Generalized Schwarzians and Normal Families
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914115776151552 |
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| author | Grätsch, Matthias |
| author_facet | Grätsch, Matthias |
| contents | We study families of analytic and meromorphic functions with bounded generalized Schwarzian derivative $S_k(f)$. We show that these families are quasi-normal. Further, we investigate associated families, such as those formed by derivatives and logarithmic derivatives, and prove several (quasi-)normality results. Moreover, we derive a new formula for $S_k(f)$, which yields a result for families $\mathcal{F}\subseteq\mathcal{H}(\mathbb{D})$ of locally univalent functions that satisfy $$S_k(f)(z)\neq b(z)\qquad \text{for some }b\in\mathcal{M}(\mathbb{D})\text{ and all } f\in\mathcal{F},\,z\in\mathbb{C}$$ and for entire functions $f$ with $S_k(f)(z)\neq0$ and $S_k(f)(z)\neq\infty$ for all $z\in\mathbb{C}$.\\ The classical Schwarzian derivative $S_f$ is contained as the case $k=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_17878 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Schwarzians and Normal Families Grätsch, Matthias Complex Variables 30D45, 30D30, 30C45 We study families of analytic and meromorphic functions with bounded generalized Schwarzian derivative $S_k(f)$. We show that these families are quasi-normal. Further, we investigate associated families, such as those formed by derivatives and logarithmic derivatives, and prove several (quasi-)normality results. Moreover, we derive a new formula for $S_k(f)$, which yields a result for families $\mathcal{F}\subseteq\mathcal{H}(\mathbb{D})$ of locally univalent functions that satisfy $$S_k(f)(z)\neq b(z)\qquad \text{for some }b\in\mathcal{M}(\mathbb{D})\text{ and all } f\in\mathcal{F},\,z\in\mathbb{C}$$ and for entire functions $f$ with $S_k(f)(z)\neq0$ and $S_k(f)(z)\neq\infty$ for all $z\in\mathbb{C}$.\\ The classical Schwarzian derivative $S_f$ is contained as the case $k=2$. |
| title | Generalized Schwarzians and Normal Families |
| topic | Complex Variables 30D45, 30D30, 30C45 |
| url | https://arxiv.org/abs/2505.17878 |