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

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 \(\frac{\root n-1 \of {p(n-1)}}{\root n \of {p(n)}}\).