Abstract
For , the power mean of order of two positive numbers and is defined by . In this paper, we establish two sharp inequalities as follows: and for all . Here and denote the geometric mean and harmonic mean of and respectively.
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1. Introduction
For , the power mean of order of two positive numbers and is defined by
Recently, the power mean has been the subject of intensive research. In particular, many remarkable inequalities for can be found in literature [1–12]. It is well known that is continuous and increasing with respect to for fixed and . If we denote by and the arithmetic mean, geometric mean and harmonic mean of and , respectively, then
In [13], Alzer and Janous established the following sharp double-inequality (see also [14,page 350]):
for all
In [15], Mao proved
for all , and is the best possible lower power mean bound for the sum .
The purpose of this paper is to answer the questions: what are the greatest values and , and the least values and , such that and for all ?
2. Main Results
Theorem 2.1.
for all , equality holds if and only if , and is the best possible lower power mean bound for the sum .
Proof.
If , then we clearly see that .
If and , then simple computation leads to
Next, we prove that is the best possible lower power mean bound for the sum .
For any and , one has
where .
Let , then the Taylor expansion leads to
Equations (2.2) and (2.3) imply that for any there exists , such that for .
Remark 2.2.
For any , one has
Therefore, is the best possible upper power mean bound for the sum .
Theorem 2.3.
for all , equality holds if and only if , and is the best possible lower power mean bound for the sum .
Proof.
If , then we clearly see that
If and , then elementary calculation yields
Next, we prove that is the best possible lower power mean bound for the sum .
For any and , one has
where .
Let , then the Taylor expansion leads to
Equations (2.6) and (2.7) imply that for any there exists , such that
for .
Remark 2.4.
For any , one has
Therefore, is the best possible upper power mean bound for the sum .
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Acknowledgments
This research is partly supported by N S Foundation of China under Grant 60850005 and the N S Foundation of Zhejiang Province under Grants Y7080185 and Y607128.
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Chu, YM., Xia, WF. Two Sharp Inequalities for Power Mean, Geometric Mean, and Harmonic Mean. J Inequal Appl 2009, 741923 (2009). https://doi.org/10.1155/2009/741923
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DOI: https://doi.org/10.1155/2009/741923