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 +:=\z∈ C |z|<1,~Im(z)>0\ with C∞-smooth extension up to (-1,1) such that f maps (-1,1) into a cone C:=\z∈ C |Im (z)| ≤ C|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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