$\ell$-Log-momotonic and Laguerre Inequality of P-recursive Sequences

Abstract: We consider $\ell$-log-momotonic sequences and Laguerre inequality of order two for sequences ${a_n}{n \ge 0}$ such that [ \frac{a{n-1}a_{n+1}}{a_n^2} = 1 + \sum_{i=1}^m \frac{r_i(\log n)}{n^{\alpha_i}} + o\left( \frac{1}{n^{\beta}} \right), ] where $m$ is a nonnegative integer, $\alpha_i$ are real numbers, $r_i(x)$ are rational functions of $x$ and [ 0 < \alpha_1 < \alpha_2 < \cdots < \alpha_m < \beta. ] We will give a sufficient condition on $\ell$-log-momotonic sequences and Laguerre inequality of order two for $n$ sufficiently large. Many P-recursive sequences fall in this frame. At last, we will give a method to find the $N$ such that for any $n\geq N$, log-momotonic inequality of order three and Laguerre inequality of order two holds.