The periodicity of meromorphic functions when they share two sets with finite weights

For a non-constant meromorphic function f and a set S subset of C boolean OR {infinity}, we define E-f(S) = boolean OR(a is an element of S){z vertical bar f(z) - a = 0, counting multiplicities}. Two non-constant meromorphic functions f and g are said to share the set S CM (counting multiplicities) if E-f(S) = E-g(S). In this paper, we investigate the periodicity of non-constant meromorphic functions f when the general expressions Psi(f(z) )= a(2)f(2)(z) + a(1)f(z) and Psi(f(z+c) )= a(2)f(2)(z + c) + a(1)f(z + c) off share two sets with finite weights. The main results obtained in the paper establish a version of that of Zhang in [J Math Anal Appl 367(2):401-408 (2010)]. In addition, it is shown that a more accurate period estimate can be obtained if the sets considered by Zhang are replaced by more general ones.