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