Geometric subfamily of locally univalent functions, Blaschke products and quasidisk
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914254428307456 |
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| author | Ahamed, Molla Basir Hossain, Rajesh |
| author_facet | Ahamed, Molla Basir Hossain, Rajesh |
| contents | In this article, we consider the family $\mathcal{F}(α)$ defined for $α\in (0, 3]$ by
\begin{align*}
{\rm Re}\left(1+\frac{zf''(z)}{f'(z)}\right) > 1 - \fracα{2} \quad \text{for } z \in \mathbb{D}.
\end{align*}
Our primary objective is to show that this family possesses significant geometric and analytic properties, including connections with Blaschke products and the Schwarzian derivative, as well as its sharp bounds. Furthermore, we prove that if $f \in \mathcal{F}(α)$, then the image $f(\mathbb{D})$ is a quasidisk. We also show that if $f \in \mathcal{F}(α)$, then $\|S_f\| = 2α(2-α)$. Moreover, we establish the sharp estimate $\|P_{f}\| \leq 2α+1$ for the pre-Schwarzian derivative of harmonic mappings $f = h + \bar{g} \in \mathcal{F}_{\mathcal{H}}(α)$, where the analytic part $h$ belongs to $\mathcal{F}(α)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07842 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometric subfamily of locally univalent functions, Blaschke products and quasidisk Ahamed, Molla Basir Hossain, Rajesh Complex Variables Primary: 30C45, 30C62, 30C80, 30J10, 31C05, Secondary: 30C20, 30C55, 31A05 In this article, we consider the family $\mathcal{F}(α)$ defined for $α\in (0, 3]$ by \begin{align*} {\rm Re}\left(1+\frac{zf''(z)}{f'(z)}\right) > 1 - \fracα{2} \quad \text{for } z \in \mathbb{D}. \end{align*} Our primary objective is to show that this family possesses significant geometric and analytic properties, including connections with Blaschke products and the Schwarzian derivative, as well as its sharp bounds. Furthermore, we prove that if $f \in \mathcal{F}(α)$, then the image $f(\mathbb{D})$ is a quasidisk. We also show that if $f \in \mathcal{F}(α)$, then $\|S_f\| = 2α(2-α)$. Moreover, we establish the sharp estimate $\|P_{f}\| \leq 2α+1$ for the pre-Schwarzian derivative of harmonic mappings $f = h + \bar{g} \in \mathcal{F}_{\mathcal{H}}(α)$, where the analytic part $h$ belongs to $\mathcal{F}(α)$. |
| title | Geometric subfamily of locally univalent functions, Blaschke products and quasidisk |
| topic | Complex Variables Primary: 30C45, 30C62, 30C80, 30J10, 31C05, Secondary: 30C20, 30C55, 31A05 |
| url | https://arxiv.org/abs/2601.07842 |