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Optimal Inequalities for Generalized Logarithmic, Arithmetic, and Geometric Means
Journal of Inequalities and Applications volume 2010, Article number: 806825 (2010)
Abstract
For , the generalized logarithmic mean
, arithmetic mean
, and geometric mean
of two positive numbers
and
are defined by
, for
,
, for
,
, and
,
, for
, and
,
, for
, and
,
, and
, respectively. In this paper, we find the greatest value
(or least value
, resp.) such that the inequality
(or
, resp.) holds for
(or
, resp.) and all
with
.
1. Introduction
For , the generalized logarithmic mean
and power mean
with parameter
of two positive numbers
and
are defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ1_HTML.gif)
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ2_HTML.gif)
respectively. It is well known that both means are continuous and increasing with respect to for fixed
and
. Recently, both means have been the subject of intensive research. In particular, many remarkable inequalities involving
and
can be found in the literature [1–9]. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ3_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_IEq40_HTML.gif)
, and be the arithmetic, identric, logarithmic, geometric, and harmonic means of two positive numbers
and
, respectively. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ4_HTML.gif)
for all .
In [10], Carlson proved that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ5_HTML.gif)
for all with
.
The following inequality is due to Sándor [11, 12]:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ6_HTML.gif)
In [13], Lin established the following results: (1) implies that
for all
with
; (2)
implies that
for all
with
; (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
with
.
Pittenger [14] established that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ7_HTML.gif)
for all , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ8_HTML.gif)
Here, and
are sharp and inequality (1.7) becomes equality if and only if
or
or
. The case
reduces to Lin's results [13]. Other generalizations of Lin's results were given by Imoru [15].
Recently, some monotonicity results of the ratio between generalized logarithmic means were established in [16–18].
The aim of this paper is to prove the following Theorem 1.1.
Theorem 1.1.
Let and
with
, then
(1) for
;
(2) for
, and
for
, moreover, in each case, the bound
for the sum
is optimal.
2. Proof of Theorem 1.1
In order to prove our Theorem 1.1 we need a lemma, which we present in this section.
Lemma 2.1.
For and
one has
(1)If , then
for
;
(2)If , then
for
,
for
, and
for
.
Proof.
-
(1)
If
, then we clearly see that
(2.1)
for .
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ10_HTML.gif)
for .
Therefore, Lemma 2.1() follows from (2.1) and (2.2).
-
(2)
If
, then
(2.3)
Therefore, Lemma 2.1(2) follows from (2.3).
Proof of Theorem 1.1.
Proof.
() If
, then (1.1) leads to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ12_HTML.gif)
() We divide the proof into two cases.
Case 1.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_IEq105_HTML.gif)
or . From inequalities (1.5) and (1.6) we clearly see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ13_HTML.gif)
for , and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ14_HTML.gif)
for .
Case 2.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_IEq109_HTML.gif)
. Without loss of generality, we assume that . Let
, then (1.1) leads to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ15_HTML.gif)
Let , then simple computations yield
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ16_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ17_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ18_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ19_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ20_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ21_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ22_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ23_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ24_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ25_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ26_HTML.gif)
where is defined as in Lemma 2.1.
We divide the proof into five subcases.
Subcase 2 A.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_IEq114_HTML.gif)
. From (2.18) and Lemma 2.1() we clearly see that
for
and
for
, then we know that
is strictly decreasing in
and strictly increasing in
. Now from the monotonicity of
and (2.17) together with the fact that
we clearly see that
for
, then from (2.7)–(2.15) and
for
we get
for
.
Subcase 2 B.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_IEq131_HTML.gif)
. Then (2.18) and Lemma 2.1(1) lead to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ27_HTML.gif)
for .
From (2.7)–(2.17) and (2.19) together with the fact that for
we know that
for
.
Subcase 2 C.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_IEq137_HTML.gif)
. Then (2.18) and Lemma 2.1(1) imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ28_HTML.gif)
for .
From (2.7)–(2.17), (2.20) and for
we know that
for
.
Subcase 2 D.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_IEq143_HTML.gif)
. Then (2.19) again yields, and for
follows from (2.7)–(2.17) and (2.19) together with
.
Subcase 2 E.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_IEq147_HTML.gif)
. Then (2.20) is also true, and for
follows from (2.7)–(2.17), (2.20) and the fact that
.
Next, we prove that the bound for the sum
is optimal in each case. The proof is divided into six cases.
Case 1.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_IEq153_HTML.gif)
. For any and
, then (1.1) leads to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ29_HTML.gif)
where
Let making use of Taylor expansion, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ30_HTML.gif)
Equations (2.21) and (2.22) imply that for any , there exists
, such that
for any
and
.
Case 2.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_IEq163_HTML.gif)
. For any and
, from (1.1) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ31_HTML.gif)
where
Let making use of Taylor expansion, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ32_HTML.gif)
Equations (2.23) and (2.24) imply that for any , there exists
, such that
for
and
.
Case 3.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_IEq173_HTML.gif)
. For and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ33_HTML.gif)
where
Let making use of Taylor expansion, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ34_HTML.gif)
Equations (2.25) and (2.26) imply that for any and any
, there exists
, such that
for
.
Case 4.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_IEq183_HTML.gif)
. For any and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ35_HTML.gif)
where
Let using Taylor expansion we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ36_HTML.gif)
Equations (2.27) and (2.28) show that for any and any
, there exists
, such that
for
.
Case 5.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_IEq193_HTML.gif)
. For any and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ37_HTML.gif)
where
Let making use of Taylor expansion we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ38_HTML.gif)
Equations (2.29) and (2.30) imply that for any and any
, there exists
, such that
for
.
Case 6.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_IEq203_HTML.gif)
. For any and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ39_HTML.gif)
where
Let , using Taylor expansion we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ40_HTML.gif)
From (2.31) and (2.32) we know that for any and any
, there exists
, such that
for
.
At last, we propose two open problems as follows.
Open Problem 1
What is the least value such that the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ41_HTML.gif)
holds for and all
with
?
Open Problem 2
What is the greatest value such that the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F806825/MediaObjects/13660_2009_Article_2255_Equ42_HTML.gif)
holds for and all
with
?
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Acknowledgments
The authors wish to thank the anonymous referee for their very careful reading of the manuscript and fruitful comments and suggestions. 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 Y607128.
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Long, BY., Chu, YM. Optimal Inequalities for Generalized Logarithmic, Arithmetic, and Geometric Means. J Inequal Appl 2010, 806825 (2010). https://doi.org/10.1155/2010/806825
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DOI: https://doi.org/10.1155/2010/806825