## Abstract and Applied Analysis

### On Growth of Meromorphic Solutions of Complex Functional Difference Equations

#### Abstract

The main purpose of this paper is to investigate the growth order of the meromorphic solutions of complex functional difference equation of the form $({\sum }_{\lambda \in I}‍{\alpha }_{\lambda }(z)({\prod }_{\nu =1}^{n}‍f(z+{c}_{\nu }{)}^{{l}_{\lambda ,\nu }}))/({\sum }_{\mu \in J}‍{\beta }_{\mu }(z)({\prod }_{\nu =1}^{n}‍f(z+$ ${c}_{\nu }{)}^{{m}_{\mu ,\nu }}))=$ $Q(z,f(p(z)))$, where $I=$ $\{\lambda =({l}_{\lambda ,1},{l}_{\lambda ,2},\dots ,{l}_{\lambda ,n})\mid {l}_{\lambda ,\nu }\in \Bbb N\bigcup ‍\{0\}\text{,\hspace\{0.17em\}\hspace\{0.17em\}}\nu =$ $1,2,\dots ,n\}$ and $J=\{\mu =({m}_{\mu ,1},{m}_{\mu ,2},\dots ,{m}_{\mu ,n})\mid {m}_{\mu ,\nu }\in \Bbb N\bigcup ‍\{0\}\text{,\hspace\{0.17em\}\hspace\{0.17em\}}\nu =1,2,\dots ,n\}$ are two finite index sets, ${c}_{\nu }\text{\hspace\{0.17em\}}(\nu =1,2,\dots ,n)$ are distinct complex numbers, ${\alpha }_{\lambda }(z)(\lambda \in I)$ and ${\beta }_{\mu }(z)(\mu \in J)$ are small functions relative to $f(z),$ and $Q(z,u)$ is a rational function in $u$ with coefficients which are small functions of $f(z)$, $p(z)={p}_{k}{z}^{k}+{p}_{k-1}{z}^{k-1}+\cdots +{p}_{0}\in \Bbb C[z]$ of degree $k\ge 1$. We also give some examples to show that our results are sharp.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 828746, 6 pages.

Dates
First available in Project Euclid: 6 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412607234

Digital Object Identifier
doi:10.1155/2014/828746

Mathematical Reviews number (MathSciNet)
MR3176772

Zentralblatt MATH identifier
07023151

#### Citation

Li, Jing; Zhang, Jianjun; Liao, Liangwen. On Growth of Meromorphic Solutions of Complex Functional Difference Equations. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 828746, 6 pages. doi:10.1155/2014/828746. https://projecteuclid.org/euclid.aaa/1412607234

#### References

• W. Cherry and Z. Ye, Nevanlinna's Theory of Value Distribution, Springer Monographs in Mathematics, Springer, Berlin, Germany, 2001.
• W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, UK, 1964.
• Y. Z. He and X. Z. Xiao, Algebroid Functions and Ordinary Differential Equations, Beijing, China, 1988.
• I. Laine, Nevanlinna Theory and Complex Differential Equations, vol. 15 of de Gruyter Studies in Mathematics, Walter de Gruyter, Berlin, Germany, 1993.
• I. Laine, J. Rieppo, and H. Silvennoinen, “Remarks on complex difference equations,” Computational Methods and Function Theory, vol. 5, no. 1, pp. 77–88, 2005.
• J. Rieppo, “On a class of complex functional equations,” Annales Academiæ Scientiarum Fennicæ, vol. 32, no. 1, pp. 151–170, 2007.
• X.-M. Zheng, Z.-X. Chen, and J. Tu, “Growth of meromorphic solutions of some difference equations,” Applicable Analysis and Discrete Mathematics, vol. 4, no. 2, pp. 309–321, 2010.
• A. Z. Mokhon'ko, “The Nevanlinna characteristics of certain meromorphic functions,” Teorija Funkciĭ, Funkcional'nyĭ Analiz i ih Priloženija, vol. 14, pp. 83–87, 1971 (Russian).
• A. A. Mokhon'ko and V. D. Mokhon'ko, “Estimates of the Nevanlinna characteristics of certain classes of meromorphic functions, and their applications to differential equations,” Akademija Nauk SSSR, vol. 15, pp. 1305–1322, 1974.
• R. Goldstein, “Some results on factorisation of meromorphic functions,” Journal of the London Mathematical Society, vol. 4, pp. 357–364, 1971.
• R. Goldstein, “On meromorphic solutions of certain functional equations,” Aequationes Mathematicae, vol. 18, no. 1-2, pp. 112–157, 1978.
• G. G. Gundersen, J. Heittokangas, I. Laine, J. Rieppo, and D. Yang, “Meromorphic solutions of generalized Schröder equations,” Aequationes Mathematicae, vol. 63, no. 1-2, pp. 110–135, 2002. \endinput