Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity

This research is a continuation of a recent paper due to the first four authors. Shared value problems related to a meromorphic function f ( z ) and its shift f ( z + c ) , where c ∈ C , are studied. It is shown, for instance, that if f ( z ) is of finite order and shares two values CM and one value IM with its shift f ( z + c ) , then f is a periodic function with period c. The assumption on the order of f can be dropped if f shares two shifts in different directions, leading to a new way of characterizing elliptic functions. The research findings also include an analogue for shifts of a well-known conjecture by Brück concerning the value sharing of an entire function f with its derivative f ′ .