I am working on the following problem:
Let
be a holomorphic polynomial of degree at least 1 so that
in the upper half plane
. Prove that
on
.
I wrote
where
are the roots of the polynomial and
are the multiplicity. Then, taking the real and complex parts, I have
Since
by assumption, if we input
where
, then all of the terms in sum the complex part will be negative, so
and therefore
. But there is a little bit of a problem with this: it is possible that
. If this is the case, then we must have
. However, I do not see how to guarantee that
. It almost seems possible to construct a counterexample to the problem with this in mind.
Does anybody have any suggestions?