A note on a conjecture concerning boundary uniqueness

Abstract

We consider the following conjecture (from Huang, et al): Let ${\Delta }^{+}$ denote the upper half disc in $\mathbb{C}$ and let $\gamma =(-1,1)$ (viewed as an interval in the real axis in $\mathbb{C}$). Assume that $F$ is a holomorphic function on ${\Delta }^{+}$ with continuous extension up to $\gamma$ such that $F$ maps $\gamma$ into $\{|\mbox{Im} z|\leq C|\mbox{Re} z|\},$ for some positive $C.$ If $F$ vanishes to infinite order at $0$ then $F$ vanishes identically. We show that given the conditions of the conjecture, either $F\equiv 0$ or there is a sequence in ${\Delta }^{+}$, converging to $0,$ along which $\mbox{Im} F/\mbox{Re} F$ (defined where $\mbox{Re} F\neq 0$) is unbounded.