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The paper aims to establish the Tur\'an inequalities, the Laguerreinequalities (order $2$), and the determinantal inequalities (order $3$) for$\Delta p(n)$ and $\Delta \bar{p}(n)$, where $\Delta f(n)$ is the first-orderforward 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 positivityof $m$-order determinants are proposed, extending to $\Delta^k p(n)$ and$\Delta^k \bar{p}(n)$, with $1 \leq m \leq 11$ and $1 \leq k \leq 5$.

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