Uniqueness of Meromorphic Functions and Differential Polynomials

Let f be a nonconstant meromorphic function defined in the whole complex plane . It is assumed that the reader is familiar with the notations of the Nevanlinna theory such as T r, f , m r, f ,N r, f , and S r, f , that can be found, for instance, in 1–3 . Let f and g be two nonconstant meromorphic functions. Let a be a finite complex number. We say that f and g share the value a CM counting multiplicities if f − a and g − a have the same zeros with the same multiplicities, and we say that f and g share the value a IM ignoring multiplicities if we do not consider the multiplicities. When f and g share 1 IM, let z0 be a 1-point of f of order p and a 1-points of g of order q; we denote by N11 r, 1/ f − 1 the counting function of those 1-points of f and g, where p q 1 and by N 2 E r, 1/ f − 1 the counting function of those 1-points of f and g, where p q ≥ 2. NL r, 1/ f − 1 is the counting function of those 1-points of both f and g, where p > q. In the same way, we can define N11 r, 1/ g − 1 , N 2 E r, 1/ g − 1 , and NL r, 1/ g − 1 . If f and g share 1 IM, it is easy to see that