A unified framework to prove multiplicative inequalities for the partition function

In this paper, we consider a certain class of inequalities for the partition function of the following form:$\prod _{i=1}^{T}p(n+s_i)\ge \prod _{i=1}^{T}p(n+r_i),$ which we call multiplicative inequalities. Given a multiplicative inequality with the condition that ${\sum }_{i=1}^T{s}_i^m\ne {\sum }_{i=1}^T{r}_i^m$ for at least one $m\ge 1$, we shall construct a unified framework so as to decide whether such a inequality holds or not. As a consequence, we will see that study of such inequalities has manifold applications. For example, one can retrieve log-concavity property, strong log-concavity, and the multiplicative inequality for $p(n)$ considered by Bessenrodt and Ono, to name a few. Furthermore, we obtain an asymptotic expansion for the finite difference of the logarithm of $p(n)$, denoted by ${(-1)}^{r-1}{\mathrm{\Delta }}^r\log p(n)$, which generalizes a result by Chen, Wang, and Xie.