Abstract
This paper gives a precise asymptotic relation between higher-order logarithmic difference and logarithmic derivatives for meromorphic functions with order strictly less then one. This allows us to formulate a useful Wiman–Valiron type estimate for logarithmic difference of meromorphic functions of small order. We then apply this estimate to prove a classical analogue of Valiron about entire solutions to linear differential equations with polynomial coefficients for linear difference equations.
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Acknowledgments
The authors would like to thank their colleague Dr. T. K. Lam for pointing out an inaccuracy on the exceptional set, now labelled as \(E^{(\eta )}\), in an earlier version of this paper. The first author would also like to acknowledge hospitality that he received during his visits to the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, in the preparation of this paper. Finally, both authors thank the referees for their constructive and detailed comments that helped to improve the readability of this paper.
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Communicated by Edward B. Saff.
Dedicated to the memory of J. Milne Anderson.
The first and second authors were partially supported by the Research Grants Council of the Hong Kong Special Administrative Region, China (GRF600609 and GRF601111) and the HKUST PDF Matching Fund. The second author was also partially supported by the National Natural Science Foundation of China (Grant No. 11271352).
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Chiang, YM., Feng, S. On the Growth of Logarithmic Difference of Meromorphic Functions and a Wiman–Valiron Estimate. Constr Approx 44, 313–326 (2016). https://doi.org/10.1007/s00365-016-9324-8
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DOI: https://doi.org/10.1007/s00365-016-9324-8