- Research Article
- Open access
- Published:
Jensen Type Inequalities Involving Homogeneous Polynomials
Journal of Inequalities and Applications volume 2010, Article number: 850215 (2010)
Abstract
By means of algebraic, analytical and majorization theories, and under the proper hypotheses, we establish several Jensen type inequalities involving th homogeneous polynomials as follows:
,
, and
, and display their applications.
1. Introduction
The following notation and hypotheses in [1–4] will be used throughout the paper:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ1_HTML.gif)
Also let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ2_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ3_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_IEq5_HTML.gif)
is a nonempty and finite subset of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ4_HTML.gif)
and the permanent of matrix
is given by (see [2, 4])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ5_HTML.gif)
here, the sum extends over all elements of the
symmetric group
.
If , then
is called
homogeneous polynomial; if
, then
is called
homogeneous symmetric polynomial (see [3]).
The famous Jensen inequality can be stated as follows: if is a convex function, then for any
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ6_HTML.gif)
A large number of generalizations and applications of the inequality (1.6) had been obtained in [1] and [5–8]. An interesting generalization of (1.6) was given by Chen et al., in [8]: Let and
. If
with
and
, then we have the following Jensen type inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ7_HTML.gif)
In this paper, by means of algebraic, analytical, and majorization theories, and under the proper hypotheses, we will establish several Jensen type inequalities involving homogeneous polynomials and display their applications.
2. Jensen Type Inequalities Involving Homogeneous Polynomials
In this section, we will use the following notation (see [1, 4, 9]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ8_HTML.gif)
2.1. A Jensen Type Inequality Involving Homogeneous Polynomials
We begin a Jensen type inequality involving homogeneous polynomials as follows.
Theorem 2.1.
Let . If
with
and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ9_HTML.gif)
The equality holds in (2.2) if there exists , such that
.
Lemma 2.2.
(Hölder's inequality, see [1, 10]). Let with
and
. If
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ10_HTML.gif)
The equality in (2.3) holds if for
Lemma 2.3.
(Power mean inequality, see [1, 10–11]). Let and
. If
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ11_HTML.gif)
The inequality is reversed for . The equality in (2.4) holds if and only if
, or
Lemma 2.4.
Let and
. If
and
with
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ12_HTML.gif)
The equality in (2.5) holds if, or there exists
, such that
.
Proof.
According to ,
and Lemmas 2.2-2.3, we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ13_HTML.gif)
From
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ14_HTML.gif)
we deduce to the inequality (2.5). Lemma 2.4 is proved.
Proof of Theorem 2.1.
First of all, we assume that . According to
,
and Lemmas 2.3-2.4, we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ15_HTML.gif)
That is, the inequality (2.2) holds.
Secondly, for some of with
satisfing
, we have the following cases.
(1)If , then the inequality (2.2) holds from the above proof.
(2)If , then there exists
that satisfies
. By the result in (1), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ16_HTML.gif)
which implies that inequality (2.2) is also true.
(3)If , then there exist sequences
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ17_HTML.gif)
We get by the case in (2) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ18_HTML.gif)
and taking in (2.11), we can get the inequality (2.2). The proof of Theorem 2.1 is thus completed.
2.2. Jensen Type Inequalities Involving Difference Substitution
Exchange the row and
row in
unit matrix
, then this matrix, written
, is called
exchange matrix. If
are
exchange matrixes, then the
matrix
is called
difference matrix, and the substitution
is difference substitution, where
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ19_HTML.gif)
Let . If
is true for any difference matrix
, then
(see [11]), and the homogeneous polynomial
is called positive semidefinite with difference substitution.
If we let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ20_HTML.gif)
then is a finite set and the count of elements of
is
, and
.
We have the following Jensen type inequality involving homogeneous polynomials and difference substitution.
Theorem 2.5.
Let . If
, and
with
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ21_HTML.gif)
The equality holds in (2.14) if there exists , such that
, and
.
Lemma 2.6.
(Jensen's inequality, see [12]). For any and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ22_HTML.gif)
The equality in (2.33) holds if and only if , or at least
numbers equal zero among the set
Lemma 2.7.
If and
, then for the difference substitution
, one has the following double inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ23_HTML.gif)
The equality holds if and only if
, or
, or
Proof.
From , it is easy to know that
. By
and Lemma 2.6, we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ24_HTML.gif)
This shows that the double inequality (2.16) holds.
Proof of Theorem 2.5.
Consider the difference substitution . Since
,
. From
, we have that
, for all
. Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ25_HTML.gif)
According to Theorem 2.1, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ26_HTML.gif)
In view of and with Lemma 2.7, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ27_HTML.gif)
By noting that , it implies that
is increasing with respect to
. Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ28_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ29_HTML.gif)
This evidently completes the proof of Theorem 2.5.
As an application of Theorem 2.5, we have the following.
Theorem 2.8.
Let If
with
then the inequality (2.14) holds. The equality holds in (2.14) if there exists
, such that
.
Proof.
First of all, we prove that . If the function
satisfies the condition that
is continuous, then we have the following identity:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ30_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ31_HTML.gif)
In fact,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ32_HTML.gif)
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ33_HTML.gif)
That is, the identity (2.23) holds.
Setting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ34_HTML.gif)
in (2.23), we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ35_HTML.gif)
Since ,
if and only if
Consider the difference substitution
. From
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ36_HTML.gif)
for arbitrary , it is easy to see that
if
.
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ37_HTML.gif)
for arbitrary . Therefore, we get that
. It follows that the inequality (2.14) holds by using Theorem 2.5. Since
, the equality holds in (2.14) if there exists
, such that
.
The proof of Theorem 2.8 is thus completed.
Remark 2.9.
Theorem 2.8 has significance in the theory of matrices. Let be an
positive definite Hermitian matrix and
its eigenvalues, let
be the diagonal matrix with the components of
as its diagonal elements, and also let
Then
for some unitary matrix
(where
is the conjugate transpose of
and
see [9, 13]). If
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ38_HTML.gif)
Write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ39_HTML.gif)
then Theorem 2.8 can be rewritten as follows, let and
. If
are
positive definite Hermitian matrix,
,
with
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ40_HTML.gif)
In fact, if are
positive definite Hermitian matrix and
, there exists a unitary matrix
such that (see [13])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ41_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ42_HTML.gif)
From with
, we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ43_HTML.gif)
According to Theorem 2.8, the inequality (2.33) holds.
Remark 2.10.
Theorem 2.8 has also significance in statistics. By using the same proving method of Theorems 2.1–2.8, we can prove the following: under the hypotheses of the Theorem 2.8, if , then
and the inequality (2.14) also holds, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ44_HTML.gif)
Let be a random variable,
let
be the probability of random events
with
. If
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ45_HTML.gif)
is the variance of random variable . The
is called
variance of random variable
and
for arbitrary
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ46_HTML.gif)
Let be also a random variable,
with
, and let the function
be increasing with
. Then the inequality (2.14) can be rewritten as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ47_HTML.gif)
where ,
.
2.3. Applications of Jensen Type Inequalities
By (1.7) and the same proving method of Theorem 2.1, we can obtain the following result.
Corollary 2.11.
Let . If
with
and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ48_HTML.gif)
One gives several integral analogues of (2.2) and (2.41) as follows.
Corollary 2.12.
Let be bounded closed region in
, and let the functions
and
be continuous, and
. If
and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ49_HTML.gif)
If , and
is an ordered set, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ50_HTML.gif)
for arbitrary and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ51_HTML.gif)
As an application of the proof of Theorem 2.8, one has the following.
Corollary 2.13.
Let If
with
then one has the following Jensen type inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ52_HTML.gif)
where
Proof.
We can suppose that , and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ53_HTML.gif)
Since , from Lemma 2.3, we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ54_HTML.gif)
By using (2.28), we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ55_HTML.gif)
The proof of Corollary 2.13 is thus completed.
Corollary 2.14.
If with
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ56_HTML.gif)
Proof.
The Vandermonde determinant is wellknown (see [14]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ57_HTML.gif)
By Theorem 2.1, for arbitrary , we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ58_HTML.gif)
Letting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ59_HTML.gif)
in inequality (2.51), it implies that the inequality (2.49) holds. The proof is completed.
Example 2.15.
Given -inscribed-polygon
with
. Defining the summation of them is an
-inscribed-polygon
, and its sides lengths are given by
with
. Also defining
, and
with
.
Wen and Zhang in [15] raised a conjecture: prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ60_HTML.gif)
where is the area of the
-inscribed-polygon
.
Now, we prove that the inequality (2.53) holds for by using Theorem 2.1.
Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ61_HTML.gif)
If , we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ62_HTML.gif)
Setting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ63_HTML.gif)
in Theorem 2.1, then inequality (2.2) is just (2.53).
For , we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ64_HTML.gif)
Taking
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ65_HTML.gif)
in Theorem 2.1, it is clear to see that inequality (2.2) deduces to (2.53).
Remark 2.16.
The following result was obtained in [15]. Let with
and
all be
-inscribed-polygons. If
, then for
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ66_HTML.gif)
This inequality can also be deduced from inequality (1.7).
3. Jensen Type Inequalities Involving Homogeneous SymmetricPolynomials
In this section, we will also use the following notation (see [4, 16]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ67_HTML.gif)
Definition 3.1.
(see [17, 18]). is called the control ordered set if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ68_HTML.gif)
for arbitrary and
The well-known Chebyshev inequality states: let ,
and
. If
and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ69_HTML.gif)
The inequality is reversed for .
We remark here that Wen and Wang generalized the inequality (3.3) in [4]: if , and
, then we have the following Chebyshev type inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ70_HTML.gif)
3.1. Jensen Type Inequalities Involving Homogeneous Symmetric Polynomials
In this subsection, we first present a Jensen type inequality involving homogeneous symmetric polynomials as follows.
Theorem 3.2.
Let let
be a control ordered set. If
with
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ71_HTML.gif)
where
Proof.
By using the same proving method of Theorem 2.1, we can suppose that . If
, then inequality (3.5) holds. So we just need to prove the following.
Let be
is a control ordered set. If
with
and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ72_HTML.gif)
We will verify inequality (3.6) by induction.
For , we find from the inequality (3.4) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ73_HTML.gif)
Since the control ordered set is nonempty and finite set by using Definition 3.1, we can suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ74_HTML.gif)
From Hardy's inequality (see [17, page 74]), we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ75_HTML.gif)
By means of , inequality (3.7), and Chebyshev's inequality (3.3), it is easy to obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ76_HTML.gif)
which implies that the inequality (3.6) holds for .
Assume that the inequality (3.6) is true for , that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ77_HTML.gif)
For , from
and
, we have
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ78_HTML.gif)
The inequality (3.6) is proved by induction. The proof of Theorem 3.2 is hence completed.
As an application of the inequality (3.6), we have the following.
Theorem 3.3.
Let , let
be a control ordered set, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ79_HTML.gif)
If then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ80_HTML.gif)
Proof.
The right-hand inequality of (3.14) is proved in [4]. Now, we will give the demonstration of the left-hand inequality in (3.14).
We can suppose that . By means of
,
, and
, we find from the inequality (3.6) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ81_HTML.gif)
From
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ82_HTML.gif)
and Hardy's inequality (see [17, page 74]), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ83_HTML.gif)
Therefore, we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ84_HTML.gif)
The proof of Theorem 3.3 is thus completed.
3.2. Remarks
Remark 3.4.
If , then Theorems 3.2 and 3.3 are also true.
Remark 3.5.
If and
, then
is a control ordered set.
In fact,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ85_HTML.gif)
Remark 3.6.
By using the proof of Theorem 3.2 and Remark 3.5, we know the following: if with
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ86_HTML.gif)
then the inequality (3.6) holds.
Remark 3.7.
The inequality (3.6) is also a Chebyshev type inequality involving homogeneous symmetric polynomials.
3.3. An Open Problem
According to Theorem 3.3, we pose the following open problem.
Conjecture 3.8.
Under the hypotheses of Theorem 3.3, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F850215/MediaObjects/13660_2009_Article_2276_Equ87_HTML.gif)
References
Bullen PS, Mitrinović PS, Vasić PM: Means and Their Inequalities. Reidel, Dordrecht, The Netherlands; 1988.
Minc H: Permanents, Encyclopedia of Mathematics and Its Applications. Volume 9999. Addison-Wesley, Reading, Mass, USA; 1978:xviii+205.
Timofte V: On the positivity of symmetric polynomial functions. I. General results. Journal of Mathematical Analysis and Applications 2003, 284(1):174–190. 10.1016/S0022-247X(03)00301-9
Wen J-J, Wang W-L: Chebyshev type inequalities involving permanents and their applications. Linear Algebra and Its Applications 2007, 422(1):295–303. 10.1016/j.laa.2006.10.014
Pečarić J, Svrtan D: New refinements of the Jensen inequalities based on samples with repetitions. Journal of Mathematical Analysis and Applications 1998, 222(2):365–373. 10.1006/jmaa.1997.5839
Xiao Z-G, Srivastava HM, Zhang Z-H: Further refinements of the Jensen inequalities based upon samples with repetitions. Mathematical and Computer Modelling 2010, 51(5–6):592–600.
Gao C, Wen J: Inequalities of Jensen-Pečarić-Svrtan-Fan type. Journal of Inequalities in Pure and Applied Mathematics 2008, 9(3, article 74):-8.
Chen Y-X, Luo J-Y, Yang J-K: A class of Jensen inequalities for homogeneous and symmetric polynomials. Journal of Sichuan Normal University 2007, 30: 481–484.
Wen J-J, Cheng SS, Gao C: Optimal sublinear inequalities involving geometric and power means. Mathematica Bohemica 2009, 134(2):133–149.
Wen J-J, Wang W-L: The optimization for the inequalities of power means. Journal of Inequalities and Applications 2006, 2006:-25.
Yang L, Feng Y, Yao Y: A class of mechanically decidable problems beyond Tarski's model. Science in China. Series A 2007, 50(11):1611–1620. 10.1007/s11425-007-0090-8
Jensen WV: Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Mathematica 1906, 30(1):175–193. 10.1007/BF02418571
Mond B, Pečarić JE: Generalization of a matrix inequality of Ky Fan. Journal of Mathematical Analysis and Applications 1995, 190(1):244–247. 10.1006/jmaa.1995.1074
Wen J-J, Wang W-L: Inequalities involving generalized interpolation polynomials. Computers & Mathematics with Applications 2008, 56(4):1045–1058. 10.1016/j.camwa.2008.01.032
Wen J-J, Zhang R-X: Two conjectured inequalities involving the sums of inscribed polygons in some circles. Journal of Shanxi Normal University 2002, 30(supplement 1):12–17.
Pečarić J, Wen J-J, Wang W-L, Lu T: A generalization of Maclaurin's inequalities and its applications. Mathematical Inequalities and Applications 2005, 8(4):583–598.
Wang B-Y: An Introduction to the Theory of Majorizations. Beijing Normal University Press, Beijing, China; 1990.
Marshall AW, Olkin I: Inequalities: Theory of Majorization and Its Applications, Mathematics in Science and Engineering. Volume 143. Academic Press, New York, NY, USA; 1979:xx+569.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Wen, JJ., Zhang, ZH. Jensen Type Inequalities Involving Homogeneous Polynomials. J Inequal Appl 2010, 850215 (2010). https://doi.org/10.1155/2010/850215
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/850215