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 α1 and α2 and the least values β1 and β2 such that the double inequalities α1T (a, b) + (1 -α1)H(a, b) < A(a, b) < β1T (a, b) + (1 -β1)H(a, b) and α2T (a, b) + (1 -α2)G(a, b) < A(a, b) < β2T (a, b) + (1 -β2)G(a, b) holds for all a, b > 0 with a ̸ = 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.