Sharp bounds for the difference between the arithmetic and geometric means (Q1930877)

Let \(A_n(x)\), and \(G_n(x)\) denote the arithmetic, resp. geometric means of \(x= (x_1,\dots, x_n)\), where \(x_i> 0\) \((i=1,\dots, n)\), and put \(V= A_n(x)- (A_n(\sqrt{x}))^2\). The double-inequality \({n\over n-1}\cdot V\leq A_n(x)- G_n(x)\leq n\cdot V\) holds true. A weighted version is proved, too. The proof uses essentially Lagrange multipliers.