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 $\Delta^+:={z\in \mathbb C \colon |z|<1,~\mathrm{Im}(z)>0}$ with $\mathcal{C}^\infty$-smooth extension up to $(-1,1)$ such that $f$ maps $(-1,1)$ into a cone $\Gamma_C:={z\in \mathbb C\colon |\mathrm{Im} (z)| \leq C|\mathrm{Re} (z)|}$, 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.