The Log-Behavior of $\sqrt[n]{p(n)}$ and $\sqrt[n]{p(n)/n}$

William Y. C. Chen, Ken Y. Zheng

Published 2015-11-09Version 1

Let $p(n)$ denote the partition function. Desalvo and Pak proved the log-concavity of $p(n)$ for $n>25$ and the inequality $\frac{p(n-1)}{p(n)}\left(1+\frac{1}{n}\right)>\frac{p(n)}{p(n+1)}$ for $n>1$. Let $r(n)=\sqrt[n]{p(n)/n}$ and $\Delta$ be the difference operator respect to $n$. Desalvo and Pak pointed out that their approach to proving the log-concavity of $p(n)$ may be employed to prove a conjecture of Sun on the log-convexity of $\{r(n)\}_{n\geq 61}$, as long as one finds an appropriate estimate of $\Delta^2 \log r(n-1)$. In this paper, we obtain a lower bound for $\Delta^2\log r(n-1)$, leading to a proof of this conjecture. From the log-convexity of $\{r(n)\}_{n\geq61}$ and $\{\sqrt[n]{n}\}_{n\geq4}$, we are led to a proof of another conjecture of Sun on the log-convexity of $\{\sqrt[n]{p(n)}\}_{n\geq27}$. Furthermore, we show that $\lim\limits_{n \rightarrow +\infty}n^{\frac{5}{2}}\Delta^2\log\sqrt[n]{p(n)}=3\pi/\sqrt{24}$. Finally, by finding an upper bound of $\Delta^2 \log\sqrt[n-1]{p(n-1)}$, we prove an inequality on the ratio $\frac{\sqrt[n-1]{p(n-1)}}{\sqrt[n]{p(n)}}$ analogous to the above inequality on the ratio $\frac{p(n-1)}{p(n)}$.