The Briggs inequality for partitions and overpartitions

Abstract

A sequence of $\{a_n\}_{n\ge 0}$ satisfies the Briggs inequality if \begin{align*} a_n^2(a_n^2-a_{n-1}a_{n+1})>a_{n-1}^2(a_{n+1}^2-a_na_{n+2}) \end{align*} holds for any $n\ge 1$. In this paper we show that both the partition function $\{p(n+N_0)\}_{n\geq 0}$ and the overpartition function $\{\overline{p}(n+\overline{N}_0)\}_{n\ge 0}$ satisfy the Briggs inequality for some $N_0$ and $\overline{N}_{0}$. Based on Chern's formula for $\eta$-quotients, we further prove that the $k$-regular partition function $\{p_k(n+N_{k})\}_{n\geq 0}$ and the $k$-regular overpartition function $\{\overline{p}_k(n+\overline{N}_k)\}_{n\ge 0}$ also satisfy the Briggs inequality for $2\le k\le 9$ and some $N_k,\overline{N}_{k}$.