Abstract
Let f be a meromorphic function in the complex plane, \(p_j\) polynomials for (\(j=0, 1, 2, \ldots , n\)) and R(z, f) an irreducible rational function in f with small meromorphic functions relative to f as coefficients. Let n be a positive integer and I, J two index sets in \(\mathbb {Z}^n\). In this paper, we systematically study the growth order of meromorphic solutions to the functional equations of the form:
We not only obtain estimates of the growth order of its meromorphic solutions in all possible cases, but also give examples to show these estimates are the best possible.
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Chern, P.T.Y.: On meromorphic functions with finite logarithmic order. Trans. Am. Math. Soc. 358(2), 473–489 (2006)
Cherry, W., Ye, Z.: Nevanlinna’s Theory of Value Distribution. Springer, Berlin (2001)
Gundersen, G.: Finite order solutions of second order linear differential equations. Trans. Am. Math. Soc. 305, 415–429 (1988)
Gundersen, G., Heittokangas, J., Laine, I., Rieppo, J., Yang, D.G.: Meromorphic solutions of generalized Schr\(\ddot{\rm o}\)der equations. Aequationes Math. 63, 110–135 (2002)
Goldstein, R.: Some results on factorization of meromorphic functions. J. Lond. Math. Soc. 4(2), 357–364 (1971)
Goldstein, R.: On meromorphic solutions of certain functional equations. Aequationes Math. 18, 112–157 (1978)
Hayman, W.K.: Angular value distribution of power series with gaps. Proc. Lond. Math. Soc. 72, 590–642 (1972)
Huang, Z., Zhang, R.: Some properties on complex functional difference equations. Abstr. Appl. Anal. 2014, 1–10 (2014)
Laine, I.: Nevanlinna Theory and Complex Differential Equation. Walter de Gruyter, Berlin-New York (1993)
Laine, I., Rieppo, J., Silvennoinen, H.: Remarks on complex difference equations. Comput. Methods Funct. Theory 5(1), 77–88 (2005)
Li, J., Zhang, J., Liao, L.: On growth of meromorphic solutions of complex functional difference equations. Abstr. Appl. Anal. 2014, 1–6 (2014)
Peng, C.-W., Chen, Z.-X.: On properties of meromorphic solutions for difference Painlevé equations. Adv. Differ. Equ. 2015, 123 (2015)
Rieppo, J.: On a class of complex functional equations. Ann. Acad. Sci. Fenn. 32(1), 151–170 (2007)
Wang, J.: Growth and poles of meromorphic solutions of some complex difference equations. J. Math. Anal. Appl. 379, 367–377 (2011)
Wen, Z.-T.: Finite logarithmic order solutions of linear \(q\)-difference equations. Bull. Korean Math. Soc. 51(1), 83–98 (2014)
Yang, C.C., Yi, H.X.: Uniqueness Theory of Meromorphic Functions. Kluwer Academic Publishers Group, Dordrecht (2003)
Zheng, X.M., Chen, Z.X.: On properties of q-difference equations. Acta Math. Sci. 32B(2), 724–734 (2012)
Zheng, X.M., Tu, J.: Existence and growth of meromorphic solutions of some nonlinear q-difference equations. Adv. Differ. Equ. 2013, 33 (2013)
Acknowledgments
The authors are grateful to the referee whose suggestions and comments have improved the clarity of the paper. The first two authors also would like to express their thanks for the hospitality of Mathematics Seminar, University of Kiel in Germany, when they visited there.
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Communicated by Ilpo Laine.
The first author was supported by the Research Project Supported by Shanxi Scholarship Council of China (No. 2013-045) and the Foundation Research Project of Shanxi Province (No. 2014021009-3). The second author was supported by the NNSF of China (No. 10671109 and No. 11001057).
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Ding, J., Wang, J. & Ye, Z. Growth of Meromorphic Solutions to Some Complex Functional Equations. Comput. Methods Funct. Theory 16, 489–502 (2016). https://doi.org/10.1007/s40315-016-0157-z
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DOI: https://doi.org/10.1007/s40315-016-0157-z