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

We consider $\mathrm{\ell }$-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 $$0<{\alpha }_1<{\alpha }_2<\cdots <{\alpha }_m<\beta \mathrm{.}$$ We will give a sufficient condition on $\mathrm{\ell }$-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.