Schur-convex functions of the $2$nd order on $ \mathbb{R}^n$

In the author's earlier paper [Revyakov M., J. Multivariate Anal. 116 (2013) 25-34] concerning mathematical statistics, a need arose to employ functions called "Schur-convex functions of the $2$nd order with respect to two variables". In the present paper, the class of Schur-convex functions of the $2$nd order in $ n$ variables is introduced. Necessary and sufficient conditions (in the form of analogs of the Sylvester criterion) are established for a function to belong to this class. Examples are given of using Schur-convex functions of the $2$nd order for achieving maximal system reliability on the set of all possible allocations of elements into its subsystems.