- Research Article
- Open access
- Published:
Some Equivalent Forms of the Arithematic-Geometric Mean Inequality in Probability: A Survey
Journal of Inequalities and Applications volume 2008, Article number: 386715 (2008)
Abstract
We link some equivalent forms of the arithmetic-geometric mean inequality in probability and mathematical statistics.
1. Introduction
The arithmetic-geometric mean inequality (in short, AG inequality) has been widely used in mathematics and in its applications. A large number of its equivalent forms have also been developed in several areas of mathematics. For probability and mathematical statistics, the equivalent forms of the AG inequality have not been linked together in a formal way. The purpose of this paper is to prove that the AG inequality is equivalent to some other renowned inequalities by using probabilistic arguments. Among such inequalities are those of Jensen, Hölder, Cauchy, Minkowski, and Lyapunov, to name just a few.
2. The Equivalent Forms
Let be a random variable, we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ1_HTML.gif)
where denotes the expected value of
.
Throughout this paper, let be a positive integer and we consider only the random variables which have finite expected values.
In order to establish our main results, we need the following lemma which is due to Infantozzi [1, 2], Marshall and Olkin [3, Page 457], and Maligranda [4, 5]. For other related results, we refer to [6–19].
Lemma 2.1.
The following inequalities are equivalent.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq5_HTML.gif)
AG inequality: , where
is a nonnegative random variable.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq8_HTML.gif)
if and
for
with
. The arithmetic-geometric mean inequality is usually applied in a simple version of
with
for each
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq16_HTML.gif)
if and
, and the opposite inequality holds if
or
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq21_HTML.gif)
if and
, and the opposite inequality holds if
or
and
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq27_HTML.gif)
for if
with
, and the opposite inequality holds if
with
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq33_HTML.gif)
if and
for
, and the opposite inequality holds if
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq38_HTML.gif)
if for
and
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq42_HTML.gif)
Let be a measure space. If
is finitely
,
and let
. Then
is finitely integrable and
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq50_HTML.gif)
If and
is
-integrable, where
is a probability space, then
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq56_HTML.gif)
Artin's theorem. Let be an open convex subset of
and
satisfy
(a) is Borel-measurable in
for each fixed
(b) is convex in
for each fixed
.
If is a measure on the Borel subsets of
such that
is
integrable for each
then
is a convex function on
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq73_HTML.gif)
Jensen's inequality. Let be a probability space and
be a random variable taking values in the open convex set
with finite expectation
. If
is convex, then
.
Proof.
The proof of the equivalent relations of can be found in [1, 2, 4, 5].
The proof of the equivalent relations of , and
can be found in [3].
Theorem 2.2.
The following inequalities are equivalent.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq83_HTML.gif)
if are random variables and
with
and
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq88_HTML.gif)
if are random variables and
with
and
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq93_HTML.gif)
if are random variables and
with
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq97_HTML.gif)
if are random variables and
with
, that is,
if
with
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq104_HTML.gif)
if are random variables and
with
and
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq109_HTML.gif)
if are random variables and
with
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq113_HTML.gif)
if are random variables and
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq116_HTML.gif)
if are random variables and
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq119_HTML.gif)
if is a random variable and
, that is,
is nondecreasing on
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq124_HTML.gif)
(see [10, 18]) , where
is a random variable if
or
, and the opposite inequality holds if
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq130_HTML.gif)
if are random variables and
with
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq134_HTML.gif)
if are random variables and
with
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq138_HTML.gif)
if are random variables and
with
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq142_HTML.gif)
if are random variables and
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq145_HTML.gif)
if are random variables and
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq148_HTML.gif)
if are random variables and
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq151_HTML.gif)
Cauchy-Bunyakovski and Schwarz's (CBS) inequality: if
are random variables.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq154_HTML.gif)
if are random variables.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq156_HTML.gif)
if are random variables and
(the inequality is reversed if
or
).
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq161_HTML.gif)
for any
if
are random variables and
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq166_HTML.gif)
if are random variables and either
or
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq170_HTML.gif)
if are random variables and
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq173_HTML.gif)
if are random variables and either
or
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq177_HTML.gif)
if are random variables and either
or
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq181_HTML.gif)
increases with if
are random variables and
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq185_HTML.gif)
Minkowski's inequality: if
are random variables,
, and the opposite inequality holds if
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq190_HTML.gif)
Triangle inequality: if
are random variables,
and the opposite inequality holds if
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq195_HTML.gif)
if is a random variable,
and
are two continuous and strictly increasing functions such that
is convex.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq200_HTML.gif)
for any if
is a random variable.
The above listed inequalities are also equivalent to the inequalities in Lemma 2.1.
Proof.
The sketch of the proof of this theorem is illustrated by the following maps of equivalent circles:
(1);
(2);
(3);
(4);
(5);
(6);
(7);
(8);
(9).
Now, we are in a position to give the proof of this theorem as follows.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq212_HTML.gif)
, see Casella and Berger [7, page 187].
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq213_HTML.gif)
is clear.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq214_HTML.gif)
: If and
, then
and
. This and
imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ2_HTML.gif)
Replacing by
in the above inequality, we obtain
.
Similarly, we can prove the case that and
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq225_HTML.gif)
is proved similarly.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq226_HTML.gif)
is clear.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq227_HTML.gif)
. Letting , and
be replaced by
and
in
, respectively, we obtain
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq234_HTML.gif)
is similarly proved.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq235_HTML.gif)
: Let . Then
. It follows from
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ3_HTML.gif)
That is, holds.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq240_HTML.gif)
is similarly proved.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq241_HTML.gif)
. Taking in
, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ4_HTML.gif)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ5_HTML.gif)
Replacing by
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ6_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ7_HTML.gif)
This proves .
Next, let . Then
and
. This and
imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ8_HTML.gif)
Replacing and
by
and
in the above inequality, respectively, and using
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ9_HTML.gif)
This proves holds.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq257_HTML.gif)
is proved similarly.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq258_HTML.gif)
. (a) Taking and
in
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ10_HTML.gif)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ11_HTML.gif)
-
(b)
Taking
and
in
,
(2.12)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ13_HTML.gif)
-
(c)
Taking
and
in
,
(2.14)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ15_HTML.gif)
-
(d)
It follows from (a), (b), and (c) that
holds.
Thus, we complete the proof.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq269_HTML.gif)
is similarly proved.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq270_HTML.gif)
is clear.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq271_HTML.gif)
by using the technique of .
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq273_HTML.gif)
. Letting and
in
, we obtain
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq278_HTML.gif)
. It follows from and
that
. This and
imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ16_HTML.gif)
Replacing and
by
and
in the above inequality, respectively, we obtain
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq288_HTML.gif)
and are similarly proved.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq290_HTML.gif)
and follow by taking
and
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq294_HTML.gif)
and follow by taking
in
and
, respectively.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq299_HTML.gif)
Casella and Berger [7, page 188].
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq300_HTML.gif)
(see [5]): Let with
and
. It follows from Benoulli's inequality
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ17_HTML.gif)
This and imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ18_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ19_HTML.gif)
This proves holds.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq307_HTML.gif)
follows by replacing and
by
and
in
, respectively.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq313_HTML.gif)
follows by replacing and
in
with
and
, respectively.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq319_HTML.gif)
. Let for
. Then, it follows from
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ20_HTML.gif)
Thus, is midconvex on
, and hence
is convex on
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ21_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ22_HTML.gif)
Letting in the both sides of the above inequality,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ23_HTML.gif)
This shows (see [13]).
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq329_HTML.gif)
. First note that, as shown above, and
are equivalent. It follows from
that, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ24_HTML.gif)
These imply for the case
.
Similarly, we can prove the case for or
by using
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq339_HTML.gif)
follows by replacing in
by
if
or
if
, respectively.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq347_HTML.gif)
follows by replacing in
by
, respectively.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq351_HTML.gif)
follows by replacing by
or
in
, respectively.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq356_HTML.gif)
follows by taking with
in
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq360_HTML.gif)
follows by taking and
in
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq364_HTML.gif)
is clear.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq365_HTML.gif)
follows by letting and
in
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq369_HTML.gif)
follows by replacing and
in
by
and
, respectively.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq375_HTML.gif)
. Replacing by
and
by
in
and changing appropriately the notation for the exponents, we obtain
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq382_HTML.gif)
is clear.
To complete our proof of equivalence of all inequalities in this theorem and in Lemma 2.1, it suffices to show further the following implications.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq383_HTML.gif)
follows by taking in
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq386_HTML.gif)
: Let , where
(hence
or
). Then it follows from
that
. Setting
, we obtain
, see [14, page 162].
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq395_HTML.gif)
follows by taking in
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_IEq398_HTML.gif)
follows by taking and replacing
by
in
.
Remark 2.3.
Letting and
with
and
in
, we obtain the inequality (5) of [18]:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ25_HTML.gif)
That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F386715/MediaObjects/13660_2007_Article_1809_Equ26_HTML.gif)
is a decreasing function of Sclove et al. [18] proved this property by means of the convexity of , see [14]. Clearly, our method is simpler than theirs.
Remark 2.4.
Each (or
) is called Hölder's inequality, each
(or
) is called CBS inequality, each
is called Lyapunov's inequality, each
is called Radon's inequality, each
is related to Jensen's inequality.
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Acknowledgments
The authors wish to thank three reviewers for their valuable suggestions that lead to substantial improvement of this paper. This work is dedicated to Professor Haruo Murakami on his 80th birthday.
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Yeh, CC., Yeh, HW. & Chan, W. Some Equivalent Forms of the Arithematic-Geometric Mean Inequality in Probability: A Survey. J Inequal Appl 2008, 386715 (2008). https://doi.org/10.1155/2008/386715
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DOI: https://doi.org/10.1155/2008/386715