  We obtain an asymptotic formula for Andrews and Paule's broken $k$-diamond
partition function $\Delta_k(n)$ where $k=1$ or $2$. Based on this asymptotic
formula, we derive that $\Delta_k(n)$ satisfies the order $d$ Tur\'an
inequalities for $d\geq 1$ and for sufficiently large $n$ when $k=1$ and $ 2$
by using a general result of Griffin, Ono, Rolen and Zagier. We also show that
Andrews and Paule's broken $k$-diamond partition function $\Delta_k(n)$ is
log-concave for $n\geq 1$ when $k=1$ and $2$. This leads to
$\Delta_k(a)\Delta_k(b)\ge\Delta_k(a+b)$ for $a,b\ge 1$ when $k=1$ and $ 2$.
