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