Laguerre inequality and determinantal inequality 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 will prove the broken $k$-diamond partition function satisfies the Laguerre inequalities of order $2$ and the determinantal inequalities of order $3$ for $k=1$ or $2$. Moreover, we conjectured the thresholds for the Laguerre inequalities of order $m$ and the positivity of $m$-order determinants for $4\le m\le 14$ for the broken $k$-diamond partition function when $k=1$ or $2$.