemmed | On Convolution and Convex Combination of Harmonic Mappings |
| title_short | On Convolution and Convex Combination of Harmonic Mappings |
| title_sort | on convolution and convex combination of harmonic mappings |
| url | http://dx.doi.org/10.1155/2021/6553600 |
| work_keys_str_mv | AT ahmadsulaimanahmadelfaqeer onconvolutionandconvexcombinationofharmonicmappings AT zhenchuanng onconvolutionandconvexcombinationofharmonicmappings AT shamanisupramaniam onconvolutionandconvexcombinationofharmonicmappings |