Extrait d'une lettre adressée à M. Hermite
The Nevanlinna-Pick problem at the zeros of a Blaschke product B having a solution of norm smaller than one is studied. All its extremal solutions are invertible in the Douglas algebra D generated by B. If B is a finite product of sparse Blaschke products (Newman Blaschke products, Frostman Blaschke products) then so are all the extremal solutions. For a Blaschke product B a formula is given for the number C(B) such that if the NP-problem has a solution of norm smaller than C(B) then all its extremal...
We give a new proof of Beurling’s result related to the equality of the extremal length and the Dirichlet integral of solution of a mixed Dirichlet-Neuman problem. Our approach is influenced by Gehring’s work in space. Also, some generalizations of Gehring’s result are presented.