Holomorphic function constant on the boundary

Question

Let $D$ be a domain and $f$ holomorphic on $D$ and continuous on its closure $\overline D$. Now, if $f$ doesn't have any zeros in $D$ one can show that if $f$ is constant on $\partial D$ it must be constant on $D$. Does this hold as well if $f$ has any zeros inside $D$? If not, can someone provide an example?

Answer 1

You need some assumptions about the boundary of $D$, namely that the boundary is not a discrete subset of the complex plane. Otherwise, your claim is clearly false (think of $D={\mathbb C}- \{a_1,...,a_n\}$ and a certain degree $n$ polynomial on ${\mathbb C}$).

The relevant theorem than is due to Rado and Cartan:

Theorem. Suppose that $D$ is a domain in ${\mathbb C}$ and $p\in \partial D$ is a non-isolated boundary point. Suppose that $U\subset \partial D$ is a neighborhood of $p$ (in the subspace topology). Assume that $f: D\to {\mathbb C}$ is holomorphic and has continuous extension to $D\cup U$ which is constant on $U$. Then $f$ is constant.

See Ch. 11, section 8, Theorem 2 in

R.Narasimhan, Complex Analysis in One Variable, Springer Verlag.