On the maximum value of Jacobi polynomials

A remarkable inequality, with utterly explicit constants, established by Erdélyi, Magnus, and Nevai, states that for α⩾β>-12, the orthonormal Jacobi polynomials Pk(α,β)(x) satisfymax|x|⩽1(1-x)α+1/2(1+x)β+1/2Pk(α,β)(x)2=O(α)[Erdélyi et al., Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994), 602-614]. They conjectured that the real order of the maximum is O(α1/2). Here we will make half a way towards this conjecture by proving a new inequality which improves their result by a factor of order (1α+1k)-1/3. We also confirm the conjecture, even in a stronger form, in some limiting cases.