Pre-Schwarzian norm estimate for certain Ma-Minda Class of functions
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914960672555008 |
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| author | Ali, Md Firoz Nurezzaman, Md Pal, Sanjit |
| author_facet | Ali, Md Firoz Nurezzaman, Md Pal, Sanjit |
| contents | Let $\mathcal{S}^*(φ)$ be the class of all analytic functions $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$, normalized by $f(0)=f'(0)-1=0$ that satisfy the subordination relation $zf'(z)/f(z)\precφ(z)$, where $φ$ is an analytic and univalent in $\mathbb{D}$ with ${\rm Re\,}φ(z)>0$ such that $φ(\mathbb{D})$ is symmetric with respect to the real axis and stralike with respect to $1$. In the present article, we obtain the sharp estimates of the pre-Schwarzian norm of $f$ and the Alexander transformation $J[f]$ for functions $f(z)$ in the class $\mathcal{S}^*(φ)$ when $φ(z)=e^{λz}$, $0<λ\leπ/2$ and $φ(z)=\sqrt{1+cz}$, $0<c\le1.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_20216 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pre-Schwarzian norm estimate for certain Ma-Minda Class of functions Ali, Md Firoz Nurezzaman, Md Pal, Sanjit Complex Variables 30C45, 30C55 Let $\mathcal{S}^*(φ)$ be the class of all analytic functions $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$, normalized by $f(0)=f'(0)-1=0$ that satisfy the subordination relation $zf'(z)/f(z)\precφ(z)$, where $φ$ is an analytic and univalent in $\mathbb{D}$ with ${\rm Re\,}φ(z)>0$ such that $φ(\mathbb{D})$ is symmetric with respect to the real axis and stralike with respect to $1$. In the present article, we obtain the sharp estimates of the pre-Schwarzian norm of $f$ and the Alexander transformation $J[f]$ for functions $f(z)$ in the class $\mathcal{S}^*(φ)$ when $φ(z)=e^{λz}$, $0<λ\leπ/2$ and $φ(z)=\sqrt{1+cz}$, $0<c\le1.$ |
| title | Pre-Schwarzian norm estimate for certain Ma-Minda Class of functions |
| topic | Complex Variables 30C45, 30C55 |
| url | https://arxiv.org/abs/2409.20216 |