论文标题
某些类别的谐波映射的尖锐精致的Bohr-Rogosinski不平等现象
The sharp refined Bohr-Rogosinski inequalities for certain classes of harmonic mappings
论文作者
论文摘要
一个$ f（z）= \ sum_ {n = 0}^{\ infty} a_nz^n $组成的$ f（z）= \ sum_ {z）= \ sum_ {n = 0} a_nz^n $的类$ \ mathcal {f} $类r_f> 0 $，这样 \ begin {equation*} i_f（r）：= \ sum_ {n = 1}^{\ infty} | a_n | r^n \ leq {d} \ left（f（f（f（f（0），\ partial \ partial \ mathbb {d} \ end {equation*}对于\ Mathcal {f} $中的每个函数$ f \，以及$ | z | = r \ leq r_f $。最大的半径$ r_f $是bohr半径和不等式$ i_f（r）\ leq {d} \ left（f（f（0），\ partial \ mathbb {d} \ right）$是bohr bohr bohr bohr bohr bohr bohr不平等，对于class $ \ mathcal {f} $如果存在一个正实数$ r_0 $，以至于$ i_f（r）\ leq {d} \ left（f（f（0），\ partial \ mathbb {d} \ right）$ class $ \ mathcal {f} $ for $ 0 \ leq r_0 $ and $ r_0 $ and $ r_ 0 $ r_ 0不平等W.R.T.类$ \ MATHCAL {F} $。在本文中，我们证明了Bohr-Rogosinski不平等的某些谐波映射的不平等现象。
A class $ \mathcal{F} $ consisting of analytic functions $ f(z)=\sum_{n=0}^{\infty}a_nz^n $ in the unit disc $ \mathbb{D}=\{z\in\mathbb{C}:|z|<1\} $ satisfies a Bohr phenomenon if there exists an $ r_f>0 $ such that \begin{equation*} I_f(r):=\sum_{n=1}^{\infty}|a_n|r^n\leq{d}\left(f(0),\partial \mathbb{D}\right) \end{equation*} for every function $ f\in\mathcal{F} $, and $ |z|=r\leq r_f $. The largest radius $ r_f $ is the Bohr radius and the inequality $ I_f(r)\leq{d}\left(f(0),\partial \mathbb{D}\right) $ is Bohr inequality for the class $ \mathcal{F} $, where `$ d $' is the Euclidean distance. If there exists a positive real number $ r_0 $ such that $ I_f(r)\leq {d}\left(f(0),\partial \mathbb{D}\right) $ holds for every element of the class $ \mathcal{F} $ for $ 0\leq r<r_0 $ and fails when $ r>r_0 $, then we say that $ r_0 $ is sharp bound for the inequality w.r.t. the class $ \mathcal{F} $. In this paper, we prove sharp refinement of the Bohr-Rogosinski inequality for certain classes of harmonic mappings.
