Convolutions of harmonic right half-plane mappings
We first prove that the convolution of a normalized right half-plane mapping with another subclass of normalized right half-plane mappings with the dilatation −z(a+z)/(1+az)$ - z(a + z)/(1 + az)$ is CHD (convex in the horizontal direction) provided a=1$a = 1$ or −1≤a≤0$ - 1 \le a \le 0$. Secondly, w...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2016-01-01
|
| Series: | Open Mathematics |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/math-2016-0069 |
| _version_ | 1818597835719835648 |
|---|---|
| author | Li YingChun Liu ZhiHong |
| author_facet | Li YingChun Liu ZhiHong |
| author_sort | Li YingChun |
| collection | DOAJ |
| description | We first prove that the convolution of a normalized right half-plane mapping with another subclass of normalized right half-plane mappings with the dilatation −z(a+z)/(1+az)$ - z(a + z)/(1 + az)$ is CHD (convex in the horizontal direction) provided a=1$a = 1$ or −1≤a≤0$ - 1 \le a \le 0$. Secondly, we give a simply method to prove the convolution of two special subclasses of harmonic univalent mappings in the right half-plane is CHD which was proved by Kumar et al. [1, Theorem 2.2]. In addition, we derive the convolution of harmonic univalent mappings involving the generalized harmonic right half-plane mappings is CHD. Finally, we present two examples of harmonic mappings to illuminate our main results. |
| first_indexed | 2024-12-16T11:54:07Z |
| format | Article |
| id | doaj.art-abfbb4a3c11e4ebeb6c8ce9f0377035d |
| institution | Directory Open Access Journal |
| issn | 2391-5455 |
| language | English |
| last_indexed | 2024-12-16T11:54:07Z |
| publishDate | 2016-01-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Open Mathematics |
| spelling | doaj.art-abfbb4a3c11e4ebeb6c8ce9f0377035d2022-12-21T22:32:37ZengDe GruyterOpen Mathematics2391-54552016-01-0114178980010.1515/math-2016-0069math-2016-0069Convolutions of harmonic right half-plane mappingsLi YingChun0Liu ZhiHong1College of Mathematics, Honghe University, Mengzi 661199, Yunnan, ChinaCollege of Mathematics, Honghe University, Mengzi 661199, Yunnan, ChinaWe first prove that the convolution of a normalized right half-plane mapping with another subclass of normalized right half-plane mappings with the dilatation −z(a+z)/(1+az)$ - z(a + z)/(1 + az)$ is CHD (convex in the horizontal direction) provided a=1$a = 1$ or −1≤a≤0$ - 1 \le a \le 0$. Secondly, we give a simply method to prove the convolution of two special subclasses of harmonic univalent mappings in the right half-plane is CHD which was proved by Kumar et al. [1, Theorem 2.2]. In addition, we derive the convolution of harmonic univalent mappings involving the generalized harmonic right half-plane mappings is CHD. Finally, we present two examples of harmonic mappings to illuminate our main results.https://doi.org/10.1515/math-2016-0069harmonic univalent mappingsharmonic convolutiongeneralized right half-plane mappings30c4558e20 |
| spellingShingle | Li YingChun Liu ZhiHong Convolutions of harmonic right half-plane mappings Open Mathematics harmonic univalent mappings harmonic convolution generalized right half-plane mappings 30c45 58e20 |
| title | Convolutions of harmonic right half-plane mappings |
| title_full | Convolutions of harmonic right half-plane mappings |
| title_fullStr | Convolutions of harmonic right half-plane mappings |
| title_full_unstemmed | Convolutions of harmonic right half-plane mappings |
| title_short | Convolutions of harmonic right half-plane mappings |
| title_sort | convolutions of harmonic right half plane mappings |
| topic | harmonic univalent mappings harmonic convolution generalized right half-plane mappings 30c45 58e20 |
| url | https://doi.org/10.1515/math-2016-0069 |
| work_keys_str_mv | AT liyingchun convolutionsofharmonicrighthalfplanemappings AT liuzhihong convolutionsofharmonicrighthalfplanemappings |