This past semester I took a graduate course in complex analysis which I completed moderately well in spite of my expectations (that is I honestly think I deserved a lower grade than I received). I had one assigned question which caused me to join MSE on the subject of Rouché's theorem. This problem comes from Chapter 5 section three of Conway's Functions of a single Complex Variable, the section is on the Argument Principle and Rouché's theorem.
The problem is number 2 of this chapter and proceeds as such:
"Suppose $f$ is analytic on $\bar B(0 ; 1) $ and satisfies $\left| f(z) \right|<1 $ for $ |z|=1$. Find the number of solutions (counting multiplicities) of the equation $f(z) = z^n$ where $n$ is an integer larger than or equal to $1$."
Now my question isn't about solving this problem but more of a hint at where to start. As I understood Rouché's theorem, I am required to compare two functions, say $f$ and $g$, so I can compare the number of poles and zeroes. Every example I could find on MSE however included an equation with more than one term and seemed (at least to me) easier than the one I was given. I unfortunately do not have any work of my own to provide as I was definitely stumped. Any discussion of this will help, or references to MSE questions I might have missed would be greatly appreciated
Best Answer
Hint:
We want to find the number of roots of the (holomorphic) function $h(z)=f(z)-z^n$ inside the unit circle. The main way Rouche's theorem is used in this context is by splitting up the function you're interested in into two parts, one of which is strictly less than the other on a given simple closed contour (here the boundary of the unit circle). Since $h$ is already a sum of two functions, there is a natural way to split it up...