  We consider the higher order Tur\'an inequality and higher order
log-concavity 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 the higher order Tur\'an inequality and
the $r$-log-concavity for $n$ sufficiently large.
  Most $P$-recursive sequences fall in this frame. At last, we will give a
method to find the exact $N$ such that for any $n>N$, the higher order Tur\'an
inequality holds.
