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Abstract
We verify the following conjecture (from Huang et al.): Let denote the upper half disc in
and let
(viewed as an interval in the real axis in
). Assume that
is a holomorphic function on
with continuous extension up to
such that
maps
into
for some positive
If
vanishes to infinite order at
then
vanishes identically. This result is already known to hold true for
Keywords:
AMS Subject Classifications:
Acknowledgements
The authors thank the referees and the editor.
Notes
1 Lakner [Citation2] defines vanishing to infinite order at , by
for all
.
2 The original theorem also ensures that there exists a such that
where
is the volume of
, and
is the surface measure on
3 Here denotes the least upper integer.
4 For example it follows from the Luzin-Privalov theorem.[Citation8]