Extremal solutions of Nevanlinna-Pick problems and certain classes of inner functions

Consider a scaled Nevanlinna-Pick interpolation problem and let a be the Blaschke product whose zeros are the nodes of the problem. It is proved that if a belongs to a certain class of inner functions, then the extremal solutions of the problem or most of them are in the same class. Three different classical classes are considered: inner functions whose derivative is in a certain Hardy space, exponential Blaschke products and the well-known class of alpha-Blaschke products, for 0 < alpha < 1.