The Briggs inequality of Boros-Moll sequences

Abstract

Briggs conjectured that if a polynomial $a_0+a_1x+\cdots+a_nx^n$ with real coefficients has only negative zeros, then $$a^2_k(a^2_k - a_{k-1}a_{k+1}) > a^2_{k-1}(a^2_{k+1} - a_ka_{k+2})$$ for any $1\leq k\leq n-1$. The Boros-Moll sequence $\{d_i(m)\}_{i=0}^m$ arises in the study of evaluation of certain quartic integral, and a lot of interesting inequalities for this sequence have been obtained. In this paper we show that the Boros-Moll sequence $\{d_i(m)\}_{i=0}^m$, its normalization $\{d_i(m)/i!\}_{i=0}^m$, and its transpose $\{d_i(m)\}_{m\ge i}$ satisfy the Briggs inequality. For the first two sequences, we prove the Briggs inequality by using a lower bound for $(d_{i-1}(m)d_{i+1}(m))/d_i^2(m)$ due to Chen and Gu and an upper bound due to Zhao. For the transposed sequence, we derive the Briggs inequality by establishing its strict ratio-log-convexity. As a consequence, we also obtain the strict log-convexity of the sequence $\{\sqrt[n]{d_i(i+n)}\}_{n\ge 1}$ for $i\ge 1$.