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Value distribution of meromorphic solutions and their derivatives of complex differential equations. (English) Zbl 1291.34148
Summary: We deal with the relationship between the small functions and the derivatives of solutions of higher-order linear differential equations $f^{(k)}+A_{k-1}f^{(k-1)}+\cdots +A_0 f = 0,\quad k\geq 2,$ where $$A_j(z)(j=0,1,\dots,k-1)$$ are meromorphic functions. The theorems of this paper improve the previous results given by El Farissi, Belaïdi, Wang, Lu, Liu, and Zhang.
##### MSC:
 34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain 34M03 Linear ordinary differential equations and systems in the complex domain 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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##### References:
 [1] W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, UK, 1964. · Zbl 0115.06203 [2] R. Nevanlinna, Eindeutige Analytische Funktionen, Springer, Berlin, Germany, Zweite Auflage, Reprint, Die Grundlehren der mathematischen Wissenschaften, Band 46, 1974. · Zbl 0278.30002 · eudml:203720 [3] Z. X. Chen, “The fixed points and hyper order of solutions of second order complex differential equations,” Acta Mathematica Scientia, vol. 20, no. 3, pp. 425-432, 2000 (Chinese). · Zbl 0980.30022 [4] M. S. Liu and X. M. Zhang, “Fixed points of meromorphic solutions of higher order linear differential equations,” Annales Academiæ Scientiarum Fennicæ, vol. 31, no. 1, pp. 191-211, 2006. · Zbl 1094.30036 · eudml:126327 [5] J. Wang and H. X. Yi, “Fixed points and hyper order of differential polynomials generated by solutions of differential equation,” Complex Variables, vol. 48, no. 1, pp. 83-94, 2003. · Zbl 1071.30029 · doi:10.1080/0278107021000037048 [6] H. X. Yi and C. C. Yang, The Uniqueness Theory of Meromorphic Functions, Science Press, Beijing, China, 1995 Chinese. [7] Q. T. Zhang and C. C. Yang, The Fixed Points and Resolution Theory of Meromorphic Functions, Beijing University Press, Beijing, China, 1988 Chinese. [8] J. Wang and W. R. Lü, “The fixed points and hyper-order of solutions of second order linear differential equations with meromorphic coefficients,” Acta Mathematicae Applicatae Sinica, vol. 27, no. 1, pp. 72-80, 2004. · Zbl 1064.30025 [9] B. Belaïdi and A. El Farissi, “Oscillation theory to some complex linear large differential equations,” Annals of Differential Equations, vol. 25, no. 1, pp. 1-7, 2009. · Zbl 1199.34469 [10] H. Y. Xu, J. Tu, and X. M. Zheng, “On the hyper exponent of convergence of zeros of f(j)-\varphi of higher order linear differential equations,” Advances in Difference Equations, vol. 2012, article 114, 16 pages, 2012. · Zbl 1350.34069 · doi:10.1186/1687-1847-2012-114 [11] B. Belaïdi, “Some precise estimates of the hyper order of solutions of some complex linear differential equations,” Journal of Inequalities in Pure and Applied Mathematics, vol. 8, no. 4, article 107, 14 pages, 2007. · Zbl 1140.34445 · emis:journals/JIPAM/article929.html?sid=929 · eudml:129066 [12] J. Tu and T. Long, “Oscillation of complex high order linear differential equations with coefficients of finite iterated order,” Electronic Journal of Qualitative Theory of Differential Equations, no. 66, pp. 1-13, 2009. · Zbl 1188.30043 · emis:journals/EJQTDE/2009/200966.pdf · eudml:230917
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