I don't know if this is true of conformal mapping or just mapping in general but I want to be completely sure that if I know how the contour of a region transforms then the points within the original contour will be inside the transformed contour as well.
If this isn't clear consider the opposite: I know how the contour transforms and it is a unit circle centered at the origin. Could there be a point within the original region that lies outside this contour?
Holomorphic functions are open maps; that is, if you can find an open set around it in the original region (i.e., it isn't a part of the boundary), then whatever point it gets mapped to will have a corresponding open set around it. Therefore, it can't be an element of the boundary of the image.