On a conjectured inequality for the largest zero of Jacobi polynomials

Abstract

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}.