Sharp Bohr radius involving Schwarz functions for certain classes of analytic functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913481788227584 |
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| author | Ahamed, Molla Basir Roy, Partha Pratim |
| author_facet | Ahamed, Molla Basir Roy, Partha Pratim |
| contents | The Bohr radius for an arbitrary class $\mathcal{F}$ of analytic functions of the form $f(z)=\sum_{n=0}^{\infty}a_nz^n$ on the unit disk $\mathbb{D}=\{z\in\mathbb{C} : |z|<1\}$ is the largest radius $R_{\mathcal{F}}$ such that every function $f\in\mathcal{F}$ satisfies the inequality \begin{align*}
d\left(\sum_{n=0}^{\infty}|a_nz^n|, |f(0)|\right)=\sum_{n=1}^{\infty}|a_nz^n|\leq d(f(0), \partial f(\mathbb{D})), \end{align*} for all $|z|=r\leq R_{\mathcal{F}}$ , where $d(0, \partial f(\mathbb{D}))$ is the Euclidean distance. In this paper, our aim is to determine the sharp improved Bohr radius for the classes of analytic functions $f$ satisfying differential subordination relation $zf^{\prime}(z)/f(z)\prec h(z)$ and $f(z)+βzf^{\prime}(z)+γz^2f^{\prime\prime}(z)\prec h(z)$, where $h$ is the Janowski function. We show that improved Bohr radius can be obtained for Janowski functions as root of an equation involving Bessel function of first kind. Analogues results are obtained in this paper for $α$-convex functions and typically real functions, respectively. All obtained results in the paper are sharp and are improved version of [{Bull. Malays. Math. Sci. Soc.} (2021) 44:1771-1785]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14773 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp Bohr radius involving Schwarz functions for certain classes of analytic functions Ahamed, Molla Basir Roy, Partha Pratim Complex Variables Primary 30C45, 30C80 The Bohr radius for an arbitrary class $\mathcal{F}$ of analytic functions of the form $f(z)=\sum_{n=0}^{\infty}a_nz^n$ on the unit disk $\mathbb{D}=\{z\in\mathbb{C} : |z|<1\}$ is the largest radius $R_{\mathcal{F}}$ such that every function $f\in\mathcal{F}$ satisfies the inequality \begin{align*} d\left(\sum_{n=0}^{\infty}|a_nz^n|, |f(0)|\right)=\sum_{n=1}^{\infty}|a_nz^n|\leq d(f(0), \partial f(\mathbb{D})), \end{align*} for all $|z|=r\leq R_{\mathcal{F}}$ , where $d(0, \partial f(\mathbb{D}))$ is the Euclidean distance. In this paper, our aim is to determine the sharp improved Bohr radius for the classes of analytic functions $f$ satisfying differential subordination relation $zf^{\prime}(z)/f(z)\prec h(z)$ and $f(z)+βzf^{\prime}(z)+γz^2f^{\prime\prime}(z)\prec h(z)$, where $h$ is the Janowski function. We show that improved Bohr radius can be obtained for Janowski functions as root of an equation involving Bessel function of first kind. Analogues results are obtained in this paper for $α$-convex functions and typically real functions, respectively. All obtained results in the paper are sharp and are improved version of [{Bull. Malays. Math. Sci. Soc.} (2021) 44:1771-1785]. |
| title | Sharp Bohr radius involving Schwarz functions for certain classes of analytic functions |
| topic | Complex Variables Primary 30C45, 30C80 |
| url | https://arxiv.org/abs/2408.14773 |