Inequalities for the overpartition function arising from determinants

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 $n \geq 42$.