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Inequalities for Generalized Logarithmic Means
Journal of Inequalities and Applications volume 2009, Article number: 763252 (2010)
Abstract
For , the generalized logarithmic mean
of two positive numbers
and
is defined as
, for
,
, for
,
,
,
, for
,
, and
, for
,
. In this paper, we prove that
, and
for all
, and the constants
, and
cannot be improved for the corresponding inequalities. Here
, and
denote the arithmetic, geometric, and harmonic means of
and
, respectively.
1. Introduction
For , the generalized logarithmic mean
and power mean
of two positive numbers
and
are defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_Equ1_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_Equ2_HTML.gif)
It is well known that and
are continuous and increasing with respect to
for fixed
and
. Let
,
,
,
, and
be the arithmetic, identric, logarithmic, geometric, and harmonic means of
and
, respectively. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_Equ3_HTML.gif)
In [1], the following results are established: (1) implies that
; (2)
implies that
; (3)
implies that there exist
such that
; (4)
implies that there exist
such that
. Hence the question was answered: what are the least value
and the greatest value
such that the inequality
holds for all
?
Stolarsky [2] proved that , with equality if and only if
.
In [3], Pittenger proved that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_Equ4_HTML.gif)
for all , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_Equ5_HTML.gif)
Here ,
are sharp and equality holds only if
or
or
. The case
reduces to Lin's results [1]. Other generalizations of Lin's results were given by Imoru [4].
Qi and Guo [5] established that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_Equ6_HTML.gif)
for all ,
and
. The upper bound in (1.6) is the best possible.
In [6], Chu et al. established the following result:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_Equ7_HTML.gif)
for all , where the
function is the logarithmic derivative of the gamma function.
Recently, some monotonicity results of the ratio between generalized logarithmic means were established in [7–9].
The purpose of this paper is to answer the following questions: what are the greatest values and
, and the least value
such that
,
, and
for all
?
2. Main Results
Theorem 2.1.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_IEq81_HTML.gif)
for all , with inequality if and only if
, and the constant
cannot be improved.
Proof.
If , then from (1.1) we clearly see that
Next, we assume that
and
, and then elementary computations yield
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_Equ8_HTML.gif)
To prove that is the largest number for which the inequality holds, we take
and
, and we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_Equ9_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_Equ10_HTML.gif)
where
Making use of the Taylor expansion, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_Equ11_HTML.gif)
Equations (2.3) and (2.4) imply that for any there exists
such that
for
.
Theorem 2.2.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_IEq97_HTML.gif)
for all , with equality if and only if
, and the constant
cannot be improved.
Proof.
Simple computations yield
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_Equ12_HTML.gif)
Next we prove that is the optimal value for which the inequality holds.
For and
, elementary computations yield
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_Equ13_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_Equ14_HTML.gif)
where
Using Taylor expansion we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_Equ15_HTML.gif)
Equations (2.7) and (2.8) imply that for any there exists
such that
for
.
Theorem 2.3.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_IEq109_HTML.gif)
for all , with equality if and only if
, and the constant
cannot be improved.
Proof.
Form (1.1) we clearly see that if
. If
, then simple computations yield
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_Equ16_HTML.gif)
To show that is the best possible constant for which the inequality holds, let
and
, and then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_Equ17_HTML.gif)
where
Using Taylor expansion we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_Equ18_HTML.gif)
Equations (2.10) and (2.11) imply that for any there exists
such that
for
.
Remark 2.4.
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F763252/MediaObjects/13660_2009_Article_2006_Equ19_HTML.gif)
Therefore, we cannot get inequality for any
and all
Remark 2.5.
It is easy to verify that for all
References
Lin TP: The power mean and the logarithmic mean. The American Mathematical Monthly 1974, 81: 879–883. 10.2307/2319447
Stolarsky KB: The power and generalized logarithmic means. The American Mathematical Monthly 1980,87(7):545–548. 10.2307/2321420
Pittenger AO: Inequalities between arithmetic and logarithmic means. Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika 1980, (678–715):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
Qi F, Guo B-N: An inequality between ratio of the extended logarithmic means and ratio of the exponential means. Taiwanese Journal of Mathematics 2003,7(2):229–237.
Chu YM, Zhang XM, Tang TM: An elementary inequality for psi function. Bulletin of the Institute of Mathematics. Academia Sinica 2008,3(3):373–380.
Li X, Chen C-P, Qi F: Monotonicity result for generalized logarithmic means. Tamkang Journal of Mathematics 2007,38(2):177–181.
Qi F, Chen S-X, Chen C-P: Monotonicity of ratio between the generalized logarithmic means. Mathematical Inequalities & Applications 2007,10(3):559–564.
Chen C-P: The monotonicity of the ratio between generalized logarithmic means. Journal of Mathematical Analysis and Applications 2008,345(1):86–89. 10.1016/j.jmaa.2008.03.071
Acknowledgments
This research is partly supported by N S Foundation of China under Grant 60850005 and N S Foundation of Zhejiang Province under Grants D7080080 and Y7080185.
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Chu, YM., Xia, WF. Inequalities for Generalized Logarithmic Means. J Inequal Appl 2009, 763252 (2010). https://doi.org/10.1155/2009/763252
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DOI: https://doi.org/10.1155/2009/763252