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Let $\overline{p}(n)$ denote the overpartition function. In this paper, westudy the asymptotic growth of finite difference of logarithm of$\sqrt[n]{\overline{p}(n)/n^{\alpha}}$ for $\alpha$ being a non-negative realnumber, namely $(-1)^{r}\Delta^r \log \sqrt[n]{\overline{p}(n)/n^{\alpha}}$ bypresenting an inequality of it with a symmetric upper and lower bound.Consequently, we arrive at log-convexity of $\sqrt[n]{\overline{p}(n)}$ and$\sqrt[n]{\overline{p}(n)/n}$, previously studied by the author. The anothermain objective of this paper is to introduce the notion of the reverse higherorder Tur\'{a}n inequalities and we prove this for$\sqrt[n]{\overline{p}(n)/n^{\alpha}}$, which not only generalize the study ofSun, Chen, and Zheng but also depicts the non real-rootedness of the Jensenpolynomial associated with the sequence mentioned before.

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