an:05023171
Zbl 1105.34059
Cao, Ting-Bin; Yi, Hong-Xun
The growth of solutions of linear differential equations with coefficients of iterated order in the unit disc
EN
J. Math. Anal. Appl. 319, No. 1, 278-294 (2006).
00125169
2006
j
34M10 30D30
meromorphic function in the unit disc; linear differential equation
The authors study the meromorphic solutions of a linear differential equation in the unit disc. They first give a definition of iterated order of a meromorphic function in the unit disc. They define the iterated \(n\)th-order \(\sigma_n(f)\) of the meromorphic function \(f\) as
\[
\sigma_n(f)=\limsup_{r\to\infty}\frac{\log^{[n]}T(r,f)}{\log{\frac1{1-r}}},\quad n\in{\mathbb{N}},
\]
and the growth index of the iterated order of a meromorphic function \(f(z)\) in the unit disc as
\[
i(f)= \begin{cases} 0 &\text{if \(f\) is nonadmissible}; \\ \min\{n: \sigma_n(f)<\infty\} &\text{if \(f\) is admissible};\\ \infty &\text{if } \sigma_n(f)=\infty \text{ for all } n\in \mathbb{N}. \\ \end{cases}
\]
Then, they investigate the iterated order of analytic solutions of the following linear differential equation with analytic coefficients in the unit disc
\[
f^{(k)}+a_{k-1}(z)f^{(k-1)}+\cdots +a_0(z)f=0.
\]
For example, they prove that if \(f\) is an analytic solution of this equation, then \(i(f)\leq \max\{i(a_j)\), \(j=1,\cdots, k-1\}.\)
Liangwen Liao (Nanjing)