Unicity theorems for entire functions concerning four small functions (Q1348411)

Let \(f\) be a non-constant meromorphic function in the complex plane \(\mathbb C\). A meromorphic function \(a\) is called a small function with respect to \(f\) if \(T(r,a)=S(r,f)\). Let \(S(f)\) denote the set of all small functions with respect to \(f\). It is known that \(S(f)\) is a field which contains \(\mathbb C\). Now, let \(f\) be a non-constant entire function, \(a \in S(f)\), and let \(k\) be a positive integer or \(\infty\). Then \(\overline{E}(a,k,f)\) denotes the set of distinct zeros of \(f-a\) with multiplicities at most \(k\). In particular, \(\overline{E}(a,\infty,f)\) is the set of distinct zeros of \(f-a\). With these notations the main result of this paper reads as follows. Theorem. Let \(f\) and \(g\) be non-constant entire functions, and let \(a_1\), \(a_2\), \(a_3\), \(a_4\) be four distinct elements in \(S(f) \cap S(g)\). If \(\overline{E}(a_j,k,f)=\overline{E}(a_j,k,g)\) for \(j=1,2,3,4\), where \(k\) is a positive integer or \(\infty\) with \(k \geq 8\), then \(f \equiv g\).