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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2412.17403 |
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| _version_ | 1866910759878918144 |
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| author | Kumar, S. Sivaprasad Tripathi, Arya Pannu, Snehal |
| author_facet | Kumar, S. Sivaprasad Tripathi, Arya Pannu, Snehal |
| contents | Logarithmic and inverse logarithmic coefficients play a crucial role in the theory of univalent functions. In this study, we focus on the class of starlike functions \(\mathcal{S}^*_ρ\), defined as \[ \mathcal{S}^*_ρ= \left\{ f \in \mathcal{A} : \frac{z f'(z)}{f(z)} \prec ρ(z), \; z \in \mathbb{D} \right\}, \] where \(ρ(z) := 1 + \sinh^{-1}(z)\), which maps the unit disk \(\mathbb{D}\) onto a petal-shaped domain. This investigation aims to establish bounds for the second Hankel and Toeplitz determinants, with their entries determined by the logarithmic coefficients of \(f\) and its inverse \(f^{-1}\), for functions \(f \in \mathcal{S}^*_ρ\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17403 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Coefficient problems for \textbf{$S^*_ρ$} Kumar, S. Sivaprasad Tripathi, Arya Pannu, Snehal Complex Variables Logarithmic and inverse logarithmic coefficients play a crucial role in the theory of univalent functions. In this study, we focus on the class of starlike functions \(\mathcal{S}^*_ρ\), defined as \[ \mathcal{S}^*_ρ= \left\{ f \in \mathcal{A} : \frac{z f'(z)}{f(z)} \prec ρ(z), \; z \in \mathbb{D} \right\}, \] where \(ρ(z) := 1 + \sinh^{-1}(z)\), which maps the unit disk \(\mathbb{D}\) onto a petal-shaped domain. This investigation aims to establish bounds for the second Hankel and Toeplitz determinants, with their entries determined by the logarithmic coefficients of \(f\) and its inverse \(f^{-1}\), for functions \(f \in \mathcal{S}^*_ρ\). |
| title | On Coefficient problems for \textbf{$S^*_ρ$} |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2412.17403 |