2010•Journal of Inequalities and ApplicationsOpen access

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

Bo-Yong Long, Yu‐Ming Chu

Abstract

For , the power mean of order of two positive numbers and is defined by , for , and , for . In this paper, we answer the question: what are the greatest value and the least value such that the double inequality holds for all and with ? Here , , and denote the classical arithmetic, geometric, and harmonic means, respectively.

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Key concepts: Mathematics, Harmonic mean, Geometric mean, Weighted geometric mean, Mean value, Inequality of arithmetic and geometric means, Generalized mean, Weighted arithmetic mean