If $f$ is analytic on $\overline{B(0,1)}$, and $|f(z)| < 1$ for $|z|=1$. Find the number of solutions to $f(z) = z^n$

Question

If $f$ is analytic on $\overline{B(0,1)}$, and $|f(z)| < 1$ for $|z|=1$. Find the number of solutions (including multiplicities) to $f(z) = z^n$.

I thought about applying Rouche's theorem to the function $h(z) = f(z) - z^n$, but Rouche's theorem requires us to have some other meromorphic function, $g$, so that we can equate the difference of zeros and poles of $f$ to the difference of zeros and poles of $g$. I'm not really sure which function I would want to use, or if Rouche's theorem is even the right approach.

Any thoughts on this one would be really appreciated.

Thanks in advance!

Answer 1

By Rouché's theorem, since$$|-f(z)|<1=|z^n|$$when $|z|=1$, the functions $z^n$ and $z^n-f(z)$ have the same number of zeros inside $B(0,1)$. So, $z^n-f(z)$ has $n$ zeros there.