The paper aims to establish the Turán inequalities, the Laguerre inequalities (order $2$), and the determinantal inequalities (order $3$) for $Δp(n)$ and $Δ\bar{p}(n)$, where $Δf(n)$ 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 $Δ^k p(n)$ and $Δ^k \bar{p}(n)$, with $1 \leq m \leq 11$ and $1 \leq k \leq 5$.