A note on a conjecture concerning boundary uniqueness

TL;DR

This paper investigates a boundary uniqueness conjecture for holomorphic functions, showing that under certain boundary conditions, the function either vanishes identically or exhibits unbounded behavior near the boundary point.

Contribution

It provides a partial result on the conjecture, demonstrating the dichotomy of the function's behavior under the given boundary and vanishing conditions.

Findings

Either the function is identically zero or the ratio of imaginary to real parts becomes unbounded near zero.

The result clarifies the boundary behavior of holomorphic functions under specific geometric constraints.

Supports the conjecture by establishing conditions for the function's triviality or unboundedness.

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.