Let $G\subset \mathbb C$ be a convex set and $f$ is an holomorphic function. I've proved that if $|f'(z)|<1$ for all $z\in G$ then $|f(z)-f(w)|<|z-w|$ for all $z,w\in G$.
Now I'm trying to find an example of an holomorphic function and an open connected set (which is not a convex set) that satifies $|f'(z)|<1$ for all $z\in G$ but not $|f(z)-f(w)|<|z-w|$
Thanks.