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 As part of this application, we present a proof of an unpublished result of T.J. Suffridge that generalizes a classical result of Marx and Strohhäcker for convex mappings of the disk.
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Muir, J.R. (2017). Open Problems Related to a Herglotz-Type Formula for Vector-Valued Mappings. In: Bracci, F. (eds) Geometric Function Theory in Higher Dimension. Springer INdAM Series, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-73126-1_8
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DOI: https://doi.org/10.1007/978-3-319-73126-1_8
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