Abstract

In this paper the problem of the isotonicity of a function with respect to a group majorization is discussed. For a group induced cone ordering the isotonicity is characterized via some conditions of Schur-Ostrowski type. As an application, a characterization of the isotonicity of a quadratic form and a linear form is presented. In addition, it is shown that the conditions are necessary and sufficient for a group majorization to be a group induced cone ordering. In consequence, a finite reflection group is determined by the conditions. Finally, some results on the isotonicity of a vector-valued function are derived. In particular, a necessary and sufficient condition on the matrix majorization — Loewner ordering isotonicity is established for a matrix-valued function. The last extends directly the classical S-O condition.

Author information

Contact details are reproduced from the original publication and may be historical.

Marek Niezgoda

Institute of Applied Mathematics, Agricultural University of Lublin, P.O. Box 158, Akademicka 13, 20-950 Lublin, Poland

Keywords

  • preordering
  • cone preordering
  • majorization
  • group majorization
  • group induced cone ordering
  • convex cone
  • dual cone
  • isotone function
  • Schur convex function
  • fundamental region
  • reflection group
  • Schur-Ostrowski’s condition

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Suggested citation

M. Niezgoda. “On Schur-Ostrowski Type Theorems for Group Majorizations.” Journal of Convex Analysis 5 (1998), No. 1, 81–105. https://doi.org/10.68381/jca05-7

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Published by Heldermann Verlag, 1998. Rights now held by Banach Press LLC.