Certain estimates involving the derivative f↦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↦Δf=f(z+c)−f(z).
An a-point of a meromorphic function f is said to be c-paired at
z\in\C if f(z)=a=f(z+c) for a fixed constant c\in\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.Comment: 19 page