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

By Y. Chiang, Shaoji Feng. 2006

814 citations recorded by the source corpus.

We investigate the growth of the Nevanlinna characteristic of f(z+η) for a fixed η∈C 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/610d97257d39faa77303963b305a8325f70a7071

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In Applied Mathematics

1. R. Korhonen
2. Zong-Xuan Chen
3. Kai Liu
4. R. Halburd
5. I. Laine
6. Lian-Zhong Yang
7. Jilong Zhang
8. X. Zheng

In Geometry and Topology

1. R. Korhonen
2. Zong-Xuan Chen
3. Kai Liu
4. I. Laine
5. R. Halburd
6. X. Zheng
7. Y. Chiang
8. Shaoji Feng

Citations in the corpus

• Nevanlinna theory for the $q$-difference operator and meromorphic solutions of $q$-difference equations (2007, methodology): “Recently, similar tools involving shifts have been developed to study ordinary difference equations [6, 13, 14].”
• A Difference Picard Theorem for Meromorphic Functions of Several Variables (2009, background): “Chiang and Feng [7] have shown that if f is a finite-order meromorphic function in the complex plane and η ∈ C, then (4.”
• Some properties of meromorphic solutions of q-difference equations☆ (2010, background): “Recently, a number of papers (see [5, 14, 17]) focused on complex difference equations and difference analogues of Nevanlinn’s theory.”
• On Sharing Values of Meromorphic Functions and Their Differences (2013, background): “(see [3]) Let f be a meromorphic function with a finite order σ, η be a nonzero constant.”
• Holomorphic curves with shift-invariant hyperplane preimages (2009, background): “This research was supported in part by the Academy of Finland Grant #112453, #118314 and #210245, a grant from the EPSRC, the Isaac Newton Institute for Mathematical Sciences, the Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research (C) #19540173, #22540181, and a project ”
• On Entire Solutions of Some Differential-Difference Equations (2013, background): “1) can be seen as a linear difference equation F(z+c) = −h′(z)F(z), some results can be found in [2,15].”
• Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity (2009, background): “Chiang and Feng have obtained similar estimates for the logarithmic differences in [3], and this work is independent from [9,10].”
• On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions (2006, background): “2) obtained in [8] as a consequence of our Theorem 2.”