On Ky Fan’s Inequality and Its Additive Analogue

Let xig 0, 1r2x is1, . . . , n.be real numbers. If Anand Gn respectively, AXn and GnX . denote the weighted arithmetic and geometric means of x1, . . . , xn respectively, 1yx1, . . . , 1yxn., then AX 1yGX qAX n X X n n F1q2 AyG.F GnX n n 1yAXnqGnX 1yGnqAn An F1q2 AnyGn.F1yAnqGnFGn, ). with equality holding if and only if x1s ??? sxn. The inequalities ). provide refinements of Ky Fan’s inequality AX rGX n nFAnrGn and its additive analogue AXyGXn nFAnyGn.