Boundary convergence and path divergence sets for bounded analytic functions in the disk

TL;DR

This paper characterizes boundary convergence and divergence sets for bounded analytic functions in the disk, revealing how paths approaching boundary points influence convergence behavior and identifying the structure of such sets.

Contribution

It establishes new geometric criteria for convergence and divergence sets based on paths within the disk, including complex intersecting paths, and analyzes specific functions.

Findings

Regions above or below a path are divergence sets if the function fails to converge along that path.

Between two converging paths, the region is a convergence set for the function.

The structure of convergence sets can be complex when paths intersect heavily.

Abstract

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 $\zeta$ as $z\to1$ with $z\in K$. $K$ is called a path divergence set for $f$ at $1$ if $f$ diverges along every path $\gamma$ which lies in $K$ and approaches $1$. In this article, we show that for a path $\gamma$ through the unit disk from $-1$ to $1$, if $f$ fails to converge along $\gamma$, then either the region above $\gamma$ or the region below $\gamma$ is a path divergence set for $f$. On the other hand, if $\gamma_1$ and $\gamma_2$ are two such paths, and $f$ converges along both $\gamma_1$ and $\gamma_2$, then the region between $\gamma_1$ and $\gamma_2$ is a convergence set for $f$. This latter fact is immediate when $\gamma_1$ and $\gamma_2$ do not intersect except at their end-points, but becomes non-trivial when $\gamma_1$ and $\gamma_2$ are highly intersecting. We conclude the paper with an examination of the convergence sets for the function $e^{\frac{z+1}{z-1}}$ at $1$.