Inequalities for the overpartition function

TL;DR

This paper investigates inequalities related to the overpartition function, extending known Turán inequalities and exploring higher order inequalities to deepen understanding of its mathematical properties.

Contribution

The paper proves new inequalities for the overpartition function and determines higher order Turán inequalities, advancing theoretical knowledge in partition theory.

Findings

Established several new inequalities for or(n)

Extended Ture1n inequalities to higher orders

Provided mathematical proofs for the inequalities

Abstract

Let $\overline{p}(n)$ denote the overpartition funtion. Engel showed that for $n\geq2$, $\overline{p}(n)$ satisfied the Tur\'{a}n inequalities, that is, $\overline{p}(n)^2-\overline{p}(n-1)\overline{p}(n+1)>0$ for $n\geq2$. In this paper, we prove several inequalities for $\overline{p}(n)$. Moreover, motivated by the work of Chen, Jia and Wang, we find that the higher order Tur\'{a}n inequalities of $\overline{p}(n)$ can also be determined.