  Let $\overline{p}(n)$ denote the overpartition funtion. Engel showed that for
$n\geq2$, $\overline{p}(n)$ satisfied the Tur\'{a}n inequalities, that is,
$\overline{p}(n)^2-\overline{p}(n-1)\overline{p}(n+1)>0$ for $n\geq2$. In this
paper, we prove several inequalities for $\overline{p}(n)$. Moreover, motivated
by the work of Chen, Jia and Wang, we find that the higher order Tur\'{a}n
inequalities of $\overline{p}(n)$ can also be determined.
