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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:
Also let
where
is a nonempty and finite subset of
and the permanent of matrix is given by (see [2, 4])
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
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:
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]):
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
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
The equality in (2.3) holds if for
Lemma 2.3.
(Power mean inequality, see [1, 10–11]). Let and . If , then
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
The equality in (2.5) holds if, or there exists , such that .
Proof.
According to , and Lemmas 2.2-2.3, we get that
From
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
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
which implies that inequality (2.2) is also true.
(3)If , then there exist sequences , such that
We get by the case in (2) 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 [11]), 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.
Theorem 2.5.
Let . If , and with , then
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
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:
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
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,
According to Theorem 2.1, we obtain that
In view of and with Lemma 2.7, we have
By noting that , it implies that is increasing with respect to . Thus,
Therefore,
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:
where
In fact,
and
That is, the identity (2.23) holds.
Setting
in (2.23), we have that
Since , if and only if Consider the difference substitution . From
for arbitrary , it is easy to see that if . , then
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
Write
then Theorem 2.8 can be rewritten as follows, let and . If are positive definite Hermitian matrix, , with , then
In fact, if are positive definite Hermitian matrix and , there exists a unitary matrix such that (see [13])
Thus,
From with , we get that
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
Let be a random variable, let be the probability of random events with . If , then
is the variance of random variable . The is called variance of random variable and for arbitrary , where
Let be also a random variable, with , and let the function be increasing with . Then the inequality (2.14) can be rewritten as follows:
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
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
If , and is an ordered set, that is,
for arbitrary and , then
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:
where
Proof.
We can suppose that , and
Since , from Lemma 2.3, we get that
By using (2.28), we find that
The proof of Corollary 2.13 is thus completed.
Corollary 2.14.
If with , then
Proof.
The Vandermonde determinant is wellknown (see [14]):
By Theorem 2.1, for arbitrary , we get that
Letting
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
where is the area of the -inscribed-polygon .
Now, we prove that the inequality (2.53) holds for by using Theorem 2.1.
Denote
If , we have that
Setting
in Theorem 2.1, then inequality (2.2) is just (2.53).
For , we get that
Taking
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
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]):
Definition 3.1.
(see [17, 18]). is called the control ordered set if
for arbitrary and
The well-known Chebyshev inequality states: let , and . If and , then
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:
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
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
We will verify inequality (3.6) by induction.
For , we find from the inequality (3.4) that
Since the control ordered set is nonempty and finite set by using Definition 3.1, we can suppose that
From Hardy's inequality (see [17, page 74]), we have that
By means of , inequality (3.7), and Chebyshev's inequality (3.3), it is easy to obtain that
which implies that the inequality (3.6) holds for .
Assume that the inequality (3.6) is true for , that is,
For , from and , we have Thus,
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,
If then
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
From
and Hardy's inequality (see [17, page 74]), we obtain that
Therefore, we deduce that
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,
Remark 3.6.
By using the proof of Theorem 3.2 and Remark 3.5, we know the following: if with and
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
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.
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Wen, JJ., Zhang, ZH. Jensen Type Inequalities Involving Homogeneous Polynomials. J Inequal Appl 2010, 850215 (2010). https://doi.org/10.1155/2010/850215
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DOI: https://doi.org/10.1155/2010/850215