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Optimal Power Mean Bounds for the Weighted Geometric Mean of Classical Means
Journal of Inequalities and Applications volume 2010, Article number: 905679 (2010)
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.
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%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_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%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ3_HTML.gif)
for all .
In [15], Mao proved
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ4_HTML.gif)
for all , and
is the best possible lower power mean bound for the sum
.
The following sharp bounds for and
in terms of power mean are proved in [16]:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ5_HTML.gif)
for all .
The purpose of this paper is to answer the question: what are the greatest value and the least value
such that the double inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ6_HTML.gif)
holds for all and
with
?
2. Main Result
In order to establish our main results we need the following lemma.
Lemma 2.1.
If ,
and
, then
() for
and
;
() for
and
.
Proof.
Simple computations lead to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ7_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ8_HTML.gif)
where :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ9_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ10_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ11_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ12_HTML.gif)
()If and
, then (2.6) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ13_HTML.gif)
Therefore, Lemma 2.1() follows from (2.1)–(2.3) and (2.5) together with (2.7).
()If and
, then (2.6) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ14_HTML.gif)
Therefore, Lemma 2.1(2) follows from (2.1)–(2.3) and (2.5) together with (2.8).
Theorem 2.2.
For all and
with
, one has
() for
;
(2) for
, and
for
, each equality occurs if and only if
, and
and
are the best possible power mean bounds for the product
.
Proof.
-
(1)
If
, then simple computations lead to
(2.9)
() If
and
, then we clearly see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ16_HTML.gif)
If and
, without loss of generality, we assume that
. Let
and
, then
, and simple computations lead to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ17_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ18_HTML.gif)
Therefore, for
follows from (2.11) and Lemma 2.1(1) together with (2.12), and
for
follows from (2.11) and Lemma 2.1(2) together with (2.12).
Next, we prove that and
are the best possible power mean bounds for the product
.
Firstly, we prove that is the best possible upper power mean bound for the product
if
.
For any and
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ19_HTML.gif)
Let , then the Taylor expansion leads to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ20_HTML.gif)
Equations (2.13) and (2.14) imply that if , then for any
there exists
, such that
for
.
Secondly, we prove that is the best possible lower power mean bound for the product
if
.
For any and
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ21_HTML.gif)
From (2.15) and , we clearly see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ22_HTML.gif)
Equation (2.16) implies that if , then for any
there exists
, such that
for
.
Thirdly, we prove that is the best possible lower power mean bound for the product
if
.
For any and
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ23_HTML.gif)
where −
.
Let , then the Taylor expansion leads to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ24_HTML.gif)
Equations (2.17) and (2.18) imply that if , then for any
there exists
, such that
for
.
Finally, we prove that is the best possible upper power mean bound for the product
if
.
For any and
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ25_HTML.gif)
From (2.19) and we clearly see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F905679/MediaObjects/13660_2009_Article_2295_Equ26_HTML.gif)
Equation (2.20) implies that if , then for any
there exists
, such that
for
.
References
Wu S: Generalization and sharpness of the power means inequality and their applications. Journal of Mathematical Analysis and Applications 2005, 312(2):637–652. 10.1016/j.jmaa.2005.03.050
Richards KC: Sharp power mean bounds for the Gaussian hypergeometric function. Journal of Mathematical Analysis and Applications 2005, 308(1):303–313. 10.1016/j.jmaa.2005.01.018
Wang WL, Wen JJ, Shi HN: Optimal inequalities involving power means. Acta Mathematica Sinica 2004, 47(6):1053–1062.
Hästö PA: Optimal inequalities between Seiffert's mean and power means. Mathematical Inequalities & Applications 2004, 7(1):47–53.
Alzer H: A power mean inequality for the gamma function. Monatshefte für Mathematik 2000, 131(3):179–188. 10.1007/s006050070007
Alzer H, Qiu S-L: Inequalities for means in two variables. Archiv der Mathematik 2003, 80(2):201–215. 10.1007/s00013-003-0456-2
Tarnavas CD, Tarnavas DD: An inequality for mixed power means. Mathematical Inequalities & Applications 1999, 2(2):175–181.
Bukor J, Tóth J, Zsilinszky L: The logarithmic mean and the power mean of positive numbers. Octogon Mathematical Magazine 1994, 2(1):19–24.
Pečarić JE: Generalization of the power means and their inequalities. Journal of Mathematical Analysis and Applications 1991, 161(2):395–404. 10.1016/0022-247X(91)90339-2
Chen J, Hu B: The identric mean and the power mean inequalities of Ky Fan type. Facta Universitatis. Series: Mathematics and Informatics 1989, (4):15–18.
Imoru CO: The power mean and the logarithmic mean. International Journal of Mathematics and Mathematical Sciences 1982, 5(2):337–343. 10.1155/S0161171282000313
Lin TP: The power mean and the logarithmic mean. The American Mathematical Monthly 1974, 81: 879–883. 10.2307/2319447
Alzer H, Janous W: Solution of problem . Crux Mathematicorum 1987, 13: 173–178.
Bullen PS, Mitrinović DS, Vasić PM: Means and Their Inequalities, Mathematics and Its Applications. Volume 31. D. Reidel, Dordrecht, The Netherlands; 1988:xx+459.
Mao Q-J: Power mean, logarithmic mean and Heronian dual mean of two positive number. Journal of Suzhou College of Education 1999, 16(1–2):82–85.
Chu Y-M, Xia W-F: Two sharp inequalities for power mean, geometric mean, and harmonic mean. Journal of Inequalities and Applications 2009, 2009:-6.
Acknowledgments
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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Long, BY., Chu, YM. Optimal Power Mean Bounds for the Weighted Geometric Mean of Classical Means. J Inequal Appl 2010, 905679 (2010). https://doi.org/10.1155/2010/905679
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DOI: https://doi.org/10.1155/2010/905679