Let $D$ be ad open set of $\mathbb{C}$ and let $f: D \to \mathbb{C}$ be a holomorphic function.
Let $z_0 \in D$ and let $x_0 \in \mathbb{R_+}$ such that the segment $S$ from $z_0$ to $z_0+x_0$ is in $D$
I would like to know if it is true that; $$ \max_{z,s \in S}|f(z)-f(s)|=|f(z_0)-f(z_0+x_0)| $$