an:02121920
Zbl 1064.30025
Wang, Jun; L??, Weiran
The fixed points and hyper-order of solutions of second order linear differential equations with meromorphic coefficients
ZH
Acta Math. Appl. Sin. 27, No. 1, 72-80 (2004).
00104469
2004
j
30D35 30D05
second order differential equation; meromorphic function; fixed point; hyperorder
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.
Hasi Wulan (Shantou)