Differential analysis of matrix convex functions

Frank Hansen, Jun Tomiyama

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)


We analyze matrix convex functions of a fixed order defined in a real interval by differential methods as opposed to the characterization in terms of divided differences given by Kraus [F. Kraus, Über konvekse Matrixfunktionen, Math. Z. 41 (1936) 18-42]. We obtain for each order conditions for matrix convexity which are necessary and locally sufficient, and they allow us to prove the existence of gaps between classes of matrix convex functions of successive orders, and to give explicit examples of the type of functions contained in each of these gaps. The given conditions are shown to be also globally sufficient for matrix convexity of order two. We finally introduce a fractional transformation which connects the set of matrix monotone functions of each order n with the set of matrix convex functions of the following order n + 1.

Original languageEnglish
Pages (from-to)102-116
Number of pages15
JournalLinear Algebra and Its Applications
Issue number1
Publication statusPublished - 2007 Jan 1


  • Matrix convex function
  • Polynomial

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics


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