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.
Highlights
For p ∈ R, the power mean of order p of two positive numbers a and b is defined by
We prove that M2α β−1 a, b is the best possible upper power mean bound for the product Aα a, b Gβ a, b H1−α−β a, b if 2α β > 1
We prove that M0 a, b is the best possible lower power mean bound for the product Aα a, b Gβ a, b H1−α−β a, b if 2α β > 1
Summary
Copyright q 2010 B.-Y. Long and Y.-M. Chu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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