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arXiv · 1605.01232

A note on uniqueness boundary of holomorphic mappings

Ninh Van Thu·Nguyen Ngoc Khanh
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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.

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Ninh Van Thu, Nguyen Ngoc Khanh. 2016-05-04. A note on uniqueness boundary of holomorphic mappings. https://arxiv.org/abs/1605.01232

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