Variants of an partition inequality of Bessenrodt–Ono

Abstract

In this paper we generalize and refine a partition inequality of Bessenrodt–Ono. We introduce and study the m shifted inequality $$\begin{aligned} p(a) \, p(b) \ge p(a+b+m-1) \end{aligned}$$where p(n) is the nth partition number, and $$a,b,m \in \mathbb {N}_0$$ with a, b positive. The inequality was first studied by Bessenrodt–Ono for $$m=1$$. We finally suggest another generalization involving polynomials, which dictate the vanishing properties of the Fourier coefficients of powers of the Dedekind $$\eta $$-function.