Inequalities for Higher-Order Logarithmic Differences of Boros--Moll Coefficients

Source abstract

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 $m\ge4$, settling a conjecture of Chen and Gu. For every $m\ge 3$, we show that the consecutive ratios $d_i(m)/d_{i-1}(m)$ form a strictly log-convex sequence for $1\le i\le\lfloor m/2\rfloor+1$ and a strictly log-concave sequence for $\lfloor m/2\rfloor\le i\le m$, as conjectured by Zhao. For each fixed $i\ge 1$, we establish $(-1)^{r-1}Δ_m^r\log d_i(m)>0$ for $r\ge 2$ and $m\ge i$, 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 $i\ge 0$, we prove strict infinite log-monotonicity of the sequence $\{d_i(m-1)d_i(m+1)/(\binom{m}{i+1}d_i(m)^2)\}_{m\ge i+1}$, 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 $i\ge 1$, we prove that the sequence $\{2^n d_i(i)/d_i(i+n)\}_{n\ge 0}$ is an infinitely divisible Hausdorff moment sequence and that the Hankel matrix $[1/d_i(i+j+k)^τ]_{j,k\ge 0}$ 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 $1/4$.