Let $\overline{p}(n)$ denote the overpartition funtion. Engel showed that for $n\geq2$, $\overline{p}(n)$ satisfied the Turán 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 Turán inequalities of $\overline{p}(n)$ can also be determined.