A characterization of Herglotz-Nevanlinna functions in two variables via integral representations

Abstract

We derive an integral representation for Herglotz-Nevanlinna functions in two variables which provides a complete characterization of this class in terms of a real number, two non-negative numbers and a positive measure satisfying certain conditions. Further properties of the representing measures are discussed.

15 pages, 2 figures, accepted for publication in Arkiv för Matematik

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• Geometric properties of measures related to holomorphic functions having positive imaginary or real part
• A subclass of boundary measures and the convex combination problem for Herglotz-Nevanlinna functions in several variables
• On the structure of Nevanlinna measures
• Characterizations of Herglotz-Nevanlinna functions using positive semi-definite functions and the Nevanlinna kernel in several variables
• Characterizations of the Lebesgue measure and product measures related to holomorphic functions having non-negative imaginary or real part