Meromorphic functions sharing three values with their shift - Majumder, Sujoy; Das, Pradip (Ukrains’kyi Matematychnyi Zhurnal, 2024)
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 cC{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.