Inequalities for the overpartition function arising from determinants

TL;DR

This paper investigates the overpartition function's inequalities, establishing a 2-log-concavity property through determinant inequalities that hold for all sufficiently large n.

Contribution

It introduces a new determinant-based inequality demonstrating 2-log-concavity of the overpartition function for n ≥ 42.

Findings

Proves a determinant inequality for overpartition function

Establishes 2-log-concavity property for large n

Provides bounds and conditions for inequalities

Abstract

Let $\overline{p}(n)$ denote the overpartition funtion. This paper presents the $2$-$\log$-concavity property of $\overline{p}(n)$ by considering a more general inequality of the following form \begin{equation*} \begin{vmatrix} \overline{p}(n) & \overline{p}(n+1) & \overline{p}(n+2) \\ \overline{p}(n-1) & \overline{p}(n) & \overline{p}(n+1) \\ \overline{p}(n-2) & \overline{p}(n-1) & \overline{p}(n) \end{vmatrix} > 0, \end{equation*} which holds for all $n \geq 42$.