Research Article Schur-Convexity of Two Types of One-Parameter Mean Values in n Variables

and let dμ= du1, . . . ,dun−1 be the differential of the volume in En−1. The weighted arithmetic mean A(x,u) and the power mean Mr(x,u) of order r with respect to the numbers x1,x2, . . . ,xn and the positive weights u1,u2, . . . ,un with ∑n i=1ui = 1 are defined, respectively, as A(x,u) = ∑ni=1uixi, Mr(x,u) = (∑ni=1uixr i ) for r =0, and M0(x,u)= ∏n i=1x ui i . For u=(1/n,1/n, . . . ,1/n), we denoteA(x,u) Δ =A(x),Mr(x,u) Δ =Mr(x). The well-known logarithmic mean L(x1,x2) of two positive numbers x1 and x2 is