New inequalities for p(n) and logp(n)

Abstract

Let p(n) denote the number of partitions of n. A new infinite family of inequalities for p(n) is presented. This generalizes a result by William Chen et al. From this infinite family, another infinite family of inequalities for $logp\left(n\right)$ is derived. As an application of the latter family one, for instance obtains that for $n\ge 120$, $\begin{array}{c}p{\left(n\right)}^2>(1+\frac{\pi }{\sqrt{24}{n}^{3/2}}-\frac{1}{n^2})p(n-1)p(n+1).\end{array}$

Keywords: Hardy–Ramanujan–Rademacher formula; Log-concavity; The partition function asymptotics.