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 root n-1 of {overline{p}(n-1)/(n-1)^{alpha }} 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}$$\\begin{aligned}&\\log \\biggl (1+\\frac{3\\pi }{4n^{5/2}}-\\frac{11+5\\alpha }{n^{11/4}}\\biggr )< \\Delta ^{2} \\log \\ \\root n-1 \\of {\\overline{p}(n-1)/(n-1)^{\\alpha }}\\\\&< \\log \\biggl (1+\\frac{3\\pi }{4n^{5/2}}\\biggr ) \\ \\ \ ext {for}\\ n \\ge N(\\alpha ), \\end{aligned}$$\\end{document}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 bigl {root n of {overline{p}(n)/n}bigr }_{n ge 19} and bigl {root n of {overline{p}(n)}bigr }_{n ge 4}. Moreover, it also establishes the asymptotic growth of Delta ^{2} log root n-1 of {overline{p}(n-1)/(n-1)^{alpha }} by showing underset{n rightarrow infty }{lim } Delta ^{2} log root n of {overline{p}(n)/n^{alpha }} = dfrac{3 pi }{4 n^{5/2}}.