$\ell$-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 \[ \frac{a_{n-1}a_{n+1}}{a_n^2} = 1 + \sum_{i=1}^m \frac{r_i(\log n)}{n^{α_i}} + o\left( \frac{1}{n^β} \right), \] where $m$ is a nonnegative integer, ${\alpha }_i$ are real numbers, $r_i(x)$ are rational functions of $x$ and \[ 0 < α_1 < α_2 < \cdots < α_m < β. \] 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 $n\ge N$, log-momotonic inequality of order three and Laguerre inequality of order two holds.