An extension of Schur-Ostrowski's condition, weak Eaton triples and generalized AI functions.

In this paper, for a given Eaton triple (V , G , D) , we present a characterization of G -increasing real functions by using the maximality property of their directional derivatives. It generalizes Schur-Ostrowski's condition related to Schur-convexity of a function. It is also useful in establishing theorems of von Neumann-Davis type regarding the form of group-invariant real functions with certain kinds of convexity. We introduce and investigate weak Eaton triples and their chains. We show a method for constructing new Eaton triples. We apply it to define generalized arrangement-increasing (AI) functions. We prove that the mentioned derivatives of G -increasing maps are generalized AI functions. [ABSTRACT FROM AUTHOR]