Log-convexity and the overpartition function.

Let $\overline{p}\left(n\right)$ 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 $\begin{array}{cc}& log(1+\frac{3\pi }{4n^{5/2}}-\frac{11+5\alpha }{n^{11/4}})<{\Delta }^{2}log\sqrt[n-1]{\overline{p}(n-1)/{(n-1)}^{\alpha }}\\ & <log(1+\frac{3\pi }{4n^{5/2}})\text{for}n\ge N\left(\alpha \right),\end{array}$ where $\alpha$ is a non-negative real number, $N\left(\alpha \right)$ is a positive integer depending on $\alpha$ , and $\Delta$ is the difference operator with respect to n. This inequality consequently implies $log$ -convexity of $\{\sqrt[n]{\overline{p}\left(n\right)/n}{\}}_{n\ge 19}$ and $\{\sqrt[n]{\overline{p}\left(n\right)}{\}}_{n\ge 4}$ . Moreover, it also establishes the asymptotic growth of ${\Delta }^{2}log\sqrt[n-1]{\overline{p}(n-1)/{(n-1)}^{\alpha }}$ by showing $\underset{n\to \infty }{lim}{\Delta }^{2}log\sqrt[n]{\overline{p}\left(n\right)/n^{\alpha }}=\frac{3\pi }{4n^{5/2}}.$