Particular cases of analytic extension of boundary values

Abstract

Tests are described for determining whether certain functions are the boundary values of analytic functions taking the upper half plane into itself. In one theorem the functions are real-valued and defined on an open interval and a single point. In other theorems, the functions are defined on a real interval taking values in an arc γ in the upper half plane. In every case these results are closely related to Loewner's theorem in which the putative boundary values are required to be real-valued. As in Loewner's theory, the quadratic inequalities are finite sums involving at most the value of the function and its first derivative at points. These results are closely related to the Pick-Nevanlinna inequalities for functions defined on a sequence of points in the upper half plane; the Pick-Nevanlinna inequalities characterize those functions that are restrictions of analytic functions taking the upper half plane into itself.