A theorem and an algorithm involving muirhead's inequality

Let a, b E Rn be column vectors, and (u, v) be the inner product of vectors u and v on Rn . Let G c GL(n, R) be a compact matrix group. For A E G and a continue function f on G, the integral fG f(A)dA is the invariant integral of the compact group G. In this paper, we study the inequality Vx E Rn G e(Aa,x)dA G e(Ab, x)dA. We prove that the above inequality holds if and only if b E Conv(Ga). This work follows a series of results, that is, Muirhead (1903), Hardy, Littlewood and Po`lya (1932), Rado (1952), Daykin (1971), Kimelfeld (1995) and Schulman (2009). Furthermore, We construct an determining algorithm when G is finite. Compared with other effective algorithms, this one is symbolic and easy to implement on computer.