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An Optimal Double Inequality for Means
Journal of Inequalities and Applications volume 2010, Article number: 578310 (2010)
Abstract
For , the generalized logarithmic mean
, arithmetic mean
and geometric mean
of two positive numbers
and
are defined by
,
;
,
,
,
;
,
,
;
,
,
;
and
, respectively. In this paper, we give an answer to the open problem: for
, what are the greatest value
and the least value
, such that the double inequality
holds for all
?
1. Introduction
For , the generalized logarithmic mean
of two positive numbers
and
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ1_HTML.gif)
It is wellknown that is continuous and increasing with respect to
for fixed
and
. In the recent past, the generalized logarithmic mean has been the subject of intensive research. Many remarkable inequalities and monotonicity results can be found in the literature [1–9]. It might be surprising that the generalized logarithmic mean, has applications in physics, economics, and even in meteorology [10–13].
If we denote by ,
,
,
and
the arithmetic mean, identric mean, logarithmic mean, geometric mean and harmonic mean of two positive numbers
and
, respectively, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ2_HTML.gif)
For , the
th power mean
of two positive numbers
and
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ3_HTML.gif)
In [14], Alzer and Janous established the following sharp double inequality (see also [15], Page 350):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ4_HTML.gif)
for all .
For , Janous [16] found the greatest value
and the least value
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ5_HTML.gif)
for all .
In [17–19] the authors present bounds for and
in terms of
and
.
Theorem A.
For all positive real numbers and
with
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ6_HTML.gif)
The proof of the following Theorem B can be found in [20].
Theorem B.
For all positive real numbers and
with
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ7_HTML.gif)
The following Theorems C–E were established by Alzer and Qiu in [21].
Theorem C.
The inequalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ8_HTML.gif)
hold for all positive real numbers and
with
if and only if
and
.
Theorem D.
Let and
be real numbers with
. If
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ9_HTML.gif)
And if , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ10_HTML.gif)
Theorem E.
For all real numbers and
with
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ11_HTML.gif)
with the best possible parameter .
However, the following problem is still open: for , 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%2F578310/MediaObjects/13660_2010_Article_2190_Equ12_HTML.gif)
holds for all ? The purpose of this paper is to give the solution to this open problem.
2. Lemmas
In order to establish our main result, we need two lemmas, which we present in this section.
Lemma 2.1.
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ13_HTML.gif)
Proof.
Let , then simple computation yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ14_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ15_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ16_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ17_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ18_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ19_HTML.gif)
If , then from (2.7) we clearly see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ20_HTML.gif)
Therefore, Lemma 2.1 follows from (2.3)–(2.6) and (2.8).
Lemma 2.2.
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ21_HTML.gif)
Proof.
Let , then simple computation leads to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ22_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ23_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ24_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ25_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ26_HTML.gif)
If , then from (2.14) we clearly see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ27_HTML.gif)
From (2.10)–(2.13) and (2.15) we know that for
.
3. Main Results
Theorem 3.1.
If , then
for all
, with equality if and only if
, and the constant
in
, cannot be improved.
Proof.
If , then we clearly see that
.
If , without loss of generality, we assume that
. Let
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ28_HTML.gif)
Firstly, we prove . The proof is divided into three cases.
Case 1.
. We note that (1.1) leads to the following identity:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ29_HTML.gif)
From (3.2) and Lemma 2.1 we clearly see that for
and
.
Case 2.
. Equation (1.1) leads to the following identity:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ30_HTML.gif)
From (3.3) and Lemma 2.2 we clearly see that for
and
.
Case 3.
. From (1.1) we have the following identity:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ31_HTML.gif)
Equation (3.4) and elementary computation yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ32_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ33_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ34_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ35_HTML.gif)
If , then (3.8) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ36_HTML.gif)
for . Therefore,
follows from (3.5)–(3.7) and (3.9).
If , then (3.8) leads to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ37_HTML.gif)
for . Therefore,
follows from (3.5)–(3.7) and (3.10).
Next, we prove that the constant in the inequality
cannot be improved. The proof is divided into five cases.
Case 1.
. For any
, let
, then (1.1) leads to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ38_HTML.gif)
where .
Making use of Taylor expansion we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ39_HTML.gif)
Case 2.
. For any
, let
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ40_HTML.gif)
where .
Using Taylor expansion we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ41_HTML.gif)
Case 3.
. For any
, let
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ42_HTML.gif)
where .
Making use of Taylor expansion and elaborated calculation we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ43_HTML.gif)
Case 4.
. For any
, let
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ44_HTML.gif)
where .
Using Taylor expansion and elaborated calculation we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ45_HTML.gif)
Case 5.
. For any
, let
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ46_HTML.gif)
where .
Using Taylor expansion and elaborated calculation we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ47_HTML.gif)
Cases 1–5 show that for any , there exists
, for any
there exists
such that
for
.
Theorem 3.2.
If , then
for all
, with equality if and only if
, and the constant
in
cannot be improved.
Proof.
If , then we clearly see that
.
If , without loss of generality, we assume that
. Let
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ48_HTML.gif)
Firstly, we prove for
. Simple computation leads to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ49_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ50_HTML.gif)
for and
.
From (3.23) we clearly see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ51_HTML.gif)
for .
Since , we have
for
. Therefore,
follows from (3.22) and (3.24).
Next, we prove that the constant cannot be improved.
For any , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F578310/MediaObjects/13660_2010_Article_2190_Equ52_HTML.gif)
Equation (3.25) imply that for any there exists
, such that
for
.
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Acknowledgment
This work was supported by the Natural Science Foundation of Zhejiang Broadcast and TV University (Grant no. XKT-09G21).
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Qian, WM., Zheng, NG. An Optimal Double Inequality for Means. J Inequal Appl 2010, 578310 (2010). https://doi.org/10.1155/2010/578310
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DOI: https://doi.org/10.1155/2010/578310