  Let $p(n)$ denote the partition function. DeSalvo and Pak proved that
$\frac{p(n-1)}{p(n)}\left(1+\frac{1}{n}\right)> \frac{p(n)}{p(n+1)}$ for $n\geq
2$, as conjectured by Chen. Moreover, they conjectured that a sharper
inequality $\frac{p(n-1)}{p(n)}\left( 1+\frac{\pi}{\sqrt{24}n^{3/2}}\right) >
\frac{p(n)}{p(n+1)}$ holds for $n\geq 45$. In this paper, we prove the
conjecture of Desalvo and Pak by giving an upper bound for $-\Delta^{2} \log
p(n-1)$, where $\Delta$ is the difference operator with respect to $n$. We also
show that for given $r\geq 1$ and sufficiently large $n$, $(-1)^{r-1}\Delta^{r}
\log p(n)>0$. This is analogous to the positivity of finite differences of the
partition function. It was conjectured by Good and proved by Gupta that for
given $r\geq 1$, $\Delta^{r} p(n)>0$ for sufficiently large $n$.
