- Research Article
- Open Access
Optimal Power Mean Bounds for the Weighted Geometric Mean of Classical Means
© B.-Y. Long and Y.-M. Chu. 2010
- Received: 13 November 2009
- Accepted: 25 February 2010
- Published: 8 March 2010
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.
- Simple Computation
- Sharp Bound
- Double Inequality
Recently, the power mean has been the subject of intensive research. In particular, many remarkable inequalities for can be found in literatures [1–12]. It is well known that is continuous and increasing with respect to for fixed and . Let , and be the classical arithmetic, geometric, and harmonic means of two positive numbers and , respectively. Then
In , Mao proved
The following sharp bounds for and in terms of power mean are proved in :
In order to establish our main results we need the following lemma.
Therefore, Lemma 2.1(2) follows from (2.1)–(2.3) and (2.5) together with (2.8).
This work is partly supported by the National Natural Science Foundation of China (Grant no. 60850005) and the Natural Science Foundation of Zhejiang Province (Grant no. D7080080, Y607128).
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