An upper bound on Jacobi polynomials

Abstract

Let ${\bf P}_k^{(α, β)} (x)$ be an orthonormal Jacobi polynomial of degree $k.$ We will establish the following inequality \begin{equation*} \max_{x \in [δ_{-1},δ_1]}\sqrt{(x- δ_{-1})(δ_1-x)} (1-x)^α(1+x)^β ({\bf P}_{k}^{(α, β)} (x))^2 < \frac{3 \sqrt{5}}{5}, \end{equation*} where $δ_{-1}<δ_1$ are appropriate approximations to the extreme zeros of ${\bf P}_k^{(α, β)} (x) .$ As a corollary we confirm, even in a stronger form, T. Erdélyi, A.P. Magnus and P. Nevai conjecture [Erdélyi et al., Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994), 602-614], by proving that \begin{equation*} \max_{x \in [-1,1]}(1-x)^{α+{1/2}}(1+x)^{β+{1/2}}({\bf P}_k^{(α, β)} (x))^2 < 3 α^{1/3} (1+ \fracα{k})^{1/6}, \end{equation*} in the region $k \ge 6, α, β\ge \frac{1+ \sqrt{2}}{4}.$