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 .
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?