06-06-2023
Problem: Let $U$ be an open and convex set and let $f: U\subset \mathbb{C} \to \mathbb{C} $ be an analytic function on $U$. Suppose that $\left\vert f^\prime (z) \right\vert \leq 1$ for all $z\in U$. Show that \[ \left\vert f\left( z \right) - f\left( w \right) \right\vert \leq \vert z - w \vert \] for any $z,w \in U$.
Solution: Let $z_1, z_2 \in D$. We will show that \[ \left\vert f\left( z_1 \right) - f\left( z_2 \right) \right\vert \leq \left\vert z_1 - z_2 \right\vert . \] Let $\gamma $ be the line joining $z_1$ and $z_2$. Since $D$ is convex, $\gamma \subseteq D$.
As function is analytic we can consider the following integral. \[ f\left( z_2 \right) - f\left( z_1 \right) = \int _\gamma f^\prime (z)~\mathrm{d} z \] Let $\ell(\gamma)$ denotes the length of $\gamma$. We have \begin{align*} \left\vert f\left( z_1 \right) - f\left( z_2 \right) \right\vert & \leq \left\vert \int _\gamma f^\prime (z)~\mathrm{d} z \right\vert \\ & \leq \left\vert f^\prime (z) \right\vert \cdot \ell (\gamma ) \\ & \leq 1\cdot \left\vert z_2 - z_1 \right\vert \\ & = \left\vert z_2 - z_1 \right\vert. \end{align*}