Results on difference analogues of Valiron-Mohon’ko theorem. (English) Zbl 1283.30067

Zhang, Ran Ran; Huang, Zhi Bo

Abstr. Appl. Anal. 2013, Article ID 273040, 6 p. (2013).

Authors’ abstract: The classical Valiron-Mohon’ko theorem has many applications in the study of complex equations. In this paper, we investigate rational functions and their shifts. We get some results on their characteristic functions. These results may be viewed as difference analogues of the Valiron-Mohon’ko theorem.

Reviewer: Dov Aharonov (Haifa)

MSC:

30D35

Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

References:

[1]	Hayman, W. K., Meromorphic Functions (1964), Oxford, UK: Clarendon Press, Oxford, UK
[2]	Yang, L., Value Distribution Theory and New Research (1982), Beijing, China: Science Press, Beijing, China
[3]	Laine, I., Nevanlinna theory and complex differential equations (1993), Berlin, Germany: Walter de Gruyter, Berlin, Germany
[4]	Bergweiler, W.; Langley, J. K., Zeros of differences of meromorphic functions, Mathematical Proceedings of the Cambridge Philosophical Society, 142, 1, 133-147 (2007) · Zbl 1114.30028
[5]	Chen, Z. X.; Huang, Z. B.; Zheng, X. M., On properties of difference polynomials, Acta Mathematica Scientia B, 31, 2, 627-633 (2011) · Zbl 1240.30151
[6]	Chen, Z. X.; Shon, K. H., Estimates for the zeros of differences of meromorphic functions, Science in China A, 52, 11, 2447-2458 (2009) · Zbl 1181.30016
[7]	Chiang, Y.-M.; Feng, S.-J., On the nevanlinna characteristic of \(f(z + \eta)\) and difference equations in the complex plane, Ramanujan Journal, 16, 1, 105-129 (2008) · Zbl 1152.30024
[8]	Halburd, R. G.; Korhonen, R. J., Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, Journal of Mathematical Analysis and Applications, 314, 2, 477-487 (2006) · Zbl 1085.30026
[9]	Laine, I.; Yang, C.-C., Clunie theorems for difference and \(q\)-difference polynomials, Journal of the London Mathematical Society, 76, 3, 556-566 (2007) · Zbl 1132.30013
[10]	Laine, I.; Yang, C.-C., Value distribution of difference polynomials, Japan Academy A, 83, 8, 148-151 (2007) · Zbl 1153.30030
[11]	Zhang, R. R.; Chen, Z. X., Value distribution of difference polynomials of meromorphic functions, Science China Mathematics, 42, 11, 1115-1130 (2012) · Zbl 1488.30204
[12]	Zhang, R. R.; Chen, Z. X., Value distribution of meromorphic functions and their differences, Turkish Journal of Mathematics, 36, 3, 395-406 (2012) · Zbl 1276.30050
[13]	Korhonen, R.; Ronkainen, O., Order reduction method for linear difference equations, Proceedings of the American Mathematical Society, 139, 9, 3219-3229 (2011) · Zbl 1232.39003
[14]	Yang, C.-C.; Laine, I., On analogies between nonlinear difference and differential equations, Japan Academy A, 86, 1, 10-14 (2010) · Zbl 1207.34118
[15]	Halburd, R. G.; Korhonen, R. J., Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations, Journal of Physics A, 40, 6, R1-R38 (2007) · Zbl 1115.39024
[16]	Gol’dberg, A. A.; Ostrovskii, I. V., Distribution of Values of Meromorphic Functions (1970), Moscow, Russia: Nauka, Moscow, Russia