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.


30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
39B32 Functional equations for complex functions
Full Text: DOI arXiv


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