Some inequalities for arithmetic and geometric means

Let A n and G n (respectively, A ′ n and G ′ n ) be the weighted arithmetic and geometric means of x 1 , …, x n (respectively, 1 – x 1 , …, 1 – x n ). We present sharp upper and lower bounds for the differences and . And we determine the best possible constants r and s such that holds for all x i ∈ [ a, b ] ( i = 1, …, n ; 0 < a < b < 1). Our theorems extend and sharpen results of Fan, Cartwright and Field, McGregor and the author.