圆盘上有界解析函数的边界收敛集和路径发散集
Boundary convergence and path divergence sets for bounded analytic functions in the disk
摘要：
设$f:\mathbb{D}\to\mathbb{C}$是有界解析函数。一个在其边界上包含点$1$的集合$K\subset\mathbb{D}$称为$f$在$1$的收敛集，如果$f(z)$收敛到某个值$ζ$，作为$z\to1$与$z\in K$。如果$f$沿着位于$K$中并接近$1$的每条路径$γ$发散，则$K$被称为$f$在$1$处的路径发散集。在这篇文章中，我们证明了对于从$-1$到$1$的通过单位圆盘的路径$γ$，如果$f$不能沿着$γ$收敛，则$γ$上面的区域或$γ$下面的区域是$f$的路径发散集.另一方面，如果$γ_1$和$γ_2$是两条这样的路径，并且$f$沿着$γ_1$和$γ_2$两者收敛，则$γ_1$和$γ_2$之间的区域是$f$的收敛集。当$γ_1$和$γ_2$除了在它们的端点处不相交时，后一个事实是直接的，但是当$γ_1$和$γ_2$高度相交时，后一个事实变得不平凡。最后，我们检查的收敛集的功能$e^{\frac{z+1}{z-1}}$在$1$的文件。
英文摘要：
Let $f:\mathbb{D}\to\mathbb{C}$ be a bounded analytic function. A set $K\subset\mathbb{D}$ which contains the point $1$ in its boundary is called a convergence set for $f$ at $1$ if $f(z)$ converges to some value $ζ$ as $z\to1$ with $z\in K$. $K$ is called a path divergence set for $f$ at $1$ if $f$ diverges along every path $γ$ which lies in $K$ and approaches $1$. In this article, we show that for a path $γ$ through the unit disk from $-1$ to $1$, if $f$ fails to converge along $γ$, then either the region above $γ$ or the region below $γ$ is a path divergence set for $f$. On the other hand, if $γ_1$ and $γ_2$ are two such paths, and $f$ converges along both $γ_1$ and $γ_2$, then the region between $γ_1$ and $γ_2$ is a convergence set for $f$. This latter fact is immediate when $γ_1$ and $γ_2$ do not intersect except at their end-points, but becomes non-trivial when $γ_1$ and $γ_2$ are highly intersecting. We conclude the paper with an examination of the convergence sets for the function $e^{\frac{z+1}{z-1}}$ at $1$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
