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Volume 2010, Article ID 850215, 21 pagesdoi:10.1155/2010/850215 Research Article Jensen Type Inequalities Involving Homogeneous Polynomials 1 College of Mathematics and Information Scien

Volume 2010, Article ID 850215, 21 pages

doi:10.1155/2010/850215

Research Article

Jensen Type Inequalities Involving

Homogeneous Polynomials

1 College of Mathematics and Information Science, Chengdu University, Sichuan 610106, China

2 Department of Mathematics, Shili Senior High School in Zixing, Chenzhou, Hunan 423400, China

Correspondence should be addressed to Jia-Jin Wen,wenjiajin623@163.com

Received 4 November 2009; Revised 25 January 2010; Accepted 8 February 2010

Academic Editor: Soo Hak Sung

Copyrightq 2010 J.-J Wen and Z.-H Zhang This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited

By means of algebraic, analytical and majorization theories, and under the proper hypotheses,

we establish several Jensen type inequalities involving γth homogeneous polynomials as follows:

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 γth

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

The equality holds in2.2 if there exists t ∈ 0, ∞, such that X1 X2 · · ·  X m  tI n

Lemma 2.2 (H¨older’s inequality, see [1 , 10 ]) Let a i,k ∈ 0, ∞, q i ∈ 0, ∞ with 1 ≤ i ≤ n and

The equality in2.3 holds if ai,1  a i,2  · · ·  a i,m for 1 ≤ i ≤ n.

Lemma 2.3 (Power mean inequality, see [1, 10–11]) Let x ∈ R n

Proof of Theorem 2.1 First of all, we assume that w  I m According to γ ∈ 1, ∞, fI n 

Secondly, for some of w kwith 1≤ k ≤ m satisfing w k / 1, we have the following cases.

2.2 Jensen Type Inequalities Involving Difference Substitution

Let f ∈ P γ x If fD n y  ∈ P

semidefinite with difference substitution

We have the following Jensen type inequality involving homogeneous polynomialsand difference substitution

Proof of Theorem 2.5 Consider the di fference substitution X k  Δn Y k Since X k ∈ Ωn , Y k 

Theorem 2.8 Let fx  Ax γ  − A γ x, γ ∈ N and γ ≥ 2 If w ∈ R m

, X k∈ Ωn with 1 ≤ k ≤ m,

then the inequality2.14 holds The equality holds in 2.14 if there exists t ∈ 0, ∞, such that

X1  X2 · · ·  X m  tI n

Proof First of all, we prove that f ∈ P∗

t ∈ 0, ∞, such that X1 X2 · · ·  X m  tI n.

Remark 2.9. Theorem 2.8has significance in the theory of matrices LetA  a i,jn ×n be an

ntrAγ−

1

Remark 2.10. Theorem 2.8 has also significance in statistics By using the same proving

is the variance of random variable ξ The D γ ξ is called γth variance of random variable ξ

0, ∞ → 0, ∞ be increasing with 1 ≤ k ≤ m Then the inequality 2.14 can be rewritten as

2.3 Applications of Jensen Type Inequalities

One gives several integral analogues of2.2 and 2.41 as follows

Corollary 2.12 Let E be bounded closed region in R s , and let the functions w : E → 0, ∞ and

for arbitrary t1, t2: t1∈ E and t2∈ E, then

Corollary 2.14 If X k∈ Ωn with 1 ≤ k ≤ m, then

w  I m , x k,i,j  n n−1/27

Example 2.15 Given N-inscribed-polygon Γk  Γk A k,1 , A k,2 , , A k,N  with 1 ≤ k ≤ m.

k1Γk  ΓA1, A2, , A N,

For N 4, we get that

Remark 2.16 The following result was obtained in15 Let Γk with 1 ≤ k ≤ m andm

3 Jensen Type Inequalities Involving Homogeneous

Definition 3.1. see 17,18 Bγis called the control ordered set if

3.1 Jensen Type Inequalities Involving Homogeneous Symmetric Polynomials

In this subsection, we first present a Jensen type inequality involving homogeneous ric polynomials as follows

symmet-Theorem 3.2 Let f ∈ Pγ x, fI n   1, w ∈ N m , letBγ be a control ordered set If X k ∈ Ωn

where f∗:Rn → R, f∗x  log fe x .

Proof By using the same proving method ofTheorem 2.1, we can suppose that w  I m If

Since the control ordered setBγ is nonempty and finite set by usingDefinition 3.1, wecan suppose that

3.3, it is easy to obtain that

Theorem 3.3 Let f ∈ Pγ x, let B γ be a control ordered set, that is,

Proof The right-hand inequality of 3.14 is proved in 4 Now, we will give the

3.2 Remarks

Remark 3.4 If γ ∈0, ∞, then Theorems3.2and3.3are also true

Remark 3.5 IfBγ ⊂ Nnand 1≤ γ ≤ 5, then B γis a control ordered set

Remark 3.7 The inequality3.6 is also a Chebyshev type inequality involving homogeneoussymmetric polynomials

3.3 An Open Problem

Conjecture 3.8 Under the hypotheses of Theorem 3.3 , one has

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