Optimal Power Mean Bounds for the Weighted Geometric Mean of Classical Means

For p. R, the power mean of order p of two positive numbers a and b is defined by M-p(a, b) = ((a(p) + b(p))/2)(1/p), for p not equal 0, and M-p(a, b) = root ab, for p = 0. In this paper, we answer the question: what are the greatest value p and the least value q such that the double inequality M-p(a, b) <= A(alpha)(a, b)G beta(a, b)H1-alpha-beta(a, b) <= M-q(a, b) holds for all a, b > 0 and alpha, beta > 0 with alpha + beta < 1? Here A(a, b) = (a + b) / 2, G(a, b) = root ab, and H(a, b) = 2ab/(a + b) denote the classical arithmetic, geometric, and harmonic means, respectively.