  Let $\overline{p}(n)$ denote the overpartition funtion. This paper presents
the $2$-$\log$-concavity property of $\overline{p}(n)$ by considering a more
general inequality of the following form \begin{equation*} \begin{vmatrix}
\overline{p}(n) & \overline{p}(n+1) & \overline{p}(n+2) \\ \overline{p}(n-1) &
\overline{p}(n) & \overline{p}(n+1) \\ \overline{p}(n-2) & \overline{p}(n-1) &
\overline{p}(n) \end{vmatrix} > 0, \end{equation*} which holds for all $n \geq
42$.
