Lets imagine that D - given domain (strip) on upper half plane H, and G - its image (connected) under mapping f with required properties. The image has curved side (not stright line, like, polygons!). I want to continue my domain (image) analytically twice. Can I see, likewise polygon that after two time analytical continuation, the latest domain has connection with initial withing fractual transformation? i.e. f**(z)=(af(z)+b)/(cf(z)+d), where f** - two times analytical continued.
Any help would be greatly appreciated.