An Extension Problem for the Coefficients of Riemann Mappings

By a Riemann mapping we mean an analytic function which is defined and univalent on the unit disk. Such a function is said to be normalized if it has value zero and positive derivative at the origin. We are concerned with normalized Riemann mappings of the unit disk into itself. If B(z) is such a function, the operator f(z) into f(B(z)) maps the Dirichlet space contractively into itself. Elementary examples show, however, that other functions also have this property [4], and therefore the property cannot be used to characterize the class. More generally, if B(z) is a normalized Riemann mapping of the unit disk into itself, the operator f(z) into f(B(z)) acts as a contraction on a family of spaces which generalize the Dirichlet space. A theorem of de Branges [3] shows that this property is characteristic of the class of normalized Riemann mappings of the unit disk into itself. We study de Branges’ theorem and its proof for possible generalization to the problem of characterizing initial segments of coefficients of a normalized Riemann mapping of the unit disk into itself. Such a characterization is not known at this time, but we shall pose a problem which we hope may lead in this direction. Given any real number ν, let D be the Krĕın space of generalized power series f(z) = ∑∞ n=1 anz ν+n with finite self-product