Phase retrieval on circles and lines

Let f and g be analytic functions on the open unit disc D such that $|f|=|g|$ on various sets A. We first prove that there exists c in the unit circle T such that $f=cg$ when A is the union of two lines in D intersecting at an angle that is an irrational multiple of π. The same conclusion is valid when f and g are in the Nevanlinna class and A is the union of the unit circle and an interior circle, tangential or not. We also provide sequential versions of the previous results and analyse the case A=r T. Finally we examine the situation when there is equality on two distinct circles in the disc, proving a result or counterexample for each possible configuration.