Log-convexity and the overpartition function (Q2688353)

A sequence \((x_n)_{n\geqslant 0}\) of positive real numbers is log-convex if the inequality \(x_n^2\leqslant x_{n-1}x_{n+1}\) is valid for all \(n\geqslant 1\) [\textit{T. Došlić}, J. Math. Inequal. 3, No. 3, 437--442 (2009; Zbl 1230.05046)]. Concerning the overpartition function \(\bar{p}(n)\), \textit{B. Engel} [Ramanujan J. 43, No. 2, 229--241 (2017; Zbl 1373.05018)] has proved that \(\bar{p}(n)\) is log-concave for \(n\geq 2\). This paper begins with a well-thought-out introduction and a brief review of the work done in the direction of overpartitions. The author establishes the log-convexity of \(\sqrt[n]{\frac{\bar{p}(n)}{n}}\) for \({n\geq{19}}\) and that of \(\sqrt[n]{\bar{p}(n)}\) for \(n\geq4.\) The author also accomplishes the establishment of the asymptotic growth of \(\Delta^2\log \sqrt[n-1]{\frac{\bar{p}(n-1)}{(n-1)^\alpha}}\) for a non-negative real number \(\alpha\).