Sequences with Inequalities

We consider infinite sequences of positive numbers. The connection between log-concavity and the Bessenrodt–Ono inequality had been in the focus of several papers. This has applications in the white noise distribution theory and combinatorics. We improve a recent result of Benfield and Roy and show that for the sequence of partition numbers $\{p(n)\}$ Nicolas’ log-concavity result implies the result of Bessenrodt and Ono towards $p(n)p(m)>p(n+m)$. We provide several examples. Benfield and Roy gave a conjecture related to $\mathrm{\ell }$-ary partition numbers. We prove part of this conjecture.