The Ramanujan Journal

, Volume 16, Issue 1, pp 105–129 | Cite as

On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane



We investigate the growth of the Nevanlinna characteristic of f(z+η) for a fixed ηC in this paper. In particular, we obtain a precise asymptotic relation between T(r,f(z+η)) and T(r,f), which is only true for finite order meromorphic functions. We have also obtained the proximity function and pointwise estimates of f(z+η)/f(z) which is a discrete version of the classical logarithmic derivative estimates of f(z). We apply these results to give new growth estimates of meromorphic solutions to higher order linear difference equations. This also allows us to solve an old problem of Whittaker (Interpolatory Function Theory, Cambridge University Press, Cambridge, 1935) concerning a first order difference equation. We show by giving a number of examples that all of our results are best possible in certain senses. Finally, we give a direct proof of a result in Ablowitz, Halburd and Herbst (Nonlinearity 13:889–905, 2000) concerning integrable difference equations.


Poisson–Jensen formula Meromorphic functions Order of growth Difference equations 

Mathematics Subject Classification (2000)

30D30 30D35 39A05 


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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsHong Kong University of Science and TechnologyHong KongPeople’s Republic of China
  2. 2.Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingPeople’s Republic of China

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