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

  • Jerry R. MuirJr.Email author
Part of the Springer INdAM Series book series (SINDAMS, volume 26)


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 As part of this application, we present a proof of an unpublished result of T.J. Suffridge that generalizes a classical result of Marx and Strohhäcker for convex mappings of the disk.


Herglotz formula Carathéodory class Convex mappings 

2010 Mathematics Subject Classification

Primary 32H02 32A26; Secondary 30C45 


  1. 1.
    Brickman, L., MacGregor, T.H., Wilken, D.R.: Convex hulls of some classical families of univalent functions. Trans. Am. Math. Soc. 156, 91–107 (1971)Google Scholar
  2. 2.
    Graham, I., Kohr, G.: Geometric Function Theory in One and Higher Dimensions. Marcel Dekker, New York (2003)Google Scholar
  3. 3.
    Marx, A.: Untersuchungen über schlichte Abbildungen. Math. Ann. 107, 40–67 (1932)Google Scholar
  4. 4.
    Muir, Jr., J.R.: A Herglotz-type representation for vector-valued holomorphic mappings on the unit ball of \({\mathbb C}^n\). J. Math. Anal. Appl. 440, 127–144 (2016)Google Scholar
  5. 5.
    Roper, K.A., Suffridge, T.J.: Convexity properties of holomorphic mappings in \({\mathbb C}^n\). Trans. Am. Math. Soc. 351, 1803–1833 (1999)Google Scholar
  6. 6.
    Rudin, W.: Function Theory in the Unit Ball of \(\mathbb {C}^n\). Springer, New York (1980)Google Scholar
  7. 7.
    Strohhäcker, E.: Beitrage zur Theorie der schlichten Funktionen. Math. Z. 37, 356–380 (1933)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball. Springer, New York (2005)zbMATHGoogle Scholar

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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