On Turán's inequality for Legendre polynomials

Abstract

Let ∆ n (x) = P n (x) 2 − P n−1 (x)P n+1 (x), where P n is the Legendre polynomial of degree n. A classical result of Turán states that ∆ n (x) ≥ 0 for x ∈ [−1, 1] and n = 1, 2, 3, .... Recently, Constantinescu improved this result. He established h n n(n + 1) (1 − x 2) ≤ ∆ n (x) (−1 ≤ x ≤ 1; n = 1, 2, 3, ...), where h n denotes the n-th harmonic number. We present the following refinement. Let n ≥ 1 be an integer. Then we have for all x ∈ [−1, 1]: α n (1 − x 2) ≤ ∆ n (x) with the best possible factor α n = µ [n/2] µ [(n+1)/2]. Here, µ n = 2 −2n 2n n is the normalized binomial mid-coefficient.

Turán type inequalities for hypergeometric functions

“…This classical result has been extended in several directions: ultraspherical, Laguerre and Hermite polynomials [16], Jacobi polynomials [10,11], general class of polynomials [9], Bessel functions of the first kind [15], modified Bessel functions of the first kind [6,13,18], and so forth. This inequality still attracts the attention of mathematicians, and it is worth mentioning here that recently the inequality (1.1) was improved by Alzer et al [2].…”

Termination conditions for positivity proving procedures

“…Not only did it provide the first computer proofs of some special function inequalities from the literature [14,15,18,19], but it even helped to resolve some open conjectures [1,20,19,23]. At the same time, the lack of termination conditions was unsatisfactory from a computational point of view.…”

A Hypergeometric Inequality

“…Besides the computational complexity, this may also be because the method of Gerhold and Kauers is not an algorithm in the strict sense, since termination is not guaranteed for an arbitrary input [24,28]. Despite this fact, this procedure has been applied successfully in proving different non-trivial inequalities, e.g., on orthogonal polynomials [1,19,27]. For the inequality at hand the approach does not seem to succeed and hence a different line of computer-assisted proof that also exploits CAD has been chosen.…”