Unicity theorems for meromorphic functions sharing five or six values in some sense (Q1911169)

Two meromorphic functions \(f\) and \(g\) share the complex value \(c\) IM (ignoring mulplicities) if \(f(z)= c\) implies \(g(z)= c\) and vice versa. The value \(c\) is shared CM (counting multiplicities) if, in addition, \(f\) and \(g\) have the same multiplicities at each \(c\)-point. For a meromorphic function \(f\), a complex value \(c\) and a positive integer \(k\), \(E_k(c, f)\) denotes the set of all points \(z\) where \(f(z)= c\) with multiplicity less than or equal to \(k\). Theorem 1 and 2 in the paper under review give a complete characterization of all pairs \(f\) and \(g\) of non-constant meromorphic functions in the complex plane that (i) share two values CM and satisfy \(E_2(a_j, f)= E_2(a_j, g)\) for three more values \(a_1\), \(a_2\), \(a_3\); (ii) share two values CM and satisfy \(E_1(a_j, f)= E_1(a_j, g)\) for four more values \(a_1\), \(a_2\), \(a_4\).