  In 2007, Andrews and Paule introduced the broken $k$-diamond partition
function $\Delta_k(n)$, which has received a lot of researches on the
arithmetic propertises. In this paper, we prove that $D^3\log \Delta_1(n-1)>0$
for $n\geq 5$ and $D^3 \log \Delta_2(n-1)>0$ for $n\geq 7$, where $D$ is the
difference operator with respect to $n$. We also conjecture that for any $k\geq
1$ and $r\geq 1$, there exists a positive integer $n_k(r)$ such that for $n\geq
n_{k}(r)$, $(-1)^r D^r \log \Delta_k(n)>0$. This is analogous to the positivity
of finite differences of the logarithm of the partition function, which has
been proved by Chen, Wang and Xie. Furthermore, we obtain that both
$\{\Delta_1(n)\}_{n\geq 0}$ and $\{\Delta_2(n)\}_{n\geq 0}$ satisfy the higher
order Tur\'an inequalities for $n \geq 6$.
