  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 $\frac{p(n-1)}{p(n)}$.
