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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, \(c\in\mathbb{C}\), \(\delta< 1\) and \(\varepsilon> 0\), then \[ m\Biggl(r, {f(z+ c)\over f(z)}\Biggr)= o\Biggl({T(r+|c|,f)^{1+\varepsilon}\over r^\delta}\Biggr) \] 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.

MSC:
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
39B32 Functional equations for complex functions
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References:
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