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.
Categories: math.CO