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Sharp Bohr Radius Constants for Certain Analytic Functions
Bulletin of the Malaysian Mathematical Sciences Society ( IF 1.2 ) Pub Date : 2021-02-10 , DOI: 10.1007/s40840-020-01071-x
Swati Anand , Naveen Kumar Jain , Sushil Kumar

The Bohr radius for a class \({\mathcal {G}}\) consisting of analytic functions \(f(z)=\sum _{n=0}^{\infty }a_nz^n\) in unit disc \({\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1\}\) is the largest \(r^*\) such that every function f in the class \({\mathcal {G}}\) satisfies the inequality

$$\begin{aligned} d\left( \sum _{n=0}^{\infty }|a_nz^n|, |f(0)|\right) = \sum _{n=1}^{\infty }|a_nz^n|\le d(f(0), \partial f({\mathbb {D}})) \end{aligned}$$

for all \(|z|=r \le r^*\), where d is the Euclidean distance. In this paper, our aim is to determine the Bohr radius for the classes of analytic functions f satisfying differential subordination relations \(zf'(z)/f(z) \prec h(z)\) and \(f(z)+\beta z f'(z)+\gamma z^2 f''(z)\prec h(z)\), where h is the Janowski function. Analogous results are obtained for the classes of \(\alpha \)-convex functions and typically real functions, respectively. All obtained results are sharp.



中文翻译:

某些解析函数的Sharp Bohr半径常数

玻尔半径一类\({\ mathcal {G}} \)组成的解析函数\(F(Z)= \总和_ {N = 0} ^ {\ infty} a_nz ^ N \)在单位圆\ ({\ mathbb {D}} = \ {z \ in {\ mathbb {C}}:| z | <1 \} \)是最大的\(r ^ * \),因此类\中的每个函数f ({\ mathcal {G}} \)满足不等式

$$ \ begin {aligned} d \ left(\ sum _ {n = 0} ^ {\ infty} | a_nz ^ n |,| f(0)| \ right)= \ sum _ {n = 1} ^ { \ infty} | a_nz ^ n | \ le d(f(0),\ partial f({\ mathbb {D}}))\ end {aligned} $$

对于所有\(| z | = r \ le r ^ * \),其中d是欧几里得距离。在本文中,我们的目的是确定满足微分从属关系\(zf'(z)/ f(z)\ prec h(z)\)\(f(z)的解析函数f类的玻尔半径+ \ beta z f'(z)+ \ gamma z ^ 2 f''(z)\ prec h(z)\),其中h是Janowski函数。分别对于\(\ alpha \)-凸函数类和典型的实函数类获得了相似的结果。所有获得的结果都是清晰的。

更新日期:2021-02-10
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