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

TL;DR

This paper studies $ ext{ell}$-log-momotonic and Laguerre inequalities for P-recursive sequences with specific asymptotic behaviors, providing conditions and methods to verify these inequalities for large indices.

Contribution

It introduces new sufficient conditions for $ ext{ell}$-log-momotonicity and Laguerre inequalities in P-recursive sequences, along with a method to determine the threshold index for these properties.

Findings

Provides a framework for analyzing asymptotic ratios of P-recursive sequences.

Establishes sufficient conditions for log-momotonicity and Laguerre inequalities.

Offers a method to find the index beyond which inequalities hold.

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.