An idea: use the Gauss-Lucas theorem (Gauss) , whose proof is pretty
easy and beautiful.
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?
Thanks for the theorem.
I was actually thinking about a geometric interpretation of the sums above, trying to locate in relation to those roots. I had some idea that the expressions were about taking averages somehow, but it wasn't really clear. Now I see the connection.
In any case, I would still like to know if my work above is salvageable. Any thoughts would be appreciated.