$|e^a-e^b| \leq |a-b|$ for complex numbers with non-positive real parts

Consider integrating $f'(z) dz$ along the line segment from $a$ to $b$

An interesting related article A norm inequality for Hermitian operators by Ritsuo Nakamoto

The American Mathematical Monthly; Mar 2003; 110, 3;

Prove and then use the following fact:

Let $D \subseteq \mathbb C$ be a convex region and let $f: D \to \mathbb C$ be holomorphic with $|f'|\le 1$ on $D$. Then for $a,b\in D$ we have

$$ |f(b) - f(a)| \le |b-a|$$