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 $k \ge 6, \alpha, \beta \ge \frac{ 1+ \sqrt{ 2}}{ 4}.$
Categories: math.CA