It's a little trickier than the real-variable problem, where you could just use the mean-value theorem.

Here the trick is to fix z_0 in D (which you have to assume is connected) and let w_0 = f(z_0). Let A = {z in G | f(z) = w_0}. You can show A = D by showing that A is both open and closed in D. The details are messy... I recall there is a pretty good proof in Conway though.