Since the body text shows up a little weird on mobile I'll post my main question and attempt here.

Conway asks why this result won't hold when G is any general region. My reasoning is that we need to guarantee that G is large enough and/or "round" enough (i.e. the G has no strange corners or wavy parts) so that u won't escape G ~~(i.e., u(z) always falls in a ball contained in the image of G)~~ (i.e., that for every ball B(z,R) in G, there exists a ball B(u(z),ρ) in u(G)). Is my reasoning correct?