An upper bound on Jacobi polynomials

Abstract

Let ${\bf P}_k^{(\alpha, \beta)} (x)$ be an orthonormal Jacobi polynomial of degree $k.$ We will establish the following inequality \begin{equation*} \max_{x \in [\delta_{-1},\delta_1]}\sqrt{(x- \delta_{-1})(\delta_1-x)} (1-x)^{\alpha}(1+x)^{\beta} ({\bf P}_{k}^{(\alpha, \beta)} (x))^2 < \frac{3 \sqrt{5}}{5}, \end{equation*} where $\delta_{-1}<\delta_1$ are appropriate approximations to the extreme zeros of ${\bf P}_k^{(\alpha, \beta)} (x) .$ As a corollary we confirm, even in a stronger form, T. Erd\'{e}lyi, A.P. Magnus and P. Nevai conjecture [Erd\'{e}lyi 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)^{\alpha+{1/2}}(1+x)^{\beta+{1/2}}({\bf P}_k^{(\alpha, \beta)} (x))^2 < 3 \alpha^{1/3} (1+ \frac{\alpha}{k})^{1/6}, \end{equation*} in the region $k \ge 6, \alpha, \beta \ge \frac{1+ \sqrt{2}}{4}.$