A note on uniqueness boundary of holomorphic mappings

TL;DR

This paper proves a conjecture that a holomorphic function with specific boundary behavior and vanishing order must be identically zero, and explores boundary regularity of Riemann mappings with additional examples.

Contribution

It confirms Huang et al.'s conjecture regarding the uniqueness of holomorphic functions under boundary and vanishing conditions, and discusses boundary regularity of Riemann mappings.

Findings

Holomorphic functions with boundary constraints vanish identically if they vanish to infinite order at a point.

Boundary regularity properties of Riemann mapping functions are established.

An example related to Huang et al.'s conjecture is provided.

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.