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Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. (English) Zbl 1085.30026

Let m(r,f) be the function and T(r,f) the characteristic of a meromorphic function f. The authors prove that if f is a non-Nevanlinna proximity constant meromorphic function, c, δ<1 and ε>0, then

mr , f(z+c) f(z)=oT(r+|c|,f) 1+ε r δ

for all r 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.

30D35Distribution of values (one complex variable); Nevanlinna theory
39B32Functional equations for complex functions