ON TURÁN’S INEQUALITY FOR LEGENDRE POLYNOMIALS

Abstract. Let ∆n(x) = Pn(x) 2 − Pn−1(x)Pn+1(x), where Pn 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 hn n(n + 1) (1 − x2) ≤ ∆n(x) (−1 ≤ x ≤ 1; n = 1, 2, 3,...), where hn 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]: with the best possible factor Here, µn = 2 −2n � 2n n αn (1 − x 2) ≤ ∆n(x) αn = µ [n/2] µ [(n+1)/2]. is the normalized binomial mid-coefficient. Keywords. Legendre polynomials, Turán’s inequality, normalized binomial mid-coefficient