Differentiable transformations preserving a cone preordering
The paper is concerned with vector preorderings (e.g., majorization) and real functions that preserve them. Extending to vector-valued functions a result of \textit{A. W. Marshall}, \textit{D. W. Walkup} and \textit{R. J.-B. Wets} [Pac. J. Math. 23, 569-584 (1967; Zbl 0153.38502)], the authors give a necessary and sufficient condition (in terms of directional derivatives) for a differentiable transformation to be isotone with respect to cone preorderings. The result is applied to characterize monotone matrix functions \(A\mapsto A^p\) with integer \(p\geq 2\) and isotone differentiable transformations with respect to \(G\)-majorizations, \(G\) being a finite reflection subgroup of the orthogonal group acting on \(\mathbb{R}^n\).
scientific article; zbMATH DE number 769049
scientific article