Proof of Moll’s minimum conjecture

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 ≤ i ≤ m attains its minimum at i = m with 2 − 2 m m ( m + 1 ) 2 m m 2 . This conjecture is stronger than the log-concavity conjecture of Moll proved by Kauers and Paule. We give a proof of Moll’s conjecture by utilizing the spiral property of the sequence { d i ( m ) } 0 ≤ i ≤ m , and the log-concavity of the sequence { i ! d i ( m ) } 0 ≤ i ≤ m .