Inequalities for Higher-Order Logarithmic Differences of Boros--Moll Coefficients
Abstract
Let di(m) denote the coefficient of xi in the Boros--Moll polynomial Pm(x). We prove that the sequence {di−1(m)di+1(m)/di(m)2}i=1m−1 is strictly log-concave for every m≥4, settling a conjecture of Chen and Gu. For every m≥ 3, we show that the consecutive ratios di(m)/di−1(m) form a strictly log-convex sequence for 1≤ i≤⌊ m/2⌋+1 and a strictly log-concave sequence for ⌊ m/2⌋≤ i≤ m, as conjectured by Zhao. For each fixed i≥ 1, we establish (−1)r−1Δmrlog di(m)>0 for r≥ 2 and m≥ i, where Δm denotes the forward difference with respect to m. For i=0, we obtain the opposite inequalities. We thereby prove a conjecture of Jiang and Wang and two further conjectures of Zhao. For every integer i≥ 0, we prove strict infinite log-monotonicity of the sequence {di(m−1)di(m+1)/(binommi+1di(m)2)}m≥ i+1, strengthening another conjecture of Zhao. We further show that every positive real power of the binomially normalized ratio sequence is a Hausdorff moment sequence with a strictly totally positive Hankel matrix. We also obtain asymptotic formulas for forward differences of arbitrary fixed order, resolving a conjecture of Jiang and Wang. For integers i≥ 1, we prove that the sequence {2n di(i)/di(i+n)}n≥ 0 is an infinitely divisible Hausdorff moment sequence and that the Hankel matrix [1/di(i+j+k)τ]j,k≥ 0 is strictly totally positive for every τ>0. For the integral interpolation, we show that the Hausdorff moment property of the normalized reciprocals holds precisely when the positive real coefficient index is at least 1/4.
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BibTeX
@misc{gu2026inequalities,
title = {{Inequalities for Higher-Order Logarithmic Differences of Boros--Moll Coefficients}},
author = {Cindy C. Y. Gu and Yafeng Liu and Matthew H. Y. Xie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25381},
eprint = {2610.25381},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25381},
}
RIS
TY - UNPB
TI - Inequalities for Higher-Order Logarithmic Differences of Boros--Moll Coefficients
AU - Cindy C. Y. Gu
AU - Yafeng Liu
AU - Matthew H. Y. Xie
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25381
UR - https://arxiv.org/abs/2610.25381
AB - Let d_i(m) denote the coefficient of x^i in the Boros--Moll polynomial P_m(x). We prove that the sequence {d_i−1(m)d_i+1(m)/d_i(m)^2}_i=1^m−1 is strictly log-concave for every m≥4, settling a conjecture of Chen and Gu. For every m≥ 3, we show that the consecutive ratios d_i(m)/d_i−1(m) form a strictly log-convex sequence for 1≤ i≤⌊ m/2⌋+1 and a strictly log-concave sequence for ⌊ m/2⌋≤ i≤ m, as conjectured by Zhao. For each fixed i≥ 1, we establish (−1)^r−1Δ_m^rlog d_i(m)>0 for r≥ 2 and m≥ i, where Δ_m denotes the forward difference with respect to m. For i=0, we obtain the opposite inequalities. We thereby prove a conjecture of Jiang and Wang and two further conjectures of Zhao. For every integer i≥ 0, we prove strict infinite log-monotonicity of the sequence {d_i(m−1)d_i(m+1)/(binommi+1d_i(m)^2)}_m≥ i+1, strengthening another conjecture of Zhao. We further show that every positive real power of the binomially normalized ratio sequence is a Hausdorff moment sequence with a strictly totally positive Hankel matrix. We also obtain asymptotic formulas for forward differences of arbitrary fixed order, resolving a conjecture of Jiang and Wang. For integers i≥ 1, we prove that the sequence {2^n d_i(i)/d_i(i+n)}_n≥ 0 is an infinitely divisible Hausdorff moment sequence and that the Hankel matrix [1/d_i(i+j+k)^τ]_j,k≥ 0 is strictly totally positive for every τ>0. For the integral interpolation, we show that the Hausdorff moment property of the normalized reciprocals holds precisely when the positive real coefficient index is at least 1/4.
ER -