Meromorphic function sharing two sets with its difference operator (Q432538)

The authors study the relationship between a meromorphic function \(f\) and its difference operators when they share two sets and obtain some results such as the following theorems. Theorem 1.3. Suppose that \(f\) is a nonconstant meromorphic function of finite order such that \(E_{f}(S_{1}, 2)=E_{\Delta_{c}f}(S_{1}, 2)\) and \(E_{f}(S_{2}, \infty)=E_{\Delta_{c}f}(S_{2}, \infty)\). If \(n\geq 7\), then \(\Delta_{c}f \equiv t f\), where \(t^{n}=1\) and \(t\neq -1\). Theorem 1.4. Suppose that \(f\) is a nonconstant meromorphic function of finite order satisfying \(E_{f}(S_{1}, 0)=E_{\Delta_{c}f}(S_{1}, 0)\) and \(E_{f}(S_{2}, \infty)=E_{\Delta_{c}f}(S_{2}, \infty)\). If there exists a constant \(\alpha\) with \(0<\alpha\leq 2\) such that \[ \overline{N}(r, f(z))+\overline{N}\left(r,\frac{1}{f(z)}\right)<\alpha T(r, f(z)), \] and if \(n\geq 15\alpha/2 +4\), then \(\Delta_{c}f\equiv t f\), where \(t^{n}=1\) and \(t\neq -1\). Theorem 1.5. Suppose that \(f\) is a nonconstant meromorphic function of finite order such that \(E_{f}(S_{1}, 2)=E_{\Delta_{c}f}(S_{1}, 2)\) and \(E_{f}(S_{2}, 0)=E_{\Delta_{c}f}(S_{2}, 0)\). If \[ \overline{\lim\limits_{r\rightarrow\infty}}\frac{N\left(r, \frac{1}{f(z)}\right)}{T(r, f(z))}<1, \] and if \(n\geq 7,\) then \(\Delta_{c}f\equiv t f\), where \(t^{n}=1\) and \(t\neq -1\). These interesting theorems improve the results obtained in [\textit{J. Zhang}, J. Math. Anal. Appl. 367, No. 2, 401--408 (2010; Zbl 1188.30044)].