## On the meromorphic solutions of linear differential equations on the complex plane.(English)Zbl 1194.34161

The authors study the growth of solutions of the linear differential equation $f^{(k)}+ A_{k-1} f^{(k-1)}+\cdots + A_0f= 0,\quad k\geq 2\tag{1}$
where $$A_0,\dots, A_{k-1}$$ are meromorphic functions on the complex plane and $$A_0\not\equiv 0$$. The iterated $$n$$-th-order $$\sigma_n(f)$$ of a meromorphic function $$f$$ is defined by
$\sigma_n(f):= \limsup_{r\to\infty} {\log^+_nT(r, f)\over\log r}\quad (n\in\mathbb{N}).$
The growth index of the iterated order of a meromorphic function $$f$$ is defined by
$i(f)= \begin{cases} 0 &\text{if }f\text{ is rational},\\ \min\{n\in\mathbb{N}:\sigma_n(f)< \infty\} &\text{if }\text{ is transcendental and }\sigma_n(f)<\infty\text{ for some }n,\\ \infty\;&\text{if }\sigma_n(f)= \infty\text{ for all }n\in\mathbb{N}.\end{cases}$
Let $$i_\lambda(f)$$ denote the growth index of the iterated convergence exponent of the sequence of zero points of a meromorhic function $$f$$, $$\lambda_n(f)$$ the iterated $$n$$-th-order convergence exponent of the sequence of zero points of a meromorphic function $$f$$.
The authors prove the following theorem:
Theorem 2.1. Let $$A_j$$, $$(j= 0,\dots,k- 1)$$ be meromorphic functions, and let $$i(A_0)= p$$ $$(0< p<\infty)$$. Assume that either $$i_\lambda({1\over A_0})< p$$ or $$\lambda_p({1\over A_0})< \sigma_p(A_0)$$, and that either $$\max\{i(A_j): j= 1,\dots, k-1\}< p$$ or
$\max\{\sigma_p(A_j): j= 1,\dots, k-1\}\leq\sigma_p(A_0)= \sigma \qquad (0<\sigma<\infty),$
$\max\{\tau_p(A_j): \sigma_p(A_j)= \sigma_p(A_0)\}<\tau_p(A_0)= \tau\qquad (0< \tau< \infty).$
Then every meromorphic solution $$f\not\equiv 0$$ whose poles are uniformly bounded multiplicities of (1) satisfies $$i(f)= p+1$$ and $$\sigma_{p+1}(f)=\sigma_p(A_0)$$.
The authors also investigate the case of nonhomogeneous differential equations.

### MSC:

 34M03 Linear ordinary differential equations and systems in the complex domain 34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
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