The Optimal Convex Combination Bounds of Arithmetic and Harmonic Means for the Seiffert's Mean

Abstract

We find the greatest value α and least value β such that the double inequality αA(a,b)+(1-α)H(a,b)<P(a,b)<βA(a,b)+(1-β)H(a,b) holds for all a,b>0 with a≠b. Here A(a,b), H(a,b), and P(a,b) denote the arithmetic, harmonic, and Seiffert's means of two positive numbers a and b, respectively.