Suppose that $f$ is analytic on a domain $D$, which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$, then I want to prove that either $f$ is constant or $f$ has a zero inside $\gamma$
if $f$ is not constant, then the max/min modulus principle applies ... meaning $|f|$ can not have any local max/min on D
I don't really know what to do next
Assume $|f(z)|=c$ for $z$ on $\gamma$. If $f$ has no zero inside $\gamma$, then $\frac1f$ is analytic there and $\frac1{|f|}$ attains its maximum on $\gamma$. So does $|f|$, hence we conclude that both $\frac1{|f(z)|}\le\frac1c$ and $|f(z)|\le c$ inside $\gamma$. Conclude that $|f|$ is constant and ultimately also $f$ is constant.