A Proof of Moll's Minimum Conjecture

Abstract

Let $d_i(m)$ denote the coefficients of the Boros-Moll polynomials. Moll's minimum conjecture states that the sequence $\{i(i+1)(d_i^2(m)-d_{i-1}(m)d_{i+1}(m))\}_{1\leq i \leq m}$ attains its minimum with $i=m$. This conjecture is a stronger than the log-concavity conjecture proved by Kausers and Paule. We give a proof of Moll's conjecture by utilizing the spiral property of the sequence $\{d_i(m)\}_{0\leq i \leq m}$, and the log-concavity of the sequence $\{i!d_i(m)\}_{0\leq i \leq m}$.