Asymptotic Growth of (-1)^r Δ^r log √[n]p(n)/n^α and the Reverse Higher Order Turán Inequalities for √[n]p(n)/n^α

2024/01/10 by Gargi Mukherjee, Mukherjee, Gargi
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Mathematics and Applications #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2401.05522

openalex publication_date 2024/01/10 · openalex created_date 2024/01/13 · openalex updated_date 2026/07/28

Abstract

Let p(n) denote the overpartition function. In this paper, we study the asymptotic growth of finite difference of logarithm of √[n]p(n)/n^α for α being a non-negative real number, namely (-1)^rΔ^r log √[n]p(n)/n^α by presenting an inequality of it with a symmetric upper and lower bound. Consequently, we arrive at log-convexity of √[n]p(n) and √[n]p(n)/n, previously studied by the author. The another main objective of this paper is to introduce the notion of the reverse higher order Turán inequalities and we prove this for √[n]p(n)/n^α, which not only generalize the study of Sun, Chen, and Zheng but also depicts the non real-rootedness of the Jensen polynomial associated with the sequence mentioned before.

Related

• Log-convexity and the overpartition function
• Inequalities for higher order differences of the logarithm of the overpartition function and a problem of Wang-Xie-Zhang
• Higher Order Turán Inequalities for the Partition Function
• The Log-Behavior of √[n]p(n) and √[n]p(n)/n
• Log-concavity of P-recursive sequences