Inequalities for the Broken $k$-Diamond Partition Function

In 2007, Andrews and Paule introduced the broken $k$-diamond partition function ${\mathrm{\Delta }}_k(n)$, which has received a lot of researches on the arithmetic propertises. In this paper, we prove that $D^3\log {\mathrm{\Delta }}_1(n-1)>0$ for $n\ge 5$ and $D^3\log {\mathrm{\Delta }}_2(n-1)>0$ for $n\ge 7$, where $D$ is the difference operator with respect to $n$. We also conjecture that for any $k\ge 1$ and $r\ge 1$, there exists a positive integer $n_k(r)$ such that for $n\ge n_k(r)$, $(-1{)}^r{D}^r\log {\mathrm{\Delta }}_k(n)>0$. 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 $\{{\mathrm{\Delta }}_1(n){\}}_{n\ge 0}$ and $\{{\mathrm{\Delta }}_2(n){\}}_{n\ge 0}$ satisfy the higher order Turán inequalities for $n\ge 6$.