Some Optimal Convex Combination Bounds for Arithmetic Mean

In this paper we derive some optimal convex combination bounds related to arithmetic mean. We find the greatest values alpha(1) and alpha(2) and the least values beta(1) and beta(2) such that the double inequalities T-alpha 1 (a, b) + (1 - alpha(1)) H (a, b) < A(a, b) < beta T-1 (a, b) (1 - beta(1))H(a, b) and T-alpha 2 (a, b) + (1 -alpha(2))G(a, b) < A(a, b) < beta T-2(a, b) + (1 - beta 2)G(a, b) holds for all a, b > 0 with a not equal b. Here T (a, b), H (a, b), A(a,b) and G(a,b) denote the second Seiffert, harmonic, arithmetic and geometric means of two positive numbers a and b, respectively.