Conjectured inequalities for Jacobi polynomials and their largest zeros

P. Leopardi and the author recently investigated, among other things, the validity of the inequality n\theta_n^{(\alpha,\beta)}\!<\! (n\!+\!1)\theta_{n+1}^{(\alpha,\beta)} between the largest zero x_n\!=\!\cos\theta_n^{(\alpha,\beta)} and x_{n+1}= \cos\theta_{n+1}^{(\alpha,\beta)} of the Jacobi polynomial P_n^{(\alpha,\beta)}(x) resp. P_{n+1}^{( \alpha,\beta)}(x), α > − 1, β > − 1. The domain in the parameter space (α, β) in which the inequality holds for all n ≥ 1, conjectured by us, is shown here to require a small adjustment—the deletion of a very narrow lens-shaped region in the square { − 1 < α < − 1/2,  − 1/2 < β < 0}.