Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 850215, 21 pages
doi:10.1155/2010/850215
Research Article
Jensen Type Inequalities Involving
Homogeneous Polynomials
Jia-Jin Wen
1
and Zhi-Hua Zhang
2
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, [email protected]
Received 4 November 2009; Revised 25 January 2010; Accepted 8 February 2010
Academic Editor: Soo Hak Sung
Copyright q 2010 J J. Wen and Z H. Zhang. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction 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:
m
k1
w
k
fX
k
/fI
X
γ
k
/
fN
n
1/γ
,and
m
k1
w
k
f
∗
X
k
≤ f
∗
m
k1
w
k
X
k
, and display their applications.
1. Introduction
X
k
x
k,1
,x
k,2
, ,x
k,n
†
, N
{
0, 1, 2, ,n,
}
,n∈ N,n≥ 2,
R
−∞, ∞
, R
n
0, ∞
n
, R
n
0, ∞
∈B
γ
× S
n
λ
α, σ
n
j1
x
α
j
σ
j
λ : B
γ
× S
n
→ R
n
j1
x
α
j
σ
j
λ : B
γ
× S
n
−→
0, ∞
⎫
⎬
⎭
\
{
0
λ : B
γ
→ R
⎫
⎬
⎭
\
{
0
}
,
P
γ
x
⎧
⎨
⎩
α∈B
γ
λ
α
where
x
α
i
j
x
α
i
j
n×n
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
x
α
1
1
x
α
.
.
.
.
.
.
.
.
.
.
.
x
α
n
1
x
α
n
2
x
α
n
3
··· x
α
n
n
⎤
⎥
⎥
,
1.4
and the permanent of n × n matrix A a
i,j
n×n
is given by see 2, 4
per A
σ∈S
n
n
j1
a
j,σj
;
1.5
here, the sum extends over all elements σ of the nth symmetric group S
n
.
If f ∈ P
γ
x, then f is called γth homogeneous polynomial; if f ∈ P
γ
x, then f is
called γth homogeneous symmetric polynomial see 3.
The famous Jensen inequality can be stated as follows: if f : I → R is a convex
function, then for any x ∈ I
n
γ
x.IfX
k
∈ R
n
with 1 ≤ k ≤ m and 0 ≤ X
1
≤ X
2
≤ ··· ≤ X
m
, then
we have the following Jensen type inequality:
1
m
m
k1
f
X
k
≥ f
1
m
m
}
,I
n
1, 1, ,1
†
,N
n
1, 2, ,n
†
,
x
γ
x
γ
1
,x
γ
2
, ,x
γ
n
1
, Δx
2
, ,Δx
n
†
x
1
,x
2
− x
1
,x
3
− x
2
, ,x
n
− x
n−1
†
.
2.1
2.1. A Jensen Type Inequality Involving Homogeneous Polynomials
We begin a Jensen type inequality involving homogeneous polynomials as follows.
Theorem 2.1. Let f ∈ P
f
m
k1
w
k
X
γ
k
f
I
n
⎤
⎥
⎦
1/γ
.
2.2
The equality holds in 2.2 if there exists t ∈ 0, ∞, such that X
1
X
2
··· X
m
tI
n
i1
1
m
m
k1
a
i,k
q
i
.
2.3
The equality in 2.3 holds if a
i,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
,μ ∈ R
n
and
n
2
··· x
n
.
4 Journal of Inequalities and Applications
Lemma 2.4. Let gx, α
n
j1
x
α
j
σj
and σ ∈ S
n
.Ifα ∈B
γ
and X
k
∈ R
n
with 1 ≤ k ≤ m,then
g
m
k1
X
γ
Proof. According to α ∈B
γ
,
n
j1
α
j
/γ1 ≤ 1 and Lemmas 2.2-2.3,wegetthat
g
1
m
m
k1
X
γ
k
,α
n
j1
1
m
m
j
/γ
⎤
⎦
γ
≥
⎡
⎣
1
m
m
k1
n
j1
x
α
j
k,σ
j
⎤
⎦
γ
1
m
g
m
k1
X
γ
k
,α
,
2.7
we deduce to the inequality 2.5. Lemma 2.4 is proved.
Proof of Theorem 2.1. First of all, we assume that w I
m
. According to γ ∈ 1, ∞, fI
n
α,σ∈B
γ
×S
n
λα, σ and Lemmas 2.3-2.4 ,wefindthat
f
m
k1
X
k
,α
≥
α,σ
∈B
γ
×S
n
λ
α, σ
f
I
n
m
k1
g
X
k
,α
,α
⎤
⎦
γ
m
k1
f
X
k
f
I
n
γ
.
2.8
That is, the inequality 2.2 holds.
Journal of Inequalities and Applications 5
Secondly, for some of w
k
with 1 ≤ k ≤ m satisfing w
k
⎢
⎢
⎣
f
m
k1
Nw
k
X
γ
k
f
I
n
⎤
⎥
⎥
⎥
⎦
1/γ
⇐⇒
m
k1
w
I
n
⎤
⎥
⎥
⎥
⎦
1/γ
,
2.9
which implies that inequality 2.2 is also true.
3 If w ∈ R
m
, then there exist sequences {w
i
k
}
∞
i1
, such that
w
i
k
∈ Q
1 ≤ i<∞
⎡
⎢
⎣
f
m
k1
w
i
k
X
γ
k
f
I
n
⎤
⎥
⎦
1/γ
,
2.11
and taking i →∞in 2.11, we can get the inequality 2.2. T he proof of Theorem 2.1 is thus
completed.
2.2. Jensen Type Inequalities Involving Difference Substitution
Exchange the ith row and jth row in nth unit matrix E, then this matrix, written Ei, j,is
⎢
⎢
⎢
⎢
⎢
⎣
10··· 0
11··· 0
.
.
.
.
.
.
.
.
.
.
.
.
11··· 1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
n×n
P
∗
γ
x
f
x
∈ P
γ
x
|B
γ
⊂ N
n
,f
D
n
y
∈ P
γ
m
k1
w
k
f
X
k
f
N
n
≤
⎡
⎢
⎣
f
m
k1
w
k
X
γ
k
k1
x
k
γ
≥
n
k1
x
γ
k
.
2.15
The equality in 2.33 holds if and only if γ 1, or at least n − 1 numbers equal zero among the set
{x
1
,x
2
, ,x
n
}.
Lemma 2.7. If γ ∈ 1, ∞ and x ∈ Ω
n
, then for the difference substitution x Δ
n
y, one has the
following double inequality:
0 ≤ y
γ
0 ≤ y
γ
1
x
γ
1
≤ x
γ
1
,
0 ≤ y
γ
2
x
2
− x
1
γ
≤ x
γ
2
− x
γ
1
,
.
.
∈ Ω
n
, Y
k
Δ
−1
n
X
k
ΔX
k
∈ R
n
with 1 ≤ k ≤ m.Fromf ∈ P
∗
γ
x, we have that fD
n
y ∈ P
γ
y, for all
D
n
∈D
n
. Hence,
f
I
n
≤
⎡
⎢
⎣
f
Δ
n
m
k1
w
k
Y
γ
k
f
Δ
n
I
n
⎤
⎥
⎦
k
∈ R
n
and with Lemma 2.7, we have
0 ≤ Y
γ
k
≤ ΔX
γ
k
,k 1, 2, ,m.
2.20
By noting that fΔ
n
y ∈ P
γ
y, it implies that f
m
k1
w
k
Δ
n
Y
γ
k
is increasing with
f
m
k1
w
k
X
γ
k
.
2.21
Therefore,
m
k1
w
k
f
X
k
f
N
n
Δ
n
Y
γ
k
f
N
n
⎤
⎥
⎦
1/γ
≤
⎡
⎢
⎣
f
m
k1
w
k
X
γ
k
··· X
m
tI
n
.
8 Journal of Inequalities and Applications
Proof. First of all, we prove that f ∈ P
∗
γ
x. If the function φ : I → R satisfies the condition
that φ
: I → R is continuous, then we have the following identity:
A
φ
x
− φ
A
x
1
n
2
dt
2
x
i
− x
j
2
,
2.23
where
x ∈ I
n
,φ
t
d
2
φ
dt
2
, ∇
t
1
,t
1 − t
1
− t
2
A
x
dt
1
dt
2
1
0
dt
1
1−t
1
0
φ
t
1
x
dt
1
1−t
1
0
φ
t
1
x
i
t
2
x
j
1 − t
1
− t
2
A
x
d
1
0
dt
1
φ
t
1
x
i
t
2
x
j
1 − t
1
− t
2
A
x
1−t
t
1
x
i
1 − t
1
A
x
dt
1
1
x
j
− A
x
φt
1
x
1
x
j
− A
x
φ
x
i
− φ
x
j
x
i
− x
j
−
φ
x
i
− φ
x
φ
A
x
A
x
1
φ
x
i
x
i
1
φ
2
x
j
1 − t
1
− t
2
A
x
dt
1
dt
2
x
i
− x
j
2
1≤i<j≤n
x
A
x
1
φ
x
i
x
i
1
φ
x
j
x
j
1
Journal of Inequalities and Applications 9
φ
A
x
A
x
1
φ
x
i
x
i
1
φ
x
j
x
j
1
A
x
A
x
1
φ
x
i
x
i
1
φ
x
j
x
j
1
x
1
φ
x
i
x
i
1
φ
x
j
x
j
1
⎞
⎟
⎟
A
x
A
x
1
φ
x
i
x
i
1
φ
x
j
x
j
1
x
A
x
1
φ
x
i
x
i
1
φ
x
j
x
j
1
φ
A
x
A
x
1
1
n
n
i1
φ
x
i
A
x
φ
A
x
A
x
1
φ
x
i
x
i
1
⎟
⎟
⎠
1
2
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
n
j1
n
x
j
− A
x
x
i
A
x
1
φ
x
j
x
j
1
−
1
φ
x
i
x
i
1
1
n
n
j1
φ
x
j
− φ
A
x
00
x
− φ
A
x
A
x
− x
j
x
j
− A
x
−
n
i1
A
n
2
⎧
⎨
⎩
n
j1
A
φ
x
− φ
A
x
n
i1
A
.
2.26
That is, the identity 2.23 holds.
10 Journal of Inequalities and Applications
Setting
φ :
0, ∞
−→ R,φ
t
t
γ
2.27
in 2.23, we have that
f
x
1
n
2
1≤i<j≤n
2
.
2.28
Since f ∈
P
γ
x, f ∈ P
∗
γ
x if and only if fΔ
n
y ∈ P
γ
y. Consider the difference
substitution x Δ
n
y.From
x
i
− x
j
2
j
x
i
− x
j
2
∈ P
∗
2
x
,
t
1
x
i
t
2
x
j
1 − t
1
− t
2
Ax
γ−2
dt
2
∈ P
γ−2
x
⇒
∇
γ
γ − 1
t
1
x
i
t
2
x
j
1 − t
1
− t
2
Ax
γ−2
n×n
be an
n × n positive definite Hermitian matrix and λ
1
, ,λ
n
its eigenvalues, let diagx be the
diagonal matrix with the components of x x
1
,x
2
, ,x
n
†
as its diagonal elements, and
also let λ λ
1
,λ
2
, ,λ
n
†
. Then A U diagλU
∗
for some unitary matrix U where U
∗
is
i1
λ
γ
i
.
2.31
Journal of Inequalities and Applications 11
Write
D
γ
A
1
n
tr A
γ
−
1
n
tr A
γ
A
λ
γ
with 1 ≤ i, j, k ≤ n, then
m
k1
w
k
D
γ
A
k
D
γ
diag
N
n
≤
⎡
⎢
⎣
D
γ
m
λ
B
U
∗
. 2.34
Thus,
D
γ
A B
D
γ
U diag
λ
A
λ
B
U
∗
D
γ
diag
f
λ
A
k
,D
γ
m
k1
w
k
A
γ
k
f
m
k1
w
k
λ
γ
A
k
,A
x, p
n
i1
p
i
x
i
,
n
i1
p
i
1.
2.37
Let ξ be a random variable, x ∈ Ω
n
, let Pξ x
i
p
i
be the probability of random events
ξ x
i
A
x
γ
,p
− A
γ
x, p
D
γ
x, p
2.38
12 Journal of Inequalities and Applications
is the variance of random variable ξ.TheD
γ
ξ is called γth variance of random variable ξ
and D
γ
ξ ≥ 0 for arbitrary γ ∈ R, where
D
0
ξ
2
A
x log x,p
− A
x, p
log A
x, p
.
2.39
Let ξ
0
be also a random variable, Pξ
0
ip
i
with 1 ≤ i ≤ n, and let the function f
k
:
0, ∞ → 0, ∞ be increasing with 1 ≤ k ≤ m. Then the inequality 2.14 can be rewritten as
follows:
m
k1
γ
k
ξ
D
γ
ξ
0
⎫
⎬
⎭
1/γ
,
2.40
where w ∈ R
m
,γ ∈ N, γ ≥ 2.
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 B
γ
⊂ N
n
,f ∈ P
γ
x.Ifw ∈ R
m
≥ f
m
k1
w
k
X
k
.
2.41
One gives several integral analogues of 2.2 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
g : E → R
n
be continuous, and
E
wdt 1.Iff ∈ P
γ
x and fI
n
1,then
E
≤ g
t
2
or g
t
2
≤ g
t
1
2.43
Journal of Inequalities and Applications 13
for arbitrary t
1
,t
2
: t
1
∈ E and t
2
∈ E,then
E
wf
k
∈ R
n
,
with 1 ≤ k ≤ m, then one has the following Jensen type inequality:
m
k1
w
k
D
γ
X
k
,p
≥ D
γ
m
k1
w
k
X
k
,p
1 − t
1
− t
2
A
x
. 2.46
Since 0 < 2 − γ<1, from Lemma 2.3,wegetthat
m
k1
w
k
ω
i,j
X
k
,t
1
,t
2
γ−2
−1
m
k1
w
k
|x
k,i
− x
k,j
|
2
m
k1
w
k
ω
2−γ
i,j
X
k
,t
1
,t
2
|x
k,i
− x
k,j
|
m
k1
w
k
ω
2−γ
i,j
X
k
,t
1
,t
2
⎞
⎠
2
≤
m
2
ω
2γ−4
i,j
X
k
,t
1
,t
2
|x
k,i
− x
k,j
|
2
m
k1
w
k
ω
2−γ
i,j
X
k
By using 2.28,wefindthat
14 Journal of Inequalities and Applications
D
γ
m
k1
w
k
X
k
,p
2
n
2
1≤i<j≤n
⎧
⎨
⎩
∇
ω
i,j
m
k,j
2
≤
2
n
2
1≤i<j≤n
⎧
⎨
⎩
∇
ω
i,j
m
k1
w
k
X
k
,t
1
,t
2
⎨
⎩
∇
ω
i,j
m
k1
w
k
X
k
,t
1
,t
2
γ−2
m
k
w
k
|x
k,i
i,j
X
k
,t
1
,t
2
γ−2
m
k1
w
k
|x
k,i
− x
k,j
|
2
dt
1
dt
2
⎫
⎬
|
2
dt
1
dt
2
m
k1
w
k
⎧
⎨
⎩
2
n
2
1≤i<j≤n
∇
ω
γ−2
i,j
X
k
,t
.
2.48
The proof of Corollary 2.13 is thus completed.
Corollary 2.14. If X
k
∈ Ω
n
with 1 ≤ k ≤ m,then
m
k1
n
n − 1
/2
det
X
k
i−1
j
n×n
≤
n
i−1
j
n×n
1≤i<j≤n
x
j
− x
i
.
2.50
By Theorem 2.1, for arbitrary x
k,i,j
∈ 0, ∞ with 1 ≤ i<j≤ n, 1 ≤ k ≤ m,wegetthat
m
k1
1≤i<j≤n
x
k,i,j
≤
n
n−1
− x
k,i
, 1 ≤ i<j≤ n, 1 ≤ k ≤ m 2.52
in inequality 2.51, it implies that the inequality 2.49 holds. The proof is completed.
Example 2.15. Given N-inscribed-polygon Γ
k
Γ
k
A
k,1
,A
k,2
, ,A
k,N
with 1 ≤ k ≤ m.
Defining the summation of them is an N-inscribed-polygon Γ
m
k1
Γ
k
ΓA
1
,A
2
, ,A
N
,
and its sides lengths are given by |A
i
m
k1
Γ
k
≥
m
k1
|
Γ
k
|
,
2.53
where |Γ| Area Γ is the area of the N-inscribed-polygon Γ.
Now, we prove that the inequality 2.53 holds for N 3, 4byusingTheorem 2.1.
Denote
a
k,i
|
A
k,i
1
2
N
i1
a
i
m
k1
p
k
, 1 ≤ i ≤ N, 1 ≤ k ≤ m.
2.54
If N 3, we have that
|
Γ
k
|
4
p
k
3
m
k1
p
k
3
i1
m
k1
p
k
− a
k,i
.
2.55
Setting
f ∈ P
4
− a
k,i
, 2 ≤ i ≤ 4, 1 ≤ k ≤ m
2.56
in Theorem 2.1, then inequality 2.2 is just 2.53.
16 Journal of Inequalities and Applications
For N 4, we get that
|
Γ
k
|
4
4
i1
p
k
− a
k,i
,
p
k
− a
k,i
.
2.57
Taking
f ∈ P
4
x
,f
x
4
i1
x
i
,n 4,
w I
m
,x
k,i
,i 1, 2, ,N, then for
N 3, 4, we have
m
k1
Γ
k
2
≤ m
3
m
k1
|
Γ
k
|
2
.
2.59
≤··· ≤ x
n
}
,
α
l
α
l
1
,α
l
2
, ,α
l
n
†
∈B
γ
,p max
1≤l≤N
α
l
1
,α
l
2
†
,
X
1
X
2
x
1,1
x
2,1
,
x
1,2
x
2,2
, ,
x
1,n
x
2,n
†
.
3.1
Journal of Inequalities and Applications 17
Definition 3.1. see 17, 18. B
γ
is called the control ordered set if
, then
m
k1
w
k
a
k
b
k
≥
m
k1
w
k
a
k
×
m
k1
w
k
b
k
j
n!
≥
per
X
1
α
i
j
n!
×
per
X
2
α
i
j
n!
.
3.4
3.1. Jensen Type Inequalities Involving Homogeneous Symmetric Polynomials
≤ f
∗
m
k1
w
k
X
k
,
3.5
where f
∗
: R
n
→ R,f
∗
xlog fe
x
.
Proof. By using the same proving method of Theorem 2.1, we can suppose that w I
m
.If
m 1, then inequality 3.5 holds. So we just need to prove the following.
Let f ∈
P
γ
For m 2, we find from the inequality 3.4 that
f
X
1
X
2
α∈B
γ
λ
α
n!
per
X
1
X
2
α
i
j
n×n
n!
.
3.7
18 Journal of Inequalities and Applications
Since the control ordered set B
γ
is nonempty and finite set by using Definition 3.1,we
can suppose that
B
γ
α
1
,α
2
, ,α
N
,α
1
≺ α
2
≺··· ≺ α
N
. 3.8
From Hardy’s inequality see 17, page 74, we have that
per
N
i
j
n!
,
per
X
2
α
1
i
j
n!
≤
per
X
2
α
2
i
j
X
2
≥
N
l1
λ
α
l
per
X
1
α
l
i
j
n!
×
per
X
2
n!
⎞
⎟
⎟
⎠
×
⎛
⎜
⎜
⎝
N
l1
λ
α
l
per
X
2
α
l
i
j
k1
f
X
k
.
3.11
For m q 1, from X
q1
∈ Ω
n
and X
1
,X
2
, ,X
q
∈ Ω
n
, we have
q
k1
X
k
∈ Ω
n
. Thus,
f
≥ f
X
q1
q
k1
f
X
k
q1
k1
f
X
k
.
3.12
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.
Journal of Inequalities and Applications 19
Theorem 3.3. Let f ∈
,X
2
∈ Ω
n
, and X
2
∈ R
n
, then
1
n!
per
⎡
⎣
X
1
X
2
α
1
i
j
⎤
⎦
≤
f
1. By means of X
1
/X
2
∈ Ω
n
, X
2
∈ Ω
n
,andX
2
∈ R
n
,we
find from the inequality 3.6 that
f
X
1
X
2
X
2
≥ f
X
1
γ
α
1
,α
2
, ,α
N
,α
1
≺ α
2
≺··· ≺ α
N
3.16
and Hardy’s inequality see 17, page 74,weobtainthat
per
X
1
X
2
α
i
j
X
1
X
2
α∈B
γ
λ
α
n!
per
X
1
X
2
α
i
j
n×n
≥
X
1
X
2
α
1
i
j
⎤
⎦
n×n
f
I
n
1
n!
per
⎡
⎣
X
1
X
2
α
1, 1, 1, 0, ,0
†
≺ 2, 1, 0, ,0
†
≺ 3, 0, ,0
†
;
1, 1, 1, 1, 0, ,0
†
≺ 2, 1, 1, 0, ,0
†
≺ 2, 2, 0, ,0
†
≺ 3, 1, 0, ,0
†
≺ 4, 0, ,0
†
;
1, 1, 1, 1, 1, 0, ,0
†
≺ 2, 1, 1, 1, 0, ,0
†
≺ 2, 2, 1, 0, ,0
†
≺ 3, 1, 1, 0, ,0
†
≺ 3, 2, 0, ,0
†
≺ 4, 1, 0, ,0
†
k1
x
γ
k
,γ∈ N, 1 <γ≤ 5,
3.20
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
1
n!
per
⎡
⎣
X
1
X
2
α
1
i
j
⎤
f
X
2
≤
n
i1
x
p
1,i
n
i1
x
p
2,i
γ/p
. 3.21
Journal of Inequalities and Applications 21
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