A note on uniqueness boundary of holomorphic mappings

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{I}\mathrm{m}$(z)>0}$with$$\mathcal{C}$^\infty$-smoothextensionupto$(-1,1)$suchthat$f$maps$(-1,1)$intoacone$\Gamma_C:={z\in \mathbb C\colon |$\mathrm{I}\mathrm{m}$ (z)| \leq C|$\mathrm{R}\mathrm{e}$ (z)|}$,forsomepositivenumber$C$,and$f$vanishestoinfiniteorderat$0$,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.