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

Abstract

Let $\overline{p}(n)$ denote the overpartition function. In this paper, we study the asymptotic growth of finite difference of logarithm of $\sqrt[n]{\overline{p}(n)/n^{\alpha}}$ for $\alpha$ being a non-negative real number, namely $(-1)^{r}\Delta^r \log \sqrt[n]{\overline{p}(n)/n^{\alpha}}$ by presenting 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 another main objective of this paper is to introduce the notion of the reverse higher order Tur\'{a}n inequalities and we prove this for $\sqrt[n]{\overline{p}(n)/n^{\alpha}}$, 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.