Journal of Convex Analysis, Vol. 5, No. 1, pp. 81-105 (1998)

On Schur-Ostrowski Type Theorems for Group Majorizations

Marek Niezgoda

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

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.

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

Classification (MSC2000): 26A48, 26A51; 20H15, 51F15

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