It's easy to see that the Phargmen-Lindelöf theorem from complex analysis can be generalized to non-simply-connected regions. Namely to regions $G$ with the property that for each $z \in \partial_\infty G $ there is a sphere $V$ in $\mathbb{C}_\infty $ centered at $z$ such that $V \cap G$ is simply connected.

The problem is that I can't think about any example of non-simply connected regions that have this property and of simply-connected regions that don't have this property...

Does the exterior of a unit ball can be considered as an example of non-simply connected region that don't satisfy the property above? What about regions that do satisfy this property?

BTW - $ \partial _ \infty $ is the boundary of $G$ with the boundary at infinity (and $\mathbb{C}_\infty$ is the Riemann-Sphere ) .

Thanks in advance