The Briggs inequality of Boros-Moll sequences

Briggs conjectured that if a polynomial a₀+a₁x+⋯+aₙxⁿ with real coefficients has only negative zeros, then a²ₖ(a²ₖ - aₖ₋₁aₖ₊₁) > a²ₖ₋₁(a²ₖ₊₁ - aₖaₖ₊₂) for any 1≤ k≤ n-1. The Boros-Moll sequence ᵢ(m)\ᵢ₌₀ᵐ 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 ᵢ(m)\ᵢ₌₀ᵐ, its normalization ᵢ(m)/i!\ᵢ₌₀ᵐ, and its transpose ᵢ(m)≥ i satisfy the Briggs inequality. For the first two sequences, we prove the Briggs inequality by using a lower bound for (dᵢ₋₁(m)dᵢ₊₁(m))/dᵢ²(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 \√[n]dᵢ(i+n)≥ 1 for i≥ 1.