Meromorphic functions sharing three values with their shift - Majumder, Sujoy; Das, Pradip (Ukrains’kyi Matematychnyi Zhurnal, 2024)
UDC 517.5We 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 $c\in\mathbb{C}\setminus\{0\}$. 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.