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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2403.14162 |
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| _version_ | 1866911807160975360 |
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| author | Kumar, S. Sivaprasad Yadav, Pooja |
| author_facet | Kumar, S. Sivaprasad Yadav, Pooja |
| contents | In this paper, we introduce and explore a new class of starlike functions denoted by $\mathcal{S}^*_{\mathfrak{B}}$, defined as follows:
$$\mathcal{S}^*_{\mathfrak{B}}=\{f\in \mathcal{A}:zf'(z)/f(z)\prec \sqrt{1+\tanh{z}}=:\mathfrak{B}(z)\}.$$
Here, $\mathfrak{B}(z)$ represents a mapping from the unit disk onto a bean-shaped domain. Our study focuses on understanding the characteristic properties of both $\mathfrak{B}(z)$ and the functions in $\mathcal{S}^*_{\mathfrak{B}}$. We derive sharp conditions under which $ψ(p)\prec\sqrt{1+\tanh(z)}$ implies $p(z)\prec ((1+A z)/(1+B z))^γ$, where $ψ(p)$ is defined as:
\begin{equation*} (1-α)p(z)+αp^2(z)+β\frac{zp'(z)}{p^k(z)}\quad \text{and}\quad (p(z))^δ+β\frac{zp'(z)}{(p(z))^k}. \end{equation*} Additionally, we establish inclusion relations involving $\mathcal{S}^*_{\mathfrak{B}}$ and derive precise estimates for the sharp radii constants of $\mathcal{S}^*_{\mathfrak{B}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_14162 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Starlike Functions Associated with a Bean Shaped Domain Kumar, S. Sivaprasad Yadav, Pooja Complex Variables In this paper, we introduce and explore a new class of starlike functions denoted by $\mathcal{S}^*_{\mathfrak{B}}$, defined as follows: $$\mathcal{S}^*_{\mathfrak{B}}=\{f\in \mathcal{A}:zf'(z)/f(z)\prec \sqrt{1+\tanh{z}}=:\mathfrak{B}(z)\}.$$ Here, $\mathfrak{B}(z)$ represents a mapping from the unit disk onto a bean-shaped domain. Our study focuses on understanding the characteristic properties of both $\mathfrak{B}(z)$ and the functions in $\mathcal{S}^*_{\mathfrak{B}}$. We derive sharp conditions under which $ψ(p)\prec\sqrt{1+\tanh(z)}$ implies $p(z)\prec ((1+A z)/(1+B z))^γ$, where $ψ(p)$ is defined as: \begin{equation*} (1-α)p(z)+αp^2(z)+β\frac{zp'(z)}{p^k(z)}\quad \text{and}\quad (p(z))^δ+β\frac{zp'(z)}{(p(z))^k}. \end{equation*} Additionally, we establish inclusion relations involving $\mathcal{S}^*_{\mathfrak{B}}$ and derive precise estimates for the sharp radii constants of $\mathcal{S}^*_{\mathfrak{B}}$. |
| title | On Starlike Functions Associated with a Bean Shaped Domain |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2403.14162 |