The Erdélyi--Magnus--Nevai and Krasikov Conjectures for Jacobi Polynomials

Let $p_n^{(α,β)}$ denote the Jacobi polynomial orthonormal for the weight $(1-x)^α(1+x)^β$ on $[-1,1]$, where $α,β\ge-1/2$, and put $S=α+β+1$. We prove the uniform degree--parameter estimate $$ (1-x)^{α+1/2}(1+x)^{β+1/2} \bigl|p_n^{(α,β)}(x)\bigr|^2 \le C\max\bigl\{1,S^{1/3},S^{1/2}(n+1)^{-1/6