An extremal problem and an estimation of the Wronskian of certain Jacobi polynomials

We study an extremal problem related to splitted Jacobi weights: for alpha, beta \u3e 0, find the largest value of max(xis an element of[-1,1]) [(1 + x)(beta) p(m)(x)(2) + (1 - x)(alpha)q(n)(x)(2)] among all polynomials p(m) and q(n) of degree at most m and n, respectively, satisfying integral(-1)(1) [(1 + x)(beta) p(m)(x)(2) + (1 - x)(alpha)q(n)(x)(2)] dx = 1. We show that the solution of this problem is related to an estimation of the Christoffel functions and the Wronskians associated with certain Jacobi polynomials. (C) 2002 Elsevier Science B.V. All rights reserved