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Two Sharp Inequalities for Power Mean, Geometric Mean, and Harmonic Mean
Journal of Inequalities and Applications volume 2009, Article number: 741923 (2009)
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.
1. Introduction
For , the power mean of order
of two positive numbers
and
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F741923/MediaObjects/13660_2009_Article_2003_Equ1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F741923/MediaObjects/13660_2009_Article_2003_Equ2_HTML.gif)
In [13], Alzer and Janous established the following sharp double-inequality (see also [14,page 350]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F741923/MediaObjects/13660_2009_Article_2003_Equ3_HTML.gif)
for all
In [15], Mao proved
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F741923/MediaObjects/13660_2009_Article_2003_Equ4_HTML.gif)
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.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F741923/MediaObjects/13660_2009_Article_2003_IEq37_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F741923/MediaObjects/13660_2009_Article_2003_Equ5_HTML.gif)
Next, we prove that is the best possible lower power mean bound for the sum
.
For any and
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F741923/MediaObjects/13660_2009_Article_2003_Equ6_HTML.gif)
where .
Let , then the Taylor expansion leads to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F741923/MediaObjects/13660_2009_Article_2003_Equ7_HTML.gif)
Equations (2.2) and (2.3) imply that for any there exists
, such that
for
.
Remark 2.2.
For any , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F741923/MediaObjects/13660_2009_Article_2003_Equ8_HTML.gif)
Therefore, is the best possible upper power mean bound for the sum
.
Theorem 2.3.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F741923/MediaObjects/13660_2009_Article_2003_IEq59_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F741923/MediaObjects/13660_2009_Article_2003_Equ9_HTML.gif)
Next, we prove that is the best possible lower power mean bound for the sum
.
For any and
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F741923/MediaObjects/13660_2009_Article_2003_Equ10_HTML.gif)
where .
Let , then the Taylor expansion leads to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F741923/MediaObjects/13660_2009_Article_2003_Equ11_HTML.gif)
Equations (2.6) and (2.7) imply that for any there exists
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F741923/MediaObjects/13660_2009_Article_2003_Equ12_HTML.gif)
for .
Remark 2.4.
For any , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F741923/MediaObjects/13660_2009_Article_2003_Equ13_HTML.gif)
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