Chebyshev-type quadrature and zeros of Faber polynomials

With any probability measure μ on [−1, 1] we associate a sequence of polynomials Fn(z) which are Faber polynomials of a univalent function F(z) on ¦z¦ > 1. If the zeros of Fn(z) are in the open unit disk then there exists a Chebyshev-type quadrature formula for μ with n nodes which is exact for all polynomials f(t) up to degree n − 1. e normalized Jacobi measures dμ(t) = Cλ(1 − t)−12−λ(1 + t)−12dt with λ < 12 the function F(z) can be expressed in terms of hypergeometric functions. Using this expression it is proved that the zeros of the associated Faber polynomials are in the open unit disk in case λ ∈ (0, λ0] for some λ0 > 0. This result solves to a large extent a problem of Förster.