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Zbl 1085.30026
Halburd, R.G.; Korhonen, R.J.
Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. (English)
[J] J. Math. Anal. Appl. 314, No. 2, 477-487 (2006). ISSN 0022-247X

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, $c\in\bbfC$, $\delta< 1$ 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.\par 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.
[Delfina Roux (Milano)]

MSC 2000:: *30D35 Distribution of values (one complex variable)
39B32 Functional equations for complex functions

Keywords: Logarithmic difference; Nevanlinna theory; Difference equation

Cited in: Zbl 1249.30110 Zbl 1236.30030 Zbl 1180.30039 Zbl 1172.30009

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