Open Problems Related to a Herglotz-Type Formula for Vector-Valued Mappings

Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 26)

Abstract

We review a recently found Herglotz-type formula that represents mappings in the class \(\mathcal {M}\), the generalization of the Carathéodory class to the open unit ball \({\mathbb B}\) of \({\mathbb C}^n\), as integrals of a fixed kernel with respect to a family of probability measures on \(\partial {\mathbb B}\). Since not every probability measure arises in the representation, there are resulting questions regarding the characterization of the family of representing measures. In a manner similar to how the one-variable Herglotz formula facilitates a representation of convex mappings of the disk, we apply the new transformation to convex mappings of \({\mathbb B}\) and present some additional open questions. As part of this application, we present a proof of an unpublished result of T.J. Suffridge that generalizes a classical result of Marx and Strohhäcker for convex mappings of the disk.

Keywords

Herglotz formula Carathéodory class Convex mappings 

2010 Mathematics Subject Classification

Primary 32H02 32A26; Secondary 30C45 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsThe University of ScrantonScrantonUSA

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