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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
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