## 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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