# Use the Maximum modulus principle and the zeros of an analytic function.

Let $$f$$ and $$g$$ be analytic on a connected open set $$U$$. Assume that the closed disc $$\overline{D(z_0,r)}$$ is contained in $$U$$, where $$r$$ is a positive number. Show that if $$|f(z)|=|g(z)|$$ on the circle $$|z-z_0|=r$$ then there is a constant $$\lambda$$ with $$|\lambda|=1$$ such that $$f=\lambda g$$ on $$U$$.

Use the Maximum modulus principle and the zeros of an analytic function.

• $f(z) = 1$, $g(z) = z$ is a counterexample on the unit disk. Mar 12, 2020 at 3:43