On Schur-Ostrowski Type Theorems for Group Majorizations

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.