Slice starlike functions over quaternions
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arXiv
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| Format: | Preprint |
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2016
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| _version_ | 1866929760081281024 |
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| author | Xu, Zhenghua Ren, Guangbin |
| author_facet | Xu, Zhenghua Ren, Guangbin |
| contents | In this paper, we initiate the study of the geometric function theory for slice starlike functions over quaternions and its subclasses. This allows us to answer negatively some questions about the Bieberbach conjecture, the growth, distortion, and covering theorems for slice regular functions. Precisely, we find that the Bieberbach conjecture holds true for slice starlike functions in contrast to the fact that the Bieberbach conjecture fails for biholomorphic starlike mappings in higher dimensions. We also establish some sharp versions of the growth, distortion, and covering theorems for quaternions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1611_10234 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Slice starlike functions over quaternions Xu, Zhenghua Ren, Guangbin Complex Variables 30G35, 30C50 In this paper, we initiate the study of the geometric function theory for slice starlike functions over quaternions and its subclasses. This allows us to answer negatively some questions about the Bieberbach conjecture, the growth, distortion, and covering theorems for slice regular functions. Precisely, we find that the Bieberbach conjecture holds true for slice starlike functions in contrast to the fact that the Bieberbach conjecture fails for biholomorphic starlike mappings in higher dimensions. We also establish some sharp versions of the growth, distortion, and covering theorems for quaternions. |
| title | Slice starlike functions over quaternions |
| topic | Complex Variables 30G35, 30C50 |
| url | https://arxiv.org/abs/1611.10234 |