Asymptotic growth of (−1)rΔrlog⁡p‾(n)/nαn and the reverse higher order Turán inequalities for p‾(n)/nαn

Let $\bar{p}\left(n\right)$ denote the overpartition function. In this paper, we study the asymptotic growth of finite difference of logarithm of $\sqrt[n]{\bar{p}\left(n\right)/n^{\alpha }}$ for α being a non-negative real number. Consequently, we retrieve log-convexity of $\sqrt[n]{\bar{p}\left(n\right)}$ and $\sqrt[n]{\bar{p}\left(n\right)/n}$, previously studied by the author aligned to the work of Chen and Zheng in context of the partition function $p\left(n\right)$. Generalizing the theme of log-convexity, began with the work of Sun, Chen, and Zheng, another main objective of this paper is to prove that $\sqrt[n]{\bar{p}\left(n\right)/n^{\alpha }}$ satisfies the reverse higher order Turán inequalities which depict the non real-rootedness of the Jensen polynomial associated with the sequence presented above.