Let be the function and the characteristic of a meromorphic function . The authors prove that if is a non-Nevanlinna proximity constant meromorphic function, , and , then
for all outside an exceptional set with finite logarithmic measure. This theorem is a difference analogue of the logarithmic derivative lemma, which is a useful tool in the study of complex solutions of nonlinear differential equations.
The paper contains also a number of results about the finite-order meromorphic solutions of large classes of nonlinear difference equations, obtained by using the above theorem.