An upper bound on Jacobi polynomials

Let Pk(α,β)(x) be an orthonormal Jacobi polynomial of degree kk. We will establish the following inequality:maxx∈[δ-1,δ1](x-δ-1)(δ1-x)(1-x)α(1+x)βPk(α,β)(x)2<355,where δ-1<δ1δ-1<δ1 are appropriate approximations to the extreme zeros of Pk(α,β)(x). As a corollary we confirm, even in a stronger form, T. Erdélyi, A.P. Magnus and P. Nevai conjecture [T. Erdélyi, A.P. Magnus, P. Nevai, Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994) 602–614] by proving thatmaxx∈[-1,1](1-x)α+1/2(1+x)β+1/2Pk(α,β)(x)2<3α1/31+αk1/6in the region k⩾6,α⩾β⩾1+24.