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Characterizations of r-Convex Functions

  • Y. X. ZhaoEmail author
  • S. Y. Wang
  • L. Coladas Uria
Article

Abstract

This paper discusses some properties of r-convexity and its relations with some other types of convexity. A characterization of convex functions in terms of r-convexity is given without assuming differentiability. The concept of strict r-convexity is introduced. For a twice continuously differentiable function f, it is shown that the strict r-convexity of f is equivalent to a certain condition on 2 f. Further, it is shown that this condition is satisfied by quasiconvex functions satisfying a less stringent condition.

Keywords

Strict r-convexity r-convexity Positive-semidefinite matrices Positive-definite matrices Convex functions Quasiconvex functions 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institute of Systems Science, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  2. 2.Department of Statistics and Operations Research, Faculty of MathematicsSantiago de Compostela UniversitySantiago de CompostelaSpain

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