On the extreme zeros of Jacobi polynomials

By applying the Euler--Rayleigh methods to a specific representation of the Jacobi polynomials as hypergeometric functions, we obtain new bounds for their largest zeros. In particular, we derive upper and lower bound for $1-x_{nn}^2(λ)$, with $x_{nn}(λ)$ being the largest zero of the $n$-th ultraspherical polynomial $P_n^{(λ)}$. For every fixed $λ>-1/2$, the limit of the ratio of our upper and lower bounds for $1-x_{nn}^2(λ)$ does not exceed $1.6$. This paper is a continuation of [1].

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