论文标题
Bohr现象，用于谐波映射的某些子类
Bohr phenomenon for certain Subclasses of Harmonic Mappings
论文作者
论文摘要
$ f（z）= \ sum_ {n = 0}^{\ infty} a_ {n} z^{n} $的bohr现象，用于$ f（z）= \ sum_ {n = 0}^{\ sum_ {n = 0}^{n} a_ { $ r_ {f} $，$ 0 <r_ {f} <1 $，使得不等式$ \ sum_ {n = 0}^{\ infty} | a_ {n} z^z^{n} | \ leq 1 $时，每当不平等$ | f（z）| \ leq 1 $保存在单位磁盘中 $ \ mathbb {d} = \ {z \ in \ mathbb {c}：| z | <1 \} $。此最大半径的确切值称为bohr半径，已确定为$ r_ {f} = 1/3 $。 BOHR现象\ cite {abu-2010}用于谐波函数的$ f（z）= h（z）= h（z）+\ edline {g（z）} $，其中$ h（z）= \ sum_ {z）= \ sum_ {n = 0} $ g（z）= \ sum_ {n = 1}^{\ infty} b_ {n} z^{n} $是找到最大的半径$ r_ {f} $，$ 0 <r_ {f} <1 $ （| a_ {n} |+| b_ {n} |）| z |^{n} \ leq d（f（0），\ partial f（\ mathbb {d}））％\ quad \ quad \ mbox {for} $$持有$ | z | \ leq r_ {f} $，shere $ d（f（0），\ partial f（\ mathbb {d}））$表示$ f（0）$和$ f（\ sathbb {d}）$ f（0）$之间的欧几里得距离。在本文中，我们研究了单位磁盘$ \ mathbb {d}中的几类谐波功能的BOHR半径。$
The Bohr phenomenon for analytic functions of the form $f(z)=\sum_{n=0}^{\infty} a_{n}z^{n}$, first introduced by Harald Bohr in 1914, deals with finding the largest radius $r_{f}$, $0<r_{f}<1$, such that the inequality $\sum_{n=0}^{\infty} |a_{n}z^{n}| \leq 1$ holds whenever the inequality $|f(z)|\leq 1 $ holds in the unit disk $\mathbb{D}=\{z \in \mathbb{C}: |z|<1\}$. The exact value of this largest radius known as Bohr radius, which has been established to be $r_{f}=1/3$. The Bohr phenomenon \cite{Abu-2010} for harmonic functions $f$ of the form $f(z)=h(z)+\overline {g(z)}$, where $h(z)=\sum_{n=0}^{\infty} a_{n}z^{n}$ and $g(z)=\sum_{n=1}^{\infty} b_{n}z^{n}$ is to find the largest radius $r_{f}$, $0<r_{f}<1$ such that $$\sum\limits_{n=1}^{\infty} (|a_{n}|+|b_{n}|) |z|^{n}\leq d(f(0),\partial f(\mathbb{D})) %\quad\mbox { for } |z|\leq r_{f}. $$ holds for $|z|\leq r_{f}$, here $d(f(0),\partial f(\mathbb{D})) $ denotes the Euclidean distance between $f(0)$ and the boundary of $f(\mathbb{D})$. In this paper, we investigate the Bohr radius for several classes of harmonic functions in the unit disk $\mathbb{D}.$
