Let $f(z)$ be an entire function whose modulus is constant on some circle. Show that $f(z)=f(z_0) + c(z-z_0)^n$ for some $n \geqslant 0$ and some constant $c$, where $z_0$ is the center of the circle.
I only know a sketch of the proof, using Schwarz reflection Principle. I would seriously appreciate giving details to it.