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The fixed points and hyper-order of solutions of second order linear differential equations with meromorphic coefficients. (Chinese. English summary) Zbl 1064.30025
Let \(z_1,z_2,\dots (r_i=| z_i| ,\;0<r_1\leq r_2\leq \cdots)\) be the fixed points of a transcendental meromorphic function \(f\). Define \[ \tau(f)= \inf\biggl\{ \tau>0, \sum^\infty_{i=1} r_i^{-\tau}<\infty\biggr\} \] and the index of fixed points of \(f\) as \[ \tau_2(f)=\overline{\lim_{r\to\infty}}\frac{\log\log\overline N(r,\frac1{f-z})}{\log r}. \] The authors study the index of fixed points for a nonzero meromorphic function which is a solution of a complex second order differential equation. For example, the authors show that suppose \(A(z)\) is a transcendental meromorphic function with \(\delta(\infty,A)>0\) then any non-zero solution \(f(z)\) of the second order complex differential equation \(f''+A(z)f=0\) and \(f', f''\) have infinite fixed points and their indexes satisfy \(\tau(f)= \tau(f')= \tau(f'')=\infty\) and \(\tau_2(f)=\tau_2(f')=\tau_2(f'')=\sigma_2(f)\), where \(\sigma_2(f)\) is the hyperorder of \(f\). The similar results are also investigated for some other type of second order complex differential equations.

MSC:
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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