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On the iterated order and the fixed points of entire solutions of some complex linear differential equations. (English) Zbl 1120.34070

The iterated $p$-order ${\sigma }_{p}\left(f\right)$ of an entire function $f$ is defined as

${\sigma }_{p}\left(f\right)=\underset{r\to +\infty }{lim sup}\frac{{log}_{p}T\left(r,f\right)}{logr}=\underset{r\to +\infty }{lim sup}\frac{{log}_{p+1}M\left(r,f\right)}{logr}·$

The finiteness degree of the order of an entire function $f$ is defined as $i\left(f\right)=0$ if $f$ is a polynomial or $i\left(f\right)=min\left\{j\in ℕ:{\sigma }_{j}\left(f\right)<\infty \right\}$ if $f$ is a transcendental entire function and ${\sigma }_{j}\left(f\right)<\infty$ for some $j\in ℕ$, otherwise $i\left(f\right)=\infty ·$ The iterated convergence exponent of the sequence of distinct zeros of an entire function $f$ is defined as

${\overline{\lambda }}_{p}\left(f\right)=lim sup\frac{{log}_{p}\overline{N}\left(r,\frac{1}{f}\right)}{logr},$

where $\overline{N}\left(r,\frac{1}{f}\right)$ is the counting function of the distinct zeros of $f\left(z\right)$ in ${{\Delta }}_{r}=\left\{z:|z|<1\right\}·$

In this paper, the author studies the iterated order and iterated convergence exponent of the sequence of distinct zeros of the entire solutions of the following linear differential equations

${f}^{\left(k\right)}+{A}_{k-1}\left(z\right){f}^{\left(k\right)}+\cdots +{A}_{1}\left(z\right){f}^{\text{'}}+{A}_{0}\left(z\right)f=0,\phantom{\rule{2.em}{0ex}}\left(1\right)$
${f}^{\left(k\right)}+{A}_{k-1}\left(z\right){f}^{\left(k\right)}+\cdots +{A}_{1}\left(z\right){f}^{\text{'}}+{A}_{0}\left(z\right)f=F\left(z\right),\phantom{\rule{2.em}{0ex}}\left(2\right)$

where ${A}_{0}\left(z\right),\cdots ,{A}_{k-1}\left(z\right)$ and $F\left(z\right)¬\equiv 0$ are entire functions. The author proves that if ${A}_{0}\left(z\right),\cdots ,{A}_{k-1}\left(z\right)$ are entire functions with ${\sigma }_{p}\left({A}_{j}\right)\le {\sigma }_{p}\left({A}_{s}\right)<\infty$ for some $s\left(0\le s\le k-1\right)$, then the equation (1) has at least one solution $f$ with $i\left(f\right)=p+1$ and ${\sigma }_{p+1}\left(f\right)={\sigma }_{p}\left({A}_{s}\right)·$ Furthermore, if $F\left(z\right)¬\equiv 0$ is an entire function with either $i\left(F\right)=q or $i\left(F\right)=q=p+1$ and ${\sigma }_{p+1}\left(F\right)<{\sigma }_{p}\left({A}_{s}\right)<+\infty ,$ then there exists at least one solution $g$ of the corresponding homogeneous equation (1) of the equation (2) such that all solutions $f$ in the solution subspace $\left\{cg+{f}_{0},c\in ℂ\right\}$ satisfying $i\left(f\right)=p+1$ and ${\sigma }_{p+1}\left(f\right)={\overline{\lambda }}_{p+1}\left(f\right)={\sigma }_{p}\left({A}_{s}\right),$ with at most one exception, where ${f}_{0}$ is a solution of (2).

##### MSC:
 34M10 Oscillation, growth of solutions (ODE in the complex domain) 30D35 Distribution of values (one complex variable); Nevanlinna theory
entire function