Higher order log-concavity of the overpartition function and its consequences

Let $\bar{p}(n)$ denote the overpartition function. In this paper, we study the asymptotic higher order $\log$-concavity property of the overpatition function in a similar framework done by Hou and Zhang for the partition function. This will enable us to move on further in order to prove $\log$-concavity of overpartitions, explicitly by studying the asymptotic expansion of the quotient $\bar{p}(n-1)\bar{p}(n+1)\mathrm{/}\bar{p}(n{)}^2$ upto a certain order so that one can finally ends up with the phenomena of $2$-$\log$-concavity and higher order Tur'{a}n property of $\bar{p}(n)$ by following a sort of unified approach.