## The fixed points and hyper order of solutions of second order complex differential equations.(Chinese. English summary)Zbl 0980.30022

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 entire function. Its index for fixed points is defined as $\tau(f)= \inf\left\{ \tau\left |\sum^\infty_{i=1} \right.{1\over r_i^\tau} <\infty\right\}.$ In this paper, the author studies the index of fixed points for a transcendental entire function which is a solution of a complex second order differential equation. For example, the author shows that suppose $$P(z)$$ is a polynomial of degree $$n\geq 1$$ then any non-zero solution $$f(z)$$ of the second order complex differential equation $$f''+P(z)f=0$$ has infinite fixed points and its index of fixed points $$\tau(f)= (n+2)/2$$. The index of fixed points of a solution $$f$$ of the second order complex differential equation $$f''+A(z)f=0$$, where $$A(z)$$ is a transcendental entire function, has been investigated. Furthermore, for a second order complex differential equation $$f''+P(z) f=Q(z)$$, where $$P(z)$$ and $$Q(z)$$ are polynomials, and a second order complex differential equation $$f''+A(z) f=F(z)$$, the author also studies the index of fixed points for a solution $$f$$.

### MSC:

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