Meromorphic functions sharing three values with their shift

Abstract

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., <strong>56</strong>, 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.