Meromorphic Functions Sharing Three Values with Their Shift

We discuss the problem of uniqueness of a meromorphic function f (z), which shares a₁(z), a₂(z), and a₃(z) CM with its shift f (z + c), where a₁(z), a₂(z), and a₃(z) are three c-periodic distinct small functions of f (z) and c ∈ \({\mathbb{C}}\) $\mathbb{C}$ \{0}. The obtained result improves the recent result of Heittokangas, et al. [Complex Var. Elliptic Equat., 56, No. 1-4, 81 (2011)] by omitting the assumption about the order of f (z). In addition, we introduce a way of characterization of elliptic functions in terms of meromorphic functions sharing values with two of their shifts. Moreover, by a number of illustrative examples, we show that our results are, in certain senses, best possible.