The log-behavior of $\root n \of {p(n)}$ and $\root n \of {p(n)/n}$

Abstract

Let p(n) denote the partition function and let $\Delta$ be the difference operator with respect to n. In this paper, we obtain a lower bound for $\Delta ^2\log \root n-1 \of {p(n-1)/(n-1)}$, leading to a proof of a conjecture of Sun on the log-convexity of $\{\root n \of {p(n)/n}\}_{n\ge 60}$. Using the same argument, it can be shown that for any real number $\alpha$, there exists an integer $n(\alpha )$ such that the sequence $\{\root n \of {p(n)/n^{\alpha }}\}_{n\ge n(\alpha )}$ is log-convex. Moreover, we show that $\lim \limits _{n \rightarrow +\infty }n^{\frac{5}{2}}\Delta ^2\log \root n \of {p(n)}=3\pi /\sqrt{24}$. Finally, by finding an upper bound for $\Delta ^2 \log \root n-1 \of {p(n-1)}$, we establish an inequality on the ratio $\frac{\root n-1 \of {p(n-1)}}{\root n \of {p(n)}}$.