Inequalities for Jacobi polynomials

A Bernstein type inequality is obtained for the Jacobi polynomials $P_n^{\alpha,\beta}(x)$, which is uniform for all degrees $n\ge0$, all real $\alpha,\beta\ge0$, and all values $x\in [-1,1]$. It provides uniform bounds on a complete set of matrix coefficients for the irreducible representations of $\mathrm{SU}(2)$ with a decay of $d^{-1/4}$ in the dimension $d$ of the representation. Moreover it complements previous results of Krasikov on a conjecture of Erd\'elyi, Magnus and Nevai.