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Generalized convex functions on normal decomposition systems

  • Marek NiezgodaEmail author
Article

Abstract

In this paper, the class of weakly \( \psi \)-uniformly convex functions is introduced and studied. Such functions are defined on a convex subcone D of a linear space V endowed with the structure of a normal decomposition system \( (V,G,(\cdot )_\downarrow ) \). In particular, superquadratic functions and c-strongly convex functions are recovered as special cases. Some results in Abramovich et al. (J Math Anal Appl 327: 1444–1460, 2007), intended for multidimensional superquadratic functions defined on \({{\mathbb {R}}_{+}^{m}}\), are extended to the case of weakly \( \psi \)-uniformly convex functions. The concept of \(\psi \)-almost superadditivity of functions is presented and applied. Interpretations of the obtained results are demonstrated for some matrix normal decomposition systems.

Keywords

Majorization Group majorization Normal decomposition system Eaton triple Convex function Weakly \(\psi \)-uniformly convex function Superquadratic function c-strongly convex function 

Mathematics Subject Classification

26B25 26D10 15A18 15A45 

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Computer ScienceUniversity of Life Sciences in LublinLublinPoland

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