The Briggs inequality for partitions and overpartitions

A sequence of ₙ≥ 0 satisfies the Briggs inequality if {align*} a_n^2(a_n^2-aₙ₋₁aₙ₊₁)>aₙ₋₁^2(aₙ₊₁^2-a_naₙ₊₂) {align*} holds for any n≥ 1. In this paper we show that both the partition function (n+N₀)≥ 0 and the overpartition function ̄(n+N̄₀)≥ 0 satisfy the Briggs inequality for some N₀ and N̄₀. Based on Chern's formula for η-quotients, we further prove that the k-regular partition function ₖ(n+Nₖ)≥ 0 and the k-regular overpartition function ̄ₖ(n+N̄ₖ)≥ 0 also satisfy the Briggs inequality for 2≤ k≤ 9 and some Nₖ,N̄ₖ.