A Proof of Moll's Minimum Conjecture

TL;DR

This paper proves Moll's minimum conjecture, demonstrating that a specific sequence derived from Boros-Moll polynomial coefficients reaches its minimum at the last index, using properties like spiral and log-concavity.

Contribution

The paper provides a proof of Moll's minimum conjecture, establishing the minimality of a sequence involving Boros-Moll polynomial coefficients at the endpoint.

Findings

The sequence $iglrace i(i+1)(d_i^2(m)-d_{i-1}(m)d_{i+1}(m)) igr brace$ attains its minimum at $i=m$.

The proof uses the spiral property of $iglrace d_i(m) igr brace$ and the log-concavity of $iglrace i!d_i(m) igr brace$.

Moll's minimum conjecture is confirmed as true.

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 $\{i!d_i(m)\}_{0\leq i \leq m}$.