Mathematica Eterna
Open Access

Abstract

Optimal Convex Combination Bounds of the weighted geometric and Harmonic Means for the Centroidal Mean

Shaoqin Gao and Lingling Song

We find the greatest value α and the least value β such that the double inequality αH(a, b) + (1 − α)S(a, b) < T(a, b) < βH(a, b) + (1 − β)S(a, b) holds for all a, b > 0 with a 6= b. Here S(a, b) denotes the weighted geometric mean of a and b with weights a a+b and b a+b , T(a, b) and H(a, b) denote the Centroidal and harmonic means of two positive numbers a and b, respectively.