The uniqueness of analytic functions on annuli sharing some values (Q448845)

Let \(f\) and \(g\) be analytic functions in \(A = \{ z \mid 1/R_0 < |z| < R_0 \}\), where \(1 < R_0 \leq + \infty\), and let \(a_j\), \(j = 1, 2, 3, 4\), be distinct complex numbers. If \(f\) and \(g\) share \(a_1\) and \(a_2\) CM in \(A\), \(f = a_3\) implies \(g = a_3\) in \(A\), \(f = a_4\) implies \(g = a_4\) in \(A\), and \(f\) is transcendental or admissible in \(A\), then the authors prove that \(f(z) \equiv g(z)\) holds in \(A\). If CM is replaced by IM, the authors show that \(f \equiv g\) in \(A\) or \(f \equiv (a_3 g - a_1 a_2)/(g-a_4)\), and \(a_1 + a_2 = a_3 + a_4\), and \(a_3\) and \(a_4\) are exceptional values of \(f\) and \(g\) in \(A\), respectively. These results improve a theorem of \textit{T.-B. Cao, H.-X. Yi} and \textit{H.-Y. Xu} [Comput. Math. Appl. 58, No. 7, 1457--1465 (2009; Zbl 1189.30065)].