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On the meromorphic solutions of linear differential equations on the complex plane. (English) Zbl 1194.34161

The authors study the growth of solutions of the linear differential equation \[ f^{(k)}+ A_{k-1} f^{(k-1)}+\cdots + A_0f= 0,\quad k\geq 2\tag{1} \]
where \(A_0,\dots, A_{k-1}\) are meromorphic functions on the complex plane and \(A_0\not\equiv 0\). The iterated \(n\)-th-order \(\sigma_n(f)\) of a meromorphic function \(f\) is defined by
\[ \sigma_n(f):= \limsup_{r\to\infty} {\log^+_nT(r, f)\over\log r}\quad (n\in\mathbb{N}). \]
The growth index of the iterated order of a meromorphic function \(f\) is defined by
\[ i(f)= \begin{cases} 0 &\text{if }f\text{ is rational},\\ \min\{n\in\mathbb{N}:\sigma_n(f)< \infty\} &\text{if }\text{ is transcendental and }\sigma_n(f)<\infty\text{ for some }n,\\ \infty\;&\text{if }\sigma_n(f)= \infty\text{ for all }n\in\mathbb{N}.\end{cases} \]
Let \(i_\lambda(f)\) denote the growth index of the iterated convergence exponent of the sequence of zero points of a meromorhic function \(f\), \(\lambda_n(f)\) the iterated \(n\)-th-order convergence exponent of the sequence of zero points of a meromorphic function \(f\).
The authors prove the following theorem:
Theorem 2.1. Let \(A_j\), \((j= 0,\dots,k- 1)\) be meromorphic functions, and let \(i(A_0)= p\) \((0< p<\infty)\). Assume that either \(i_\lambda({1\over A_0})< p\) or \(\lambda_p({1\over A_0})< \sigma_p(A_0)\), and that either \(\max\{i(A_j): j= 1,\dots, k-1\}< p\) or
\[ \max\{\sigma_p(A_j): j= 1,\dots, k-1\}\leq\sigma_p(A_0)= \sigma \qquad (0<\sigma<\infty), \]
\[ \max\{\tau_p(A_j): \sigma_p(A_j)= \sigma_p(A_0)\}<\tau_p(A_0)= \tau\qquad (0< \tau< \infty). \]
Then every meromorphic solution \(f\not\equiv 0\) whose poles are uniformly bounded multiplicities of (1) satisfies \(i(f)= p+1\) and \(\sigma_{p+1}(f)=\sigma_p(A_0)\).
The authors also investigate the case of nonhomogeneous differential equations.

MSC:

34M03 Linear ordinary differential equations and systems in the complex domain
34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
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[1] Belaïdi, B., On the iterated order and the fixed points of entire solutions of some complex linear differential equations, Electron. J. qual. theory differ. equ., 9, 1-11, (2006) · Zbl 1120.34070
[2] Belaïdi, B., Oscillation of fixed points of solutions of some linear differential equations, Acta math. univ. Comenian. (N.S.), 77, 2, 263-269, (2008) · Zbl 1174.34528
[3] Belaïdi, B.; Hamani, K., The rate of growth of solutions of linear differential equations with meromorphic coefficients, Southeast Asian bull. math., 30, 3, 405-414, (2006) · Zbl 1121.34089
[4] Cao, T.-B., Complex oscillation of entire solutions of higher-order linear differential equations, Electron. J. differential equations, 2006, 81, 1-8, (2006)
[5] Cao, T.-B.; Chen, Z.-X.; Zheng, X.-M.; Tu, J., On the iterated order of meromorphic solutions of higher order linear differential equations, Ann. differential equations, 21, 2, 111-122, (2005)
[6] Cao, T.-B.; Yi, H.-X., On the complex oscillation of higher order linear differential equations with meromorphic coefficients, J. syst. sci. complex., 20, 135-148, (2007) · Zbl 1134.34056
[7] Chen, Z.-X., The fixed points and hyper order of solutions of second order complex differential equations, Acta math. sci. ser. A chin. ed., 20, 3, 425-432, (2000), (in Chinese) · Zbl 0980.30022
[8] Chen, Z.-X., On the rate of growth of meromorphic solutions of higher order linear differential equations, Acta math. sinica (chin. ser.), 42, 3, 552-558, (1999), (in Chinese)
[9] Chen, Z.-X., The growth of solutions of second order linear differential equations with meromorphic coefficients, Kodai math. J., 22, 208-221, (1999) · Zbl 0940.34069
[10] Chen, Z.-X.; Gao, S.-A., The complex oscillation theory of certain non-homogeneous linear differential equations with transcendental entire coefficients, J. math. anal. appl., 179, 403-416, (1993) · Zbl 0804.30028
[11] Chen, Z.-X.; Yang, C.-C., Quantitative estimations on the zeros and growths of entire solutions of linear differential equations, Complex var. elliptic equ., 42, 119-133, (2000) · Zbl 1036.34101
[12] Chiang, Y.-M.; Hayman, W.K., Estimates on the growth of meromorphic solutions of linear differential equations, Comment. math. helv., 79, 451-470, (2004) · Zbl 1057.34110
[13] Frank, G.; Hellerstein, S., On the meromorphic solutions of non-homogeneous linear differential equations with polynomial coefficients, Proc. London math. soc., 53, 3, 407-428, (1986) · Zbl 0635.34005
[14] Gundersen, G., Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London math. soc., 37, 2, 88-104, (1988) · Zbl 0638.30030
[15] Hayman, W., Meromorphic functions, (1964), Clarendon Press Oxford · Zbl 0115.06203
[16] Hayman, W., The local growth of power series: A survey of wiman – valiron method, Canad. math. bull., 17, 3, 317-358, (1974) · Zbl 0314.30021
[17] He, Y.-Z.; Xiao, X.-Z., Algebroid functions and ordinary differential equations, (1988), Science Press Beijing, (in Chinese)
[18] Kinnunen, L., Linear differential equations with solutions of finite iterated order, Southeast Asian bull. math., 22, 4, 385-405, (1998) · Zbl 0934.34076
[19] Laine, I., Nevanlinna theory and complex differential equations, (1993), W. de Gruyter Berlin
[20] Laine, I.; Yang, R.-H., Finite order solutions of complex linear differential equations, Electron. J. differential equations, 2004, 5, 1-8, (2004)
[21] Liu, M.-S.; Zhang, X.-M., Fixed points of meromorphic solutions of higher order linear differential equations, Ann. acad. sci. fenn. math., 31, 191-211, (2006) · Zbl 1094.30036
[22] Sato, D., On the rate of growth of entire functions of fast growth, Bull. amer. math. soc. (N.S.), 69, 411-414, (1963) · Zbl 0109.30104
[23] Tu, J.; Yi, C.-F., On the growth of solutions of a class of higher order linear differential equations with coefficients having the same order, J. math. anal. appl., 340, 487-497, (2008) · Zbl 1141.34054
[24] Wang, J.; Lü, W.-R., The fixed points and hyper-order of solutions of second order linear differential equations with meromorphic coefficients, Acta math. sinica (chin. ser.), 27, 1, 72-80, (2004), (in Chinese) · Zbl 1064.30025
[25] Wang, J.; Yi, H.-X., Fixed points and hyper order of differential polynomials generated by solutions of differential equation, Complex var. elliptic equ., 48, 1, 83-94, (2003) · Zbl 1071.30029
[26] Xu, J.-F.; Zhang, Z.-L., Growth order of meromorphic solutions of higher-order linear differential equations, Kyungpook math. J., 48, 123-132, (2008) · Zbl 1158.34054
[27] Yang, L., Value distribution theory, (1982), Science Press Beijing
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