The problem of estimating certain functionals defined on a group of linear operators generating a group induced cone (GIC) ordering is studied. A result of A. Berman and R. J. Plemmons [Math. Inequal. Appl. 1, No. 1, 149–152 (1998; Zbl 0889.15009)] is extended from the sum function to Schur-convex functions. It is shown that the problem has a closed connection with both Schur type inequality and weak group majorization. Some applications are given for matrices.