Nevanlinna theory for the difference operator

Certain estimates involving the derivative f 7→ f ′ of a meromorphic function play key roles in the construction and applications of classical Nevanlinna theory. The purpose of this study is to extend the usual Nevanlinna theory to a theory for the exact difference f 7→ ∆f = f(z+ c)− f(z). An a-point of a meromorphic function f is said to be c-paired at z ∈ C if f(z) = a = f(z+c) for a fixed constant c ∈ C. In this paper the distribution of paired points of finite-order meromorphic functions is studied. One of the main results is an analogue of the second main theorem of Nevanlinna theory, where the usual ramification term is replaced by a quantity expressed in terms of the number of paired points of f . Corollaries of the theorem include analogues of the Nevanlinna defect relation, Picard’s theorem and Nevanlinna’s five value theorem. Applications to difference equations are discussed, and a number of examples illustrating the use and sharpness of the results are given.