Sharp Bohr radius involving Schwarz functions for certain classes of analytic functions

Molla Basir Ahamed, Partha Pratim Roy
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Abstract

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 $f(z)+\beta zf^{\prime}(z)+\gamma z^2f^{\prime\prime}(z)\prec h(z)$, 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 $\alpha$-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].
涉及某些类解析函数的施瓦茨函数的夏普玻尔半径
在单位圆盘$\mathbb{D}=\{z\in\mathbb{C}上,形式为$f(z)=\sum_{n=0}^{\infty}a_nz^n$的解析函数的任意类$\mathcal{F}$的玻尔半径为|z|<1\}$是最大的半径$R_{\mathcal{F}}$,使得每个函数$f\in\mathcal{F}$满足不等式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 $|z|=r\leq R_{mathcal{F}}$ , 其中 $d(0, \partialf(\mathbb{D}))$ 是欧氏距离。在本文中,我们的目的是为满足微分从属关系 $zf^{\prime}(z)/f(z)\prech(z)$ 和 $f(z)+\beta zf^{\prime}(z)+\gamma z^2f^{\prime\prime}(z)\prec h(z)$ 的解析函数类确定尖锐的改进玻尔半径,其中 $h$ 是雅诺夫斯基函数。我们证明,改进的玻尔半径可以作为涉及第一类贝塞尔函数的方程的根来获得。本文分别针对$\alpha$-凸函数和典型实函数得到了类似结果。本文得到的所有结果都很尖锐,是[{Bull.Malaysal. Math. Sci. Soc.} (2021) 44:1771-1785] 的改进版。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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