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

We study an extremal problem related to “splitted” Jacobi weights: for α,β>0, find the largest value of maxx∈[−1,1] [(1+x)βpm(x)2+(1−x)αqn(x)2] among all polynomials pm and qn of degree at most m and n, respectively, satisfying∫−11[(1+x)βpm(x)2+(1−x)αqn(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.