In 2007, Andrews and Paule introduced the broken k-diamond partition function \(\Delta _{k}(n)\) , whose arithmetic properties have received considerable attention. In this paper, we will prove that the broken k-diamond partition function satisfies the Laguerre inequality of order 2 and the determinantal inequality of order 3 for \(k=1\) or 2. Moreover, we have conjectured the thresholds for the Laguerre inequalities of order m and the positivity of m-order determinants for \(1\le m\le 14\) for the broken k-diamond partition function when \(k=1\) or 2.
