On sub- and superadditive vector-valued maps with applications to group majorization

In this paper, motivated by results in [S.S. Dragomir, Quasilinearity of some functionals associated with monotonic convex functions, RGMIA Res. Rep. Coll. 14 (2011), Artcle 24, 13 pp.] and [M. Niezgoda, Cone orderings, group majorizations and similarly separable sequences, Linear Algebra Appl. 436 (3) (2012) pp. 579–594], we investigate maps involving quotient with an ≺1-subadditive operator Φ and a positive superadditive functional φ. We pay special attention to vector-valued operators Φ. We study the subadditivity of the map with (≺1-increasing) convex f. Next, we give applications for group majorization preorders. We establish some preorder-preservation results for the map with a group majorization ≺G and a combination ≺1,2 of preorders ≺1 and ≺2. In consequence, we show that the function is ≺G-increasing.