Boundary Interpolation by Finite Blaschke Products

Given 2n distinct points z1, z′ 1, z2, z ′ 2, . . . , zn, z ′ n (in this order) on the unit circle, and n points w1, . . . , wn on the unit circle, we show how to construct a Blaschke product B of degree n such that B(zj) = wj for all j and, in addition, B(z′ j) = B(z ′ k) for all j and k. Modifying this example yields a Blaschke product of degree n− 1 that interpolates the zj ’s to the wj ’s. We present two methods for constructing our Blaschke products: one reminiscent of Lagrange’s interpolation method and the second reminiscent of Newton’s method. We show that locating the zeroes of our Blaschke product is related to another fascinating problem in complex analysis: the Sendov conjecture. We use this fact to obtain estimates on the location of the zeroes of the Blaschke product.