 Solution

The two holomorphic function functions agree throughout their overlap region strip, so they define a single holomorphic function on the union of their domains, which is the whole complex plane. The resulting entire function is bounded outside a sufficiently large disk by the assumptions at infinity. It is bounded on the disk by continuity and compactness. Liouville theorem therefore makes it constant:Agreement on an open overlap region is what permits analytic continuation; equality of unspecified boundary values alone would need additional hypotheses.