Let $U \subseteq \mathbb{C}$ be an open connected subset and $f: U \rightarrow \mathbb{C}$ an holomorphic function.  

Let $z\_{0} \in U$ and $r > 0$ with $B\_{r}(z\_{0}) \subseteq U$.  

If $\forall z \in B\_{r}(z\_{0})$ $|f(z)| \geq |f(z\_{0})|$ then $f(z\_{0})=0$.

As $f$ is an holomorphic map $f$ has the mean value property and then  

$|f(z\_{0})|\leq \max\_{\partial B\_{r}(z\_{0})}|f|$  

but I don't how to derive that is zero.

Thanks in advance.