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On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions


Authors: Yik-Man Chiang and Shao-Ji Feng
Journal: Trans. Amer. Math. Soc. 361 (2009), 3767-3791
MSC (2000): Primary 30D30, 30D35, 39A05; Secondary 46E25, 20C20
DOI: https://doi.org/10.1090/S0002-9947-09-04663-7
Published electronically: February 10, 2009
MathSciNet review: 2491899
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Abstract: A crucial ingredient in the recent discovery by Ablowitz, Halburd, Herbst and Korhonen (2000, 2007) that a connection exists between discrete Painlevé equations and (finite order) Nevanlinna theory is an estimate of the integrated average of $ \log^+\vert f(z+1)/f(z)\vert$ on $ \vert z\vert=r$. We obtained essentially the same estimate in our previous paper (2008) independent of Halburd et al. (2006). We continue our study in this paper by establishing complete asymptotic relations amongst the logarithmic differences, difference quotients and logarithmic derivatives for finite order meromorphic functions. In addition to the potential applications of our new estimates in integrable systems, they are also of independent interest. In particular, our findings show that there are marked differences between the growth of meromorphic functions with Nevanlinna order less than and greater than one. We have established a ``difference'' analogue of the classical Wiman-Valiron type estimates for meromorphic functions with order less than one, which allow us to prove that all entire solutions of linear difference equations (with polynomial coefficients) of order less than one must have positive rational order of growth. We have also established that any entire solution to a first order algebraic difference equation (with polynomial coefficients) must have a positive order of growth, which is a ``difference'' analogue of a classical result of Pólya.


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Additional Information

Yik-Man Chiang
Affiliation: Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, People’s Republic of China
Email: machiang@ust.hk

Shao-Ji Feng
Affiliation: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, People’s Republic of China
Email: fsj@amss.ac.cn

DOI: https://doi.org/10.1090/S0002-9947-09-04663-7
Keywords: Difference operators, Poisson-Jensen formula, Wiman-Valiron theory, integrable difference equations
Received by editor(s): January 5, 2007
Received by editor(s) in revised form: July 25, 2007
Published electronically: February 10, 2009
Additional Notes: This research was supported in part by the Research Grants Council of the Hong Kong Special Administrative Region, China (HKUST6135/01P and 600806). The second author was also partially supported by the National Natural Science Foundation of China (Grant No. 10501044) and by the HKUST PDF Matching Fund.
The second author thanks the Hong Kong University of Science and Technology for its hospitality during his visit from August 2004 to March 2005
Many main results in this paper were presented in the “Computational Methods and Function Theory” meeting held in Joensuu, Finland, June 13–17, 2005.
Dedicated: Dedicated to the eightieth birthday of Walter K. Hayman
Article copyright: © Copyright 2009 American Mathematical Society

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