Bounds on Tur{á}n determinants
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Let μdenote a symmetric probability measure on
[-1,1] and let (p_n) be the corresponding orthogonal polynomials normalized such that p_n(1)=1. We prove that the normalized Tur{á}n determinant Δ_n(x)/(1-x^2), where Δ_n=p_n^2-p_{n-1}p_{n+1}, is a Tur{á}n determinant of order n-1 for orthogonal polynomials with respect to (1-x^2)dμ(x). We use this to prove lower and upper bounds for the normalized Tur{á}n determinant in the interval -1<x<1.