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.
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