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 Mp a, b a b /2 , for p / 0, and Mp a, b √ 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 Mp a, b ≤ A a, b G a, b H1−α−β a, b ≤ Mq a, b holds for all a, b > 0 and α, β > 0 with α β < 1? Here A a, b a b /2, G a, b √ ab, and H a, b 2ab/ a b denote the classical arithmetic, geometric, and harmonic means, respectively.