Hyperbolically Convex Functions

  • Diego Mejía
  • Christian Pommerenke
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 10)


A conformal map f of the unit disk D of the complex plane into itself is called hyperbolically convex if the hyperbolic segment between any two points of f (D) also lies in f (D). These functions form a non-linear space invariant under Moebius transformations of D onto itself. The fact that this space is non-linear makes it impossible to use many of the standard methods.

This survey talk will concentrate on
  • Analytic characterizations of h-convex functions

  • Inequalities for h-convex functions

  • Hausdorff dimension of image sets

A few proofs will be sketched.


Hausdorff Dimension Fuchsian Group Hyperbolic Geometry Analytic Characterization Jordan Domain 


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Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • Diego Mejía
    • 1
  • Christian Pommerenke
    • 2
  1. 1.Departamento de MatemáticasUniversidad NacionalMedellínColombia
  2. 2.Fachbereich Mathematik, 8-2Technische UniversitätBerlinGermany

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