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

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 $1$.