Turán inequalities for the broken $k$-diamond partition function

We obtain an asymptotic formula for Andrews and Paule’s broken $k$-diamond partition function ${\mathrm{\Delta }}_k(n)$ where $k=1$ or $2$. Based on this asymptotic formula, we derive that ${\mathrm{\Delta }}_k(n)$ satisfies the order $d$ Turán inequalities for $d\ge 1$ 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 ${\mathrm{\Delta }}_k(n)$ is log-concave for $n\ge 1$ when $k=1$ and $2$. This leads to ${\mathrm{\Delta }}_k(a){\mathrm{\Delta }}_k(b)\ge {\mathrm{\Delta }}_k(a+b)$ for $a,b\ge 1$ when $k=1$ and $2$.