Laguerre inequalities and determinantal inequalities for the finite difference of the partition functions

The paper aims to establish the Turán inequalities, the Laguerre inequalities (order $2$), and the determinantal inequalities (order $3$) for $\mathrm{\Delta }p(n)$ and $\mathrm{\Delta }\bar{p}(n)$, where $\mathrm{\Delta }f(n)$ is the first-order forward difference of a sequence $f(n)$. The functions $p(n)$ and $\bar{p}(n)$ denote the partition function and overpartition function, respectively. Conjectures for thresholds of Laguerre inequalities (order $m$) and positivity of $m$-order determinants are proposed, extending to ${\mathrm{\Delta }}^{k}p(n)$ and ${\mathrm{\Delta }}^k\bar{p}(n)$, with $1\le m\le 11$ and $1\le k\le 5$.