On Erd\'{e}lyi-Magnus-Nevai conjecture for Jacobi polynomials

T. Erd\'{e}lyi, A.P. Magnus and P. Nevai conjectured that for $\alpha, \beta \ge - {1/2} ,$ the orthonormal Jacobi polynomials ${\bf P}_k^{(\alpha, \beta)} (x)$ satisfy the inequality \begin{equation*} \max_{x \in [-1,1]}(1-x)^{\alpha+{1/2}}(1+x)^{\beta+{1/2}}({\bf P}_k^{(\alpha, \beta)} (x) )^2 =O (\max \left\{1,(\alpha^2+\beta^2)^{1/4} \right\}), \end{equation*} [Erd\'{e}lyi et al.,Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994), 602-614]. Here we will confirm this conjecture in the ultraspherical case $\alpha = \beta \ge \frac{1+ \sqrt{2}}{4},$ even in a stronger form by giving very explicit upper bounds. We also show that \begin{equation*} \sqrt{\delta^2-x^2} (1-x^2)^{\alpha}({\bf P}_{2k}^{(\alpha, \alpha)} (x))^2 < \frac{2}{\pi} (1+ \frac{1}{8(2k+ \alpha)^2} ) \end{equation*} for a certain choice of $\delta,$ such that the interval $(- \delta, \delta)$ contains all the zeros of ${\bf P}_{2k}^{(\alpha, \alpha)} (x).$ Slightly weaker bounds are given for polynomials of odd degree.