# Strong maximum modulus principle

(Weak) maximum modulus principle states that if $f$ is a non-constant holomorphic function on some domain $D$ in $\mathbb{C}$ then $|f|$ can't have global maximum in $D$.

Strong version says that $|f|$ cannot attain local maximum in $D$.

My professor said, that strong version is a consequence of weak version and identity theorem. I don't see that. Any explanation please?

By definition, $z_0\in D$ is a local maximum of $|f|$ if it is a global maximum of $|f|$ on some neighborhood $U\subset D$ such that $z_0\in U$. Suppose such a local maximum exists at $z_0$; then by the weak version, $f$ must be constant on the neighborhood $U\ni z_0$. But since $D$ is connected and $U\subset D$, the fact that $f$ is constant on $U$ implies that $f$ is constant on all of $D$ by the identity theorem. Thus $|f|$ can only have a local maximum in $D$ if $f$ is constant, so a non-constant holomorphic function cannot attain a local maximum modulus in $D$.