On the growth and fixed points of solutions of second order differential equations with meromorphic coefficients.

*(English)*Zbl 1100.34067Two theorems are proved in this paper. First, all nontrivial meromorphic solutions of
\[
f''+A_1(z)e^{az}f'+A_0(z)e^{bz}f=0\tag{*}
\]
are of infinite order, provided \(A_0\), \(A_1\) are non-vanishing meromorphic functions of order \(<1\) and that the non-vanishing complex constants \(a,b\) satisfy either \(\arg a\neq\arg b\), or \(a=cb\) with \(0<c<1\). Secondly, it is proved that such solutions \(f\) always have infinitely many distinct fixed-points. In fact, the exponent of convergence of this sequence of fixed-points is infinite. The same assertion also holds for \(f'\), \(f''\) and for \(d_2f''+d_1f'+d_0f\) with complex constants \(d_0\), \(d_1\), \(d_2\) not all vanishing simultaneously. The basic device in the proofs is a careful analysis of derivatives of \(f\) on suitably selected sequence of radii. As for some other closely related recent papers dealing with fixed-points of solutions of linear differential equations, we refer to Z. Chen [Acta Math. Sci. (Chin. Ed.) 20, No. 3, 425–432 (2000; Zbl 0980.30022)], I. Laine and J. Rieppo [Complex Variables, Theory Appl. 49, No. 12, 897–911 (2004; Zbl 1080.34076)] and J. Wang, H. Yi and H. Cai [J. Syst. Sci. Complex. 17, No. 2, 271–280 (2004; Zbl 1090.34071)].

Reviewer: Ilpo Laine (Joensuu)

##### MSC:

34M10 | Oscillation, growth of solutions to ordinary differential equations in the complex domain |

34M05 | Entire and meromorphic solutions to ordinary differential equations in the complex domain |

30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |

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\textit{Z. Chen} and \textit{K. H. Shon}, Acta Math. Sin., Engl. Ser. 21, No. 4, 753--764 (2005; Zbl 1100.34067)

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##### References:

[1] | Yang, L.: Value Distribution Theory and New Research, Science Press, Beijing, 1982 (in Chinese) · Zbl 0633.30029 |

[2] | Hayman, W.: Meromorphic Function, Clarendon Press, Oxford, 1964 |

[3] | Laine, I.: Nevanlinna Theory and Complex Differential Equations, W. de Gruyter, Berlin, 1993 |

[4] | Hille, E.: Ordinary Differential Equations in the Complex Domain, Wiley, New York, 1976 · Zbl 0343.34007 |

[5] | Frei, M.: Uber die subnormalen losungen der differentialgleichung w” + e w’ +(konst.)w = 0. Comment. Math. Helv., 36, 1–8 (1962) · Zbl 0115.06904 · doi:10.1007/BF02566887 |

[6] | Ozawa, M.: On a solution of w” + e w’ + (az + b)w = 0. Kodai Math. J., 3, 295–309 (1980) · Zbl 0463.34028 · doi:10.2996/kmj/1138036197 |

[7] | Gundersen, G.: On the question of whether f” + e f” + B(z)f = 0 can admit a solution f0 of finite order. Proc, R.S.E., 102(A), 9–17 (1986) · Zbl 0598.34002 |

[8] | Langley, J. K.: On complex oscillation and a problem of Ozawa. Kodai Math. J., 9, 430–439 (1986) · Zbl 0609.34041 · doi:10.2996/kmj/1138037272 |

[9] | Amemiya, I., Ozawa, M.: Non–existence of finite order solutions of w” + e w’ + Q(z)w = 0. Hokkaido Math. J., 10, 1–17 (1981) · Zbl 0554.34003 |

[10] | Chen, Z. X.: The growth of solutions of f” +e f’ +Q(z)f = 0 where the order (Q) = 1. Science in China (Series A), 45(3), 290–300 (2002) · Zbl 1054.34139 |

[11] | Chen, Z. X.: The fixed points and hyper order of solutions of second order complex differential equations. Acta Math. Scientia, (in Chinese), 20(3), 425–432 (2000) · Zbl 0980.30022 |

[12] | 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 · doi:10.1112/jlms/s2-37.121.88 |

[13] | Chen, Z. X., Yang, C. C.: Some oscillation theorems for linear differetial equations with meromorphic coefficients. Southeast Asian Bull. of Math., 23, 409–417 (1999) · Zbl 0972.34072 |

[14] | Hayman, W.: The local growth of power series: a survey of the Wiman–Valiron method. Canad. Math. Bull., 17, 317–358 (1974) · Zbl 0314.30021 · doi:10.4153/CMB-1974-064-0 |

[15] | He, Y. Z., Xiao, X. Z.: Algebroid Functions and Ordinary Differential Equations. Science Press, Beijing, 1988 (in Chinese) |

[16] | Valiron, G.: Lectures on the General Theory of Integral Functions, Chelsea, New York, 1949 |

[17] | Chen, Z. X.: Zeros of meromorphic solutions of higher order linear differential equations. Analysis, 14, 425–438 (1994) · Zbl 0815.34003 |

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