Asymptotic Growth of $(-1{)}^r{\mathrm{\Delta }}^r\log \sqrt[n]{\bar{p}(n)\mathrm{/}{n}^{\alpha }}$ and the Reverse Higher Order Turán Inequalities for $\sqrt[n]{\bar{p}(n)\mathrm{/}{n}^{\alpha }}$

Let $\bar{p}(n)$ denote the overpartition function. In this paper, we study the asymptotic growth of finite difference of logarithm of $\sqrt[n]{\bar{p}(n)\mathrm{/}{n}^{\alpha }}$ for $\alpha$ being a non-negative real number, namely $(-1{)}^r{\mathrm{\Delta }}^r\log \sqrt[n]{\bar{p}(n)\mathrm{/}{n}^{\alpha }}$ by presenting an inequality of it with a symmetric upper and lower bound. Consequently, we arrive at log-convexity of $\sqrt[n]{\bar{p}(n)}$ and $\sqrt[n]{\bar{p}(n)\mathrm{/}n}$, previously studied by the author. The another main objective of this paper is to introduce the notion of the reverse higher order Turán inequalities and we prove this for $\sqrt[n]{\bar{p}(n)\mathrm{/}{n}^{\alpha }}$, which not only generalize the study of Sun, Chen, and Zheng but also depicts the non real-rootedness of the Jensen polynomial associated with the sequence mentioned before.