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.


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