Inequalities for higher order differences of the logarithm of the overpartition function and a problem of Wang–Xie–Zhang

Let \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}p¯(n) denote the overpartition function. In this paper, our primary goal is to study the asymptotic behavior of the finite differences of the logarithm of the overpartition function, i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-1)^{r-1}\Delta ^r \log {\overline{p}}(n)$$\end{document}(-1)r-1Δrlogp¯(n), by studying the inequality of the following form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \log \Bigl (1+\dfrac{C(r)}{n^{r-1/2}}-\dfrac{1+C_1(r)}{n^{r}}\Bigr ){} & {} <(-1)^{r-1}\Delta ^r \log {\overline{p}}(n) \\ {}{} & {} <\log \Bigl (1+\dfrac{C(r)}{n^{r-1/2}}\Bigr )\ \text {for}\ n \ge N(r), \end{aligned}$$\end{document}log(1+C(r)nr-1/2-1+C1(r)nr)<(-1)r-1Δrlogp¯(n)<log(1+C(r)nr-1/2)forn≥N(r),where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(r), C_1(r), \text {and}\ N(r)$$\end{document}C(r),C1(r),andN(r) are computable constants depending on the positive integer r, determined explicitly. This solves a problem posed by Wang, Xie and Zhang in the context of searching for a better lower bound of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-1)^{r-1}\Delta ^r \log {\overline{p}}(n)$$\end{document}(-1)r-1Δrlogp¯(n) than 0. By settling the problem, we are able to show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{n\rightarrow \infty }(-1)^{r-1}\Delta ^r \log {\overline{p}}(n) =\dfrac{\pi }{2}\Bigl (\dfrac{1}{2}\Bigr )_{r-1}n^{\frac{1}{2}-r}. \end{aligned}$$\end{document}limn→∞(-1)r-1Δrlogp¯(n)=π2(12)r-1n12-r.


Introduction
An overpartition of a positive integer n is a nonincreasing sequence of positive integers whose sum is n in which the first occurrence of a number may be overlined, p(n) denotes the number of overpartitions of n, and we define p(0) = 1. For example, there are 8 overpartitions of 3 enumerated by 3, 3, 2 + 1, 2 + 1, 2 + 1, 2 + 1, 1 + 1 + 1, 1 + 1 + 1. A thorough study of the overpartition function started with the work of Corteel and Lovejoy [1], although it has been studied under different nomenclature that dates back to MacMahon. Similar to the Hardy-Ramanujan-Rademacher formula for the partition function (cf. [2,3]), Zuckerman's [4] formula for p(n) states that for (h, k) ∈ Z ≥0 × Z ≥1 . Engel [5] determined an error term for p(n) and found that where R 2 (n, N ) < N 5/2 πn 3/2 sinh π √ n N , (1.3) similar to the work done by Lehmer [6] in order to obtain an error bound for the partition function p(n).
A positive sequence {a n } n≥0 is said to be log-concave (resp. log-convex) if for all n ≥ 1, a 2 n ≥ a n−1 a n+1 (resp. a 2 n ≤ a n−1 a n+1 ), and it is said to be strictly log-concave (resp. strictly log-convex) if the inequality is strict.
Using the notations above, Engel's result [5] actually states that {p(n)} n≥1 is log-concave. In fact, if one defines p(0) := 1, then {p(n)} n≥0 is actually also log-concave. Engel proved that {p(n)} n≥4 is strictly log-concave by using the asymptotic formula (1.2) with N = 3, and the error bound (1.3). Prior to Engel's work on overpartitions, the log-concavity of the partition function p(n) and its associated inequalities has been studied in a wider spectrum, details can be found in [7][8][9]. On the other hand, Liu and Zhang [10] proved a family of inequalities for the overpartition function. Higher order log-concavity and log-convexity for the overpartition function has been studied in [11,12] respectively.
Chen, Guo and Wang [13] introduced the notion of ratio log-convexity of a sequence and established that ratio log-convexity implies log-convexity under a certain initial condition. A sequence {a n } n≥k is called ratio log-convex if {a n+1 /a n } n≥k is log-convex or, equivalently, for n ≥ k + 1, 3 log a n−1 = log a n+2 − 3 log a n+1 + 3 log a n − log a n−1 ≥ 0, where be the difference operator defined by f (n) = f (n + 1) − f (n). Chen, Guo, and Wang relates the ratio log-convexity of a sequence, say {a n } n≥k , with strict log-convexity of the associated sequence { n √ a n } n≥k stated in the following theorem.
then the sequence { n √ a n } n≥k is strictly log-convex.
Similar to the work done by Chen et al. [8] for p(n), Wang, Xie and Zhang [14] proved the following two theorems.
Remark 1.4 Following Theorem 1.3, we observe that log-concavity and ratio logconvexity for p(n) correspond to the cases r = 2 and r = 3 respectively.
Wang, Xie, and Zhang raised the following question: In other words, their problem reads "Moreover, we seek a sharp lower bound for (−1) r−1 r log p(n)".
The main motivation of this paper is to give an affirmative answer to the Problem 1.5 in Theorems 1.6 and 1.8. This in turn clarifies the asymptotic growth of (−1) r−1 r log p(n), see Corollary 1.9. In Corollaries 1.10 and 1.11, we recover the log-concavity and its (shifted) companion inequality respectively. Theorem 1.6 For n ≥ 26, (1.10) (1.12) Theorem 1.8 For r ∈ Z ≥2 and n ≥ N (r), where C(r) and C 1 (r) are given in (1.7)-(1.8). (1.14) Proof Multiplying both sides of (1.4) (resp. (1.13)) by √ n (resp. by n r−1/2 ) and taking limits as n tends to infinity, we obtain (1.14).
Proof Observe that N (2) = 344 and from the lower bound of (1.13), we observe that {p(n)} n≥344 is log-concave and for the remaining cases 5 ≤ n ≤ 343, we confirm by numerical checking in Mathematica.
This paper is organized as follows. A preliminary setup for decomposing (−1) r−1 r log p(n) = H r + G r (cf. see (2.4), (2.5), and (2.6)), as done in [14], and estimations for both H r and G r are given in Sect. 2. Proofs of Theorems 1.6 and 1.8 are given in Sect. 3.

preliminary lemmas
Following the notations given in Engel [5] and Wang et al. [14], split p(n) as where and Remark 2.1 The splitting for p(n) used here is actually slightly different from what is found in [5,14].

Lemma 2.5
For all n ≥ 1, and L (1) Proof Equation (2.13) follows immediately by applying Proposition 2.4 on each of the factors in H r being presented in (2.5) for r = 1.
Proof Rewrite (2.5) as and applying Proposition 2.4, we get π 2 Since for all positive integers n, r and k, (2.19) Now we further investigate the lower bound of H r , given in (2.19).