  Bessenrodt and Ono, Chen, Wang and Jia, DeSalvo and Pak were the first to
discover the log-subadditivity, log-concavity, and the third-order Tur\'{a}n
inequality of partition function, respectively. Many other important partition
statistics are proved to enjoy similar properties. This paper focuses on the
partition function $\overline{p}_k(n)$, which counts the number of
overpartitions of $n$ with no parts divisible by $k$. We provide a
combinatorial proof to establish that for any $k\geq2$, the partition function
$\overline{p}_k(n)$ exhibits strict log-subadditivity. Specifically, we show
that $\overline{p}_k(a)\overline{p}_k(b)>\overline{p}_k(a+b)$ for integers
$a\geq b\geq1$ and $a+b\geq k$. Furthermore, we investigate the log-concavity
and the satisfaction of the third-order Tur\'{a}n inequality for
$\overline{p}_k(n)$, where $2\leq k\leq9$.
