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?