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On the growth and fixed points of solutions of second order differential equations with meromorphic coefficients. (English) Zbl 1100.34067
Two 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)].

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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