- Research Article
- Open Access
Jensen Type Inequalities Involving Homogeneous Polynomials
© J.-J.Wen and Z.-H. Zhang. 2010
- Received: 4 November 2009
- Accepted: 8 February 2010
- Published: 7 April 2010
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.
- Homogeneous Polynomial
- Type Inequality
- Closed Region
- Conjugate Transpose
- Inequality State
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 ).
The famous Jensen inequality can be stated as follows: if is a convex function, then for any we have
A large number of generalizations and applications of the inequality (1.6) had been obtained in  and [5–8]. An interesting generalization of (1.6) was given by Chen et al., in : Let and . If with and , then we have the following Jensen type inequality:
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.
2.1. A Jensen Type Inequality Involving Homogeneous Polynomials
We begin a Jensen type inequality involving homogeneous polynomials as follows.
The equality holds in (2.2) if there exists , such that .
The equality in (2.3) holds if for
The inequality is reversed for . The equality in (2.4) holds if and only if , or
The equality in (2.5) holds if , or there exists , such that .
we deduce to the inequality (2.5). Lemma 2.4 is proved.
Proof of Theorem 2.1.
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.
which implies that inequality (2.2) is also true.
(3)If , then there exist sequences , such that
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
Let . If is true for any difference matrix , then (see ), and the homogeneous polynomial is called positive semidefinite with difference substitution.
If we let
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.
The equality holds in (2.14) if there exists , such that , and .
The equality in (2.33) holds if and only if , or at least numbers equal zero among the set
The equality holds if and only if , or , or
This shows that the double inequality (2.16) holds.
Proof of Theorem 2.5.
This evidently completes the proof of Theorem 2.5.
As an application of Theorem 2.5, we have the following.
Let If with then the inequality (2.14) holds. The equality holds in (2.14) if there exists , such that .
That is, the identity (2.23) holds.
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.
According to Theorem 2.8, the inequality (2.33) holds.
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.
One gives several integral analogues of (2.2) and (2.41) as follows.
As an application of the proof of Theorem 2.8, one has the following.
The proof of Corollary 2.13 is thus completed.
in inequality (2.51), it implies that the inequality (2.49) holds. The proof is completed.
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 .
where is the area of the -inscribed-polygon .
Now, we prove that the inequality (2.53) holds for by using Theorem 2.1.
in Theorem 2.1, then inequality (2.2) is just (2.53).
in Theorem 2.1, it is clear to see that inequality (2.2) deduces to (2.53).
This inequality can also be deduced from inequality (1.7).
for arbitrary and
The inequality is reversed for .
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.
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.
We will verify inequality (3.6) by induction.
which implies that the inequality (3.6) holds for .
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.
The right-hand inequality of (3.14) is proved in . Now, we will give the demonstration of the left-hand inequality in (3.14).
The proof of Theorem 3.3 is thus completed.
If , then Theorems 3.2 and 3.3 are also true.
If and , then is a control ordered set.
then the inequality (3.6) holds.
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.
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