Chebyshev polynomials for disjoint compact sets

We are concerned with the problem of finding among all polynomials of degree n with leading coefficient 1, the one which has minimal uniform norm on the union of two disjoint compact sets in the complex plane. Our main object here is to present a class of disjoint sets where the best approximation can be determined explicitly for all n. A closely related approximation problem is obtained by considering all polynomials that have degree no larger than n and satisfy an interpolatory constraint. Such problems arise in certain iterative matrix computations. Again, we discuss a class of disjoint compact sets where the optimal polynomial is explicitly known for all n.