Log-concavity of the overpartition function

Abstract

We prove that the overpartition function p(n) is log-concave for all n ≥ 2. The proof is based on Sills Rademacher type series for p(n) and inspired by DeSalvo and Pak's proof for the partition function.

Inequalities for the overpartition function arising from determinants

“…We conclude the paper by undertaking a brief study on totally positive matrices with entries from sequences of overpartitions. Due to Engel [9], we know that for n ≥ 2, det M 2 (p(n)) := det p(n) p(n + 1) p(n − 1) p(n) > 0.…”

“…for positive integers h and k. In order to prove log-concavity of p(n), Engel [9] provided an error term for p(n)…”

Log-convexity and the overpartition function

“…for positive integers h and k. In somewhat a similar spirit as Lehmer [8] obtained an error bound for the partition function, Engel [6] provided an error term for p(n)…”

“…We start by laying out a brief outline of Engel's primary set up [6] for proving log-concavity of {p(n)} n≥2 . Setting N = 3 in (1.2), we express p(n) as…”

“…Engel [6] proved that {p(n)} n≥2 is log-concave by using the asymptotic formula (1.2) with N = 2 followed by (1.3). Prior to Engel's work on overpartitions, log-concavity of partition function p(n) and its associated inequalities has been studied in a broad spectrum, for example see [1], [2], and [5].…”