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.
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