Cone orderings, group majorizations and similarly separable vectors

We generalize some results on majorization in papers by Wu and Debnath [S. Wu, L. Debnath, Inequalities for convex sequences and their applications, Comput. Math. Appl. 54 (2007) 525–534] and Marshall et al. [A.W. Marshall, I. Olkin, F. Proschan, Monotonicity of ratios of means and other applications of majorization, in: O. Shisha (Ed.), Inequalities, Academic Press, New York, 1967, pp. 177–190]. We present sufficient conditions for some vector inequalities to hold in the case of cone orderings and group-induced cone orderings. The framework used is based on results for similarly separable vectors given in paper [M. Niezgoda, Bifractional inequalities and convex cones, Discrete Math. 306 (2006) 231–243].