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
If , then is called homogeneous polynomial; if , then is called homogeneous symmetric polynomial (see ).
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. Jensen Type Inequalities Involving Homogeneous Polynomials
2.1. A Jensen Type Inequality Involving Homogeneous Polynomials
We begin a Jensen type inequality involving homogeneous polynomials as follows.
we deduce to the inequality (2.5). Lemma 2.4 is proved.
Proof of Theorem 2.1.
That is, the inequality (2.2) holds.
which implies that inequality (2.2) is also true.
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
We have the following Jensen type inequality involving homogeneous polynomials and difference substitution.
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.
That is, the identity (2.23) holds.
The proof of Theorem 2.8 is thus completed.
According to Theorem 2.8, the inequality (2.33) holds.
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.
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).
3. Jensen Type Inequalities Involving Homogeneous SymmetricPolynomials
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.
We will verify inequality (3.6) by induction.
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.
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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