Bounds on Turán determinants

Let μμ denote a symmetric probability measure on [−1,1][−1,1] and let (pn)(pn) be the corresponding orthogonal polynomials normalized such that pn(1)=1pn(1)=1. We prove that the normalized Turán determinant Δn(x)/(1−x2)Δn(x)/(1−x2), where Δn=pn2−pn−1pn+1, is a Turán determinant of order n−1n−1 for orthogonal polynomials with respect to (1−x2)dμ(x). We use this to prove lower and upper bounds for the normalized Turán determinant in the interval −1