Let $f$ be a non-constant and analytic on a neighborhood of closure of the unit disk such that $|f(z)|=\text{constant}$ for $|z|=1$. Prove $f$ has at least one zero inside unit disk.
I thought of using Rouche's somehow. Using $f(z)-z$, and taking constant is less than $1$, I can actually conclude from Rouche's theorem that the equation $f(z)-z=0$ have a fixed point inside the unit disk. I am stuck for other constants greater than equals to one and exactly zero.
I hope there should be a little trick I am missing here. It will be awesome to see if maximum principle can be applied to conclude the result.
Best Answer
Hint: if $f$ has no zero inside the disk, consider $1/f$ and use the maximum modulus principle.