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In 2007, Andrews and Paule introduced the broken $k$-diamond partitionfunction $\Delta_k(n)$, which has received a lot of researches on thearithmetic 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 thedifference operator with respect to $n$. We also conjecture that for any $k\geq1$ and $r\geq 1$, there exists a positive integer $n_k(r)$ such that for $n\geqn_{k}(r)$, $(-1)^r D^r \log \Delta_k(n)>0$. This is analogous to the positivityof finite differences of the logarithm of the partition function, which hasbeen 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 higherorder Tur\'an inequalities for $n \geq 6$.

Inequalities for the Broken $k$-Diamond Partition Function