Sharp Coefficient Bounds for certain $q$-Starlike Functions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918520581783552 |
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| author | Kumar, S. Sivaprasad Pannu, Snehal |
| author_facet | Kumar, S. Sivaprasad Pannu, Snehal |
| contents | Geometric function theory increasingly draws on $q$-calculus to model discrete and quantum-inspired phenomena. Motivated by this, the present paper introduces new subclasses of analytic functions: the class $\mathcal{S}^{*}_{ξ_q}$ of $q$-starlike functions associated with the Ma-Minda function $ξ_q(z)$, and its limiting classical counterpart $\mathcal{S}^{*}_ξ$ associated with $ξ(z)$, where $q \in (0,1)$. We systematically establish sharp coefficient estimates including the Fekete-Szegö, Hankel and Toeplitz determinants. We establish the sharpness of the $q$-coefficient estimates using a newly derived integral representation, which offers a more effective alternative to the conventional convolution-based extremal construction. It is further shown that all $q$-results reduce to their classical counterparts as $q \to 1^{-}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_05625 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp Coefficient Bounds for certain $q$-Starlike Functions Kumar, S. Sivaprasad Pannu, Snehal Complex Variables Geometric function theory increasingly draws on $q$-calculus to model discrete and quantum-inspired phenomena. Motivated by this, the present paper introduces new subclasses of analytic functions: the class $\mathcal{S}^{*}_{ξ_q}$ of $q$-starlike functions associated with the Ma-Minda function $ξ_q(z)$, and its limiting classical counterpart $\mathcal{S}^{*}_ξ$ associated with $ξ(z)$, where $q \in (0,1)$. We systematically establish sharp coefficient estimates including the Fekete-Szegö, Hankel and Toeplitz determinants. We establish the sharpness of the $q$-coefficient estimates using a newly derived integral representation, which offers a more effective alternative to the conventional convolution-based extremal construction. It is further shown that all $q$-results reduce to their classical counterparts as $q \to 1^{-}$. |
| title | Sharp Coefficient Bounds for certain $q$-Starlike Functions |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2601.05625 |