NEVANLINNA THEORY FOR THE DIFFERENCE OPERATOR

Certain estimates involving the derivative f 7! f 0 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 2 C if f(z) = a = f(z + c) for a xed constant c 2 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 ramication 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