A note on uniqueness boundary of holomorphic mappings

Abstract

In this paper, we prove Huang et al.'s conjecture stated that if $f$ is a holomorphic function on ${\Delta }^{+}:=\{z\in \mathbb{C}:|z|<1,\mathrm{Im}(z)>0\}$ with ${\mathcal{C}}^{\infty }$-smooth extension up to $(-1,1)$ such that $f$ maps $(-1,1)$ into a cone ${\Gamma }_C:=\{z\in \mathbb{C}:|\mathrm{Im}(z)|\le C|\mathrm{Re}(z)|\}$, for some positive number $C$, and $f$ vanishes to infinite order at $0$, then $f$ vanishes identically. In addition, some regularity properties of the Riemann mapping functions on the boundary and an example concerning Huang et al.'s conjecture are also given.

12 pages