An Extension of a Criterion for Unimodality

Abstract

We prove that if $P(x)$ is a polynomial with nonnegative nondecreasing coefficients and $n$ is a positive integer, then $P(x+n)$ is unimodal. Applications and open problems are presented.

Proof of Moll’s minimum conjecture

“…The objective of this note is to give a proof of Moll's conjecture on the minimum value of a sequence involving the coefficients of the Boros-Moll polynomials which arise in the evaluation of the following quartic integral, see, [1][2][3][4][5][6]11]. It has been shown that for any a > −1 and any nonnegative integer m, ∞ 0 1 (x 4 + 2ax 2 + 1) m+1 dx = π 2 m+3/2 (a + 1) m+1/2 P m (a), where…”

Proof of a conjecture on unimodality

“…Now let j ≥ m. Then (d+1)(j +1) ≥ (d+1)(m+1) ≥ m+1 by (1). Every term in the sum ( 2) is therefore non-positive, and thus b j+1 ≤ b j .…”

“…Let P (x) be nonnegative and non-decreasing. It is shown that P (x + 1) is unimodal in [2] and more generally, that P (x + n) is unimodal when n is a positive integer in [1]. Further, the following is conjectured.…”

“…It is not difficult to see that m(d) and m(d) coincide when d is a positive integer. In [1], it is shown that m(d) is a mode of P (x + d) when d is a positive integer. The statement is not true when d is only a positive number.…”

A remarkable sequence of integers

“…that satisfies P m (x) = A(x + 1). The criterion was extended in [1] to include the shifts A(x+ j) and in [32] for arbitrary shifts. The original proof of the unimodality of P m (a) can be found in [7].…”