Pre-Schwarzian and Schwarzian norm Estimates for Robertson class
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| Format: | Preprint |
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2025
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| _version_ | 1866917096407957504 |
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| author | Ahamed, Molla Basir Hossain, Rajesh Wang, Xiaoyuan |
| author_facet | Ahamed, Molla Basir Hossain, Rajesh Wang, Xiaoyuan |
| contents | Let $\mathcal{A}$ denote the class of analytic functions $f$ on the unit disk $\mathbb{D}=\{z\in\mathbb{C} : |z|<1\}$, normalized by $f(0)=0$ and $f^{\prime}(0)=1$. For $-π/2<α<π/2$, let $\mathcal{S}_α$ be the subclass of $\mathcal{A}$ consisting of functions $f$ that satisfy the relation $\mathrm{Re}\{e^{iα}\left(1+zf^{\prime\prime}(z)/f^{\prime}(z)\right)\}>0$ for $z\in\mathbb{D}$. In this paper, we first give an equivalent characterization for a subclass of Robertson functions; then we present the distortion and growth theorems and obtain the pre-Schwarzian and Schwarzian norms for the subclass $\mathcal{S}_α$. In addition, a sharp upper bound of the Schwarzian norm for the subclass is given in terms of the value $f^{\prime \prime}(0)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_16702 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pre-Schwarzian and Schwarzian norm Estimates for Robertson class Ahamed, Molla Basir Hossain, Rajesh Wang, Xiaoyuan Complex Variables 30C45, 30C55 Let $\mathcal{A}$ denote the class of analytic functions $f$ on the unit disk $\mathbb{D}=\{z\in\mathbb{C} : |z|<1\}$, normalized by $f(0)=0$ and $f^{\prime}(0)=1$. For $-π/2<α<π/2$, let $\mathcal{S}_α$ be the subclass of $\mathcal{A}$ consisting of functions $f$ that satisfy the relation $\mathrm{Re}\{e^{iα}\left(1+zf^{\prime\prime}(z)/f^{\prime}(z)\right)\}>0$ for $z\in\mathbb{D}$. In this paper, we first give an equivalent characterization for a subclass of Robertson functions; then we present the distortion and growth theorems and obtain the pre-Schwarzian and Schwarzian norms for the subclass $\mathcal{S}_α$. In addition, a sharp upper bound of the Schwarzian norm for the subclass is given in terms of the value $f^{\prime \prime}(0)$. |
| title | Pre-Schwarzian and Schwarzian norm Estimates for Robertson class |
| topic | Complex Variables 30C45, 30C55 |
| url | https://arxiv.org/abs/2511.16702 |