Meromorphic functions sharing three values with their shift

UDC 517.5 We discuss the  problem of uniqueness of a meromorphic function f ( z ) , which shares a 1 ( z ) , a 2 ( z ) , and a 3 ( z ) CM with its shift f ( z + c ) , where a 1 ( z ) , a 2 ( z ) , and a 3 ( z ) are three c -periodic distinct small functions of f ( z ) and  c ∈ ℂ ∖ { 0 } . The obtained result improves the recent result of Heittokangas et al. [Complex Var. and Elliptic Equat., 56, No. 1–4, 81–92 (2011)]  by dropping the assumption about the order of f ( z ) .  In addition, we introduce a way of characterizing elliptic functions in terms of meromorphic functions  sharing values with two of their shifts.  Moreover, we show by  a number of illustrating examples that  our results are,  in certain senses, best possible.