On Erdélyi-Magnus-Nevai conjecture for Jacobi polynomials

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T. Erdélyi, A.P. Magnus and P. Nevai conjectured that for α, β≥ - 1/2, the orthonormal Jacobi polynomials Pₖ⁽α, β⁾ (x) satisfy the inequality equation* ₓ ∈ [₋₁,₁](1-x)α+1/2(1+x)β+1/2(Pₖ⁽α, β⁾ (x))² =O ({1,(α²+β²)¹/⁴ }), equation* [Erdélyi 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 α= β≥ 1+ √24, even in a stronger form by giving very explicit upper bounds. We also show that equation* √δ²-x² (1-x²)α(P₂ₖ⁽α, α⁾ (x))² < 2π (1+ 1/8(2k+ α)²) equation* for a certain choice of δ, such that the interval (- δ, δ) contains all the zeros of P₂ₖ⁽α, α⁾ (x). Slightly weaker bounds are given for polynomials of odd degree.