Log-convexity and the overpartition function

Abstract

Let $\overline{p}(n)$ denote the overpartition function. In this paper, we obtain an inequality for the sequence ${\Delta }^2\log \sqrt[n-1]{\overline{p}(n-1)/(n-1{)}^{\alpha }}$ which states that \begin{equation*} \log \biggl(1+\frac{3π}{4n^{5/2}}-\frac{11+5α}{n^{11/4}}\biggr) < Δ^{2} \log \ \sqrt[n-1]{\overline{p}(n-1)/(n-1)^α} < \log \biggl(1+\frac{3π}{4n^{5/2}}\biggr) \ \ \text{for}\ n \geq N(α), \end{equation*} where $\alpha$ is a non-negative real number, $N(\alpha )$ is a positive integer depending on $\alpha$ and $\Delta$ is the difference operator with respect to $n$. This inequality consequently implies $\log$-convexity of $\{\sqrt[n]{\overline{p}(n)/n}{\}}_{n\ge 19}$ and $\{\sqrt[n]{\overline{p}(n)}{\}}_{n\ge 4}$. Moreover, it also establishes the asymptotic growth of ${\Delta }^2\log \sqrt[n-1]{\overline{p}(n-1)/(n-1{)}^{\alpha }}$ by showing $\underset{n\to \infty }{\lim }{\Delta }^2\log \sqrt[n]{\overline{p}(n)/n^{\alpha }}=\frac{3\pi }{4n^{5/2}}.$