Log-convexity and the overpartition function

Let p¯(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{p}(n)$$\end{document} denote the overpartition function. In this paper, we obtain an inequality for the sequence Δ2logp¯(n-1)/(n-1)αn-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta ^{2}\log \ \root n-1 \of {\overline{p}(n-1)/(n-1)^{\alpha }}$$\end{document} which states that log(1+3π4n5/2-11+5αn11/4)<Δ2logp¯(n-1)/(n-1)αn-1<log(1+3π4n5/2)forn≥N(α),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\log \biggl (1+\frac{3\pi }{4n^{5/2}}-\frac{11+5\alpha }{n^{11/4}}\biggr )< \Delta ^{2} \log \ \root n-1 \of {\overline{p}(n-1)/(n-1)^{\alpha }}\\&< \log \biggl (1+\frac{3\pi }{4n^{5/2}}\biggr ) \ \ \text {for}\ n \ge N(\alpha ), \end{aligned}$$\end{document}where α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} is a non-negative real number, N(α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N(\alpha )$$\end{document} is a positive integer depending on α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}, and Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document} is the difference operator with respect to n. This inequality consequently implies log\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log $$\end{document}-convexity of {p¯(n)/nn}n≥19\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bigl \{\root n \of {\overline{p}(n)/n}\bigr \}_{n \ge 19}$$\end{document} and {p¯(n)n}n≥4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bigl \{\root n \of {\overline{p}(n)}\bigr \}_{n \ge 4}$$\end{document}. Moreover, it also establishes the asymptotic growth of Δ2logp¯(n-1)/(n-1)αn-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta ^{2} \log \ \root n-1 \of {\overline{p}(n-1)/(n-1)^{\alpha }}$$\end{document} by showing limn→∞Δ2logp¯(n)/nαn=3π4n5/2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\underset{n \rightarrow \infty }{\lim } \Delta ^{2} \log \ \root n \of {\overline{p}(n)/n^{\alpha }} = \dfrac{3 \pi }{4 n^{5/2}}.$$\end{document}