Restricted Subadditivity of Homogeneous Means

We study Minkowski’s inequality M xqy.FM x.qM y., x,ygIn,ns2, 3 . . . wf wf wf I:R is an interval.and its reverse for the means M x.swy1 nis1w xi.f xi. , xs x , . . . , x .gIn wf n 1 n / is1f xi. satisfying the generalized homogeneity equation M tx.stM x., xgI n , t , tx gI, is1, . . . , n, ns2, 3, . . . . wf wf i By a result of Acz´el and Dar´oczy w1x, these homogeneous means belong to three classes of two-parameter families of means. For one class the class called Gini means., Minkowski’s inequality or subadditivity. has been studied extensively by Beckenbach w2x, Dresher w4x, Danskin w3x, Losonczi w7, 8x, and P´ales w9x. The aim of this paper is to study the sub- and superadditivity of the other two classes.