Log-convexity and the overpartition function

DOI10.1007/S11139-022-00578-0arXiv2201.07836MaRDI QIDQ6388678

Gargi Mukherjee

Publication date: 19 January 2022

Abstract: Let $overlinep(n)$ denote the overpartition function. In this paper, we obtain an inequality for the sequence $Delt{a}^{2}logsqrt[n-1]overlinep(n-1)/(n-1{)}^{alpha}$ which states that �egin{equation*} log �iggl(1+frac{3pi}{4n^{5/2}}-frac{11+5alpha}{n^{11/4}}�iggr) < Delta^{2} log sqrt[n-1]{overline{p}(n-1)/(n-1)^{alpha}} < log �iggl(1+frac{3pi}{4n^{5/2}}�iggr) ext{for} n geq N(alpha), 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 $$ and $$. Moreover, it also establishes the asymptotic growth of $Delt{a}^{2}logsqrt[n-1]overlinep(n-1)/(n-1{)}^{alpha}$ by showing $undersetnightarrowinftylimDelt{a}^{2}logsqrt[n]overlinep(n)/n^{alpha}=dfrac3pi4n^{5/2}.$

Mathematics Subject Classification ID

Asymptotic results on arithmetic functions (11N37) Combinatorial inequalities (05A20)

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