Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 543250, 12 pages
doi:10.1155/2010/543250
Research Article
On Schur Convexity of Some Symmetric Functions
Wei-Feng Xia1 and Yu-Ming Chu2
1
2
School of Teachers Education, Huzhou Teachers College, Huzhou 313000, China
Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
Correspondence should be addressed to Yu-Ming Chu, chuyuming2005@yahoo.com.cn
Received 20 November 2009; Accepted 3 March 2010
Academic Editor: Shusen Ding
Copyright q 2010 W.-F. Xia and Y.-M. Chu. 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.
For x x1 , x2 , . . . , xn ∈ 0, 1n and r ∈ {1, 2,
. . . , n}, the symmetric function Fn x, r is defined as
Fn x, r Fn x1 , x2 , . . . , xn ; r 1≤i1 <i2 ···<ir ≤n rj1 1xij /1−xij , where i1 , i2 , . . . , in are positive
integers. In this paper, the Schur convexity, Schur multiplicative convexity, and Schur harmonic
convexity of Fn x, r are discussed. As consequences, several inequalities are established by use of
the theory of majorization.
1. Introduction
Throughout this paper, we use the following notation system.
For x x1 , x2 , . . . , xn , y y1 , y2 , . . . , yn ∈ Rn , and α ∈ R, let
x y x1 y1 , x2 y2 , . . . , xn yn ,
xy x1 y1 , x2 y2 , . . . , xn yn ,
αx αx1 , αx2 , . . . , αxn ,
An x Gn x n
1
xi ,
n i1
n
1.1
1/n
xi
,
i1
α x α x1 , α x2 , . . . , α xn .
2
Journal of Inequalities and Applications
If xi > 0, i 1, 2, . . . , n, then we denote by
1 1
1
1
, ,...,
x
x1 x2
xn
log x log x1 , log x2 , . . . , log xn .
1.2
Next we introduce some definitions and well-known results.
Definition 1.1. Let E ⊆ Rn be a set, and a real-valued function F on E is called a Schur convex
function if
Fx1 , x1 , . . . , xn F y1 , y2 , . . . , yn
1.3
for each pair of n-tuples x x1 , . . . , xn and y y1 , . . . , yn in E with x ≺ y, that is,
k
k
xi yi ,
i1
k 1, 2, . . . , n − 1,
i1
n
i1
xi n
1.4
yi ,
i1
where xi denotes the ith largest component of x. A function F is called Schur concave if −F
is Schur convex.
Definition 1.2. Let E ⊆ 0, ∞n be a set, and a function F : E → 0, ∞ is called a Schur
multiplicatively convex function on E if
Fx1 , x2 , . . . , xn F y1 , y2 , . . . , yn
1.5
for each pair of n-tuples x x1 , x2 , . . . , xn and y y1 , y2 , . . . , yn in E with log x ≺ log y. F
is called Schur multiplicatively concave if 1/F is Schur multiplicatively convex.
Definition 1.3. Let E ⊆ 0, ∞n be a set. A function F : E → R is called a Schur harmonic
convex or Schur harmonic concave, resp. function on E if
Fx1 , x2 , . . . , xn or , resp. F y1 , y2 , . . . , yn
1.6
for each pair of n-tuples x x1 , x2 , . . . , xn and y y1 , y2 , . . . , yn in E with 1/x ≺ 1/y.
Schur convexity was introduced by Schur 1 in 1923 and it has many applications in
analytic inequalities 2, 3, extended mean values 4, 5, graphs and matrices 6, and other
related fields. Recently, the Schur multiplicative convexity was investigated in 7–9 and the
Schur hamonic convexity was discussed in 10.
Journal of Inequalities and Applications
3
For x x1 , x2 , . . . , xn ∈ 0, 1n n 2 and r ∈ {1, 2, . . . , n}, the symmetric function
Fn x, r is defined as
Fn x, r Fn x1 , x2 , . . . , xn ; r r 1x
ij
1i1 <i2 ···<ir n j1
1 − xij
,
1.7
where i1 , i2 , . . . , in are positive integers.
The aim of this article is to discuss the Schur convexity, Schur multiplicative convexity,
and Schur harmonic convexity of the symmetric function Fn x, r.
Lemma 1.4 see 11. Let f : 0, 1n → R be a continuous symmetric function. If f is differentiable
in 0, 1n , then f is Schur convex in 0, 1n if and only if
x1 − x2 ∂fx ∂fx
−
∂x1
∂x2
1.8
0
for all x x1 , x2 , . . . , xn ∈ 0, 1n .
Lemma 1.5 see 7. Let f : 0, 1n → 0, ∞ be a continuous symmetric function. If f is
differentiable in 0, 1n , then f is Schur multiplicatively convex in 0, 1n if and only if
log x1 − log x2
x1
∂fx
∂fx
− x2
∂x1
∂x2
0
1.9
for all x x1 , x2 , . . . , xn ∈ 0, 1n .
Lemma 1.6 see 10. Let f : 0, 1n → 0, ∞ be a continuous symmetric function. If f is
differentiable in 0, 1n , then f is Schur harmonic convex in 0, 1n if and only if
x1 − x2 ∂fx
x12
∂x1
−
∂fx
x22
∂x2
0
1.10
for all x x1 , x2 , . . . , xn ∈ 0, 1n .
Lemma 1.7 see 12. Let x x1 , x2 , . . . , xn ∈ 0, ∞n and
c−x
nc/s − 1
c − x1
c − x2
c − xn
,
,...,
nc/s − 1 nc/s − 1
nc/s − 1
c x1
c x2
c xn
,
,...,
nc/s 1 nc/s 1
nc/s 1
i1
xi s. If c s, then
≺ x1 , x2 , . . . , xn x.
Lemma 1.8 see 12. Let x x1 , x2 , . . . , xn ∈ 0, ∞n and
cx
nc/s 1
n
n
i1
1.11
xi s. If c 0, then
≺ x1 , x2 , . . . , xn x.
1.12
4
Journal of Inequalities and Applications
Lemma 1.9 see 13. Suppose that x x1 , x2 , . . . , xn ∈ 0, ∞n and
then
s − λx
n−λ
s − λx1 s − λx2
s − λxn
,
,...,
n−λ
n−λ
n−λ
n
i1
xi s. If 0 λ 1,
≺ x1 , x2 , . . . , xn x.
1.13
2. Main Result
Theorem 2.1. The symmetric function Fn x, r is Schur convex, Schur multiplicatively convex, and
Schur harmonic convex in 0, 1n for all r 1, 2, . . . , n.
Proof. According to Lemmas 1.4–1.6 we only need to prove that
∂Fn x, r ∂Fn x, r
−
0,
∂x1
∂x2
∂Fn x, r
∂Fn x, r
− x2
0,
log x1 − log x2 x1
∂x1
∂x2
∂Fn x, r
∂Fn x, r
− x22
0,
x1 − x2 x12
∂x1
∂x2
x1 − x2 2.1
2.2
2.3
for all x x1 , x2 , . . . , xn ∈ 0, 1n and r ∈ {1, 2, . . . , n}.
We divided the proof into seven cases.
Case 1. If r 1 and x x1 , x2 , . . . , xn ∈ 0, 1n , then 1.7 leads to
Fn x, 1 n
1 xi
i1
1 − xi
,
2.4
∂Fn x, 1
2
, i 1, 2,
∂xi
1 − xi 2
2x1 − x2 2 2 − x1 − x2 ∂Fn x, 1 ∂Fx, 1
−
0,
2.5
x1 − x2 ∂x1
∂x2
1 − x1 2 1 − x2 2
2 log x1 − log x2 x1 − x2 1 − x1 x2 ∂Fn x, 1
∂Fn x, 1
− x2
0,
log x1 − log x2 x1
∂x1
∂x2
1 − x1 2 1 − x2 2
2.6
∂Fn x, 1
∂Fn x, 1
2x1 − x2 2 x1 x2 − 2x1 x2 − x22
0.
x1 − x2 x12
∂x1
∂x2
1 − x1 2 1 − x2 2
2.7
Journal of Inequalities and Applications
5
Case 2. If n r 2 and x x1 , x2 , . . . , xn ∈ 0, 1n , then 1.7 yields
F2 x, 2 1 x1 1 x2 ,
1 − x1 1 − x2 2.8
∂F2 x, 2
2F2 x, 2
, i 1, 2,
∂xi
1 − xi 1 xi 2x1 − x2 2 x1 x2 ∂F2 x, 2 ∂F2 x, 2
−
0,
2.9
x1 − x2 ∂x1
∂x2
1 − x1 2 1 − x2 2
2 log x1 − log x2 x1 − x2 1 x1 x2 ∂F2 x, 2
∂F2 x, 2
− x2
0,
log x1 − log x2 x1
∂x1
∂x2
1 − x1 2 1 − x2 2
2.10
x1 − x2 ∂F2 x, 2
x12
∂x1
−
∂F2 x, 2
x22
∂x2
2x1 − x2 2 x1 x2 F2 x, 2
1 − x1 2 1 − x2 2
0.
2.11
Case 3. If n 3, r 2, and x x1 , x2 , . . . , xn ∈ 0, 1n , then from 1.7 we clearly see that
1 x1 1 x2 F3 x, 2 1 − x1 1 − x2 1 x1 1 x2
1 − x1 1 − x2
1 x3
,
1 − x3
∂F3 x, 2
21 x3−i 21 x3 ,
2
∂xi
1 − xi 1 − x3−i 1 − xi 2 1 − x3 ∂F3 x, 2 ∂F3 x, 2
−
x1 − x2 ∂x1
∂x2
2x1 − x2 2
1 − x1 2 1 − x2 2
log x1 − log x2
x1
i 1, 2,
1 x3
0,
x1 x2 2 − x1 − x2 1 − x3
∂F3 x, 2
∂F3 x, 2
− x2
∂x1
∂x2
2 log x1 − log x2 x1 − x2 1 − x1 2 1 − x2 2
2.12
2.13
1 x3
1 x1 x2 1 − x1 x2 0,
1 − x3
2.14
∂F3 x, 2
∂F3 x, 2
− x22
x1 − x2 x12
∂x1
∂x2
2x1 − x2 2
1 − x1 2 1 − x2 2
1 x3
0.
x1 x2 x1 x2 − 2x1 x2 1 − x3
2.15
6
Journal of Inequalities and Applications
Case 4. If n 4, r 2, and x x1 , x2 , . . . , xn ∈ 0, 1n , then 1.7 implies that
Fn x, 2 1 x1 1 x2 1 − x1 1 − x2 n
1 xi
1 x1 1 x2 1 − x1 1 − x2 i3 1 − xi
Fn−2 x3 , x4 , . . . , xn ; 2,
2.16
n
2 1 x3−j
∂Fn x, 2
1 xi
2
,
2
2
∂xj
1 − xj 1 − x3−j
1 − xj i3 1 − xi
x1 − x2 ∂Fn x, 2 ∂Fn x, 2
−
∂x1
∂x2
x1 x2 2 − x1 − x2 1 − x1 2 1 − x2 2
x1
2x1 − x2 2
log x1 − log x2
j 1, 2,
n
1 xi
i3
∂Fn x, 2
∂Fn x, 2
− x2
∂x1
∂x2
1 − xi
2.17
0,
n
2 log x1 − log x2 x1 − x2 1 xi
1 x1 x2 1 − x1 x2 0,
1 − xi
1 − x1 2 1 − x2 2
i3
2.18
∂Fn x, 2
∂Fn x, 2
− x22
x1 − x2 x12
∂x1
∂x2
2x1 − x2 2
1 − x1 2 1 − x2 2
x1 x2 x1 x2 − 2x1 x2 n
1 xi
i3
1 − xi
2.19
0.
Case 5. If n r 3, and x x1 , x2 , . . . , xn ∈ 0, 1n , then 1.7 leads to
Fn x, n n
1 xi
i1
1 − xi
,
∂Fn x, n
2
Fn x, n, i 1, 2,
∂xi
1 − xi 1 xi ∂Fn x, n ∂Fn x, n
−
x1 − x2 ∂x1
∂x2
2x1 − x2 2 x1 x2 Fn x, n 0,
1 − x12 1 − x22
2.20
2.21
Journal of Inequalities and Applications
∂Fn x, n
∂Fn x, n
− x2
log x1 − log x2 x1
∂x1
∂x2
7
2.22
2 log x1 − log x2 x1 − x2 1 x1 x2 Fn x, n 0,
1 − x12 1 − x22
∂Fn x, n
∂Fn x, n
− x22
x1 − x2 x12
∂x1
∂x2
2.23
2x1 − x2 2 x1 x2 Fn x, n 0.
1 − x12 1 − x22
Case 6. If n 4, r n − 1, and x x1 , x2 , . . . , xn ∈ 0, 1n , then 1.7 yields
Fn x, n − 1 1 x1 1 x2 Fn−2 x3 , x4 , . . . , xn ; n − 3
1 − x1 1 − x2 n
1 xi
1 x1 1 x2 ,
1 − x1 1 − x2 i3 1 − xi
2.24
2 1 x3−j
∂Fn x, n − 1
2 Fn−2 x3 , x4 , . . . , xn ; n − 3
∂xj
1 − xj 1 − x3−j
2
1 − xj
2
n
1 xi
i3
1 − xi
,
j 1, 2,
∂Fn x, n − 1 ∂Fn x, n − 1
−
x1 − x2 ∂x1
∂x2
2x1 − x2 2
1 − x1 2 1 − x2 2
x1 x2 Fn−2 x3 , x4 , . . . , xn ; n − 3
2 − x1 − x2 n
1 xi
i3
log x1 − log x2
2.25
1 − xi
0,
∂Fn x, n − 1
∂Fn x, n − 1
− x2
x1
∂x1
∂x2
2 log x1 − log x2 x1 − x2 1 x1 x2 Fn−2 x3 , x4 , . . . , xn ; n − 3
1 − x1 2 1 − x2 2
1 − x1 x2 n
1 xi
i3
1 − xi
0,
2.26
8
Journal of Inequalities and Applications
∂Fn x, n − 1
∂Fn x, n − 1
− x22
x1 − x2 x12
∂x1
∂x2
2x1 − x2 2
x1 x2 Fn−2 x3 , x4 , . . . , xn ; n − 3
2.27
1 − x1 2 1 − x2 2
n
1 xi
0.
x1 x2 − 2x1 x2 1 − xi
i3
Case 7. If n 5, 3 r n − 2, and x x1 , x2 , . . . , xn ∈ 0, 1n , then 1.7 implies
Fn x, r 1 x1 1 x2 Fn−2 x3 , x4 , . . . , xn ; r − 2
1 − x1 1 − x2 1 x1 1 x2
Fn−2 x3 , x4 , . . . , xn ; r − 1
1 − x1 1 − x2
Fn−2 x3 , x4 , . . . , xn ; r,
2.28
∂Fn x, r
21 x3−i Fn−2 x3 , x4 , . . . , xn ; r − 2
∂xi
1 − xi 2 1 − x3−i 2
Fn−2 x3 , x4 , . . . , xn ; r − 1,
1 − xi 2
∂Fn x, r ∂Fn x, r
−
x1 − x2 ∂x1
∂x2
2x1 − x2 2
2
1 − x1 1 − x2 2
i 1, 2,
x1 x2 Fn−2 x3 , x4 , . . . , xn ; r − 2
2.29
2 − x1 − x2 Fn−2 x3 , x4 , . . . , xn ; r − 1 0,
∂Fn x, r
∂Fn x, r
− x2
log x1 − log x2 x1
∂x1
∂x2
2 log x1 − log x2 x1 − x2 1 x1 x2 Fn−2 x3 , x4 , . . . , xn ; r − 2
1 − x1 2 1 − x2 2
1 − x1 x2 Fn−2 x3 , x4 , . . . , xn ; r − 1 0,
2.30
∂Fn x, r
∂Fn x, r
− x22
x1 − x2 x12
∂x1
∂x2
2x1 − x2 2
1 − x1 2 1 − x2 2
x1 x2 Fn−2 x3 , x4 , . . . , xn ; r − 2
x1 x2 − 2x1 x2 Fn−2 x3 , x4 , . . . , xn ; r − 1 0.
2.31
Journal of Inequalities and Applications
9
Therefore, inequality 2.1 follows from inequalities 2.5, 2.9, 2.13, 2.17, 2.21,
2.25, and 2.29, inequality 2.2 follows from inequalities 2.6, 2.10, 2.14,2.18, 2.22,
2.26, and 2.30, and inequality 2.3 follows from inequalities 2.7, 2.11, 2.15, 2.19,
2.23, 2.27, and 2.31.
3. Applications
In this section, we establish several inequalities by use of Theorem 2.1 and the theory of
majorization.
It follows from Lemmas 1.7, 1.8, 1.9, and Theorem 2.1 that Theorem 3.1 is obvious.
Theorem 3.1. If x x1 , x2 , . . . , xn ∈ 0, 1n , s 1
Fn x, r Fn
2
Fn x, r Fn
i1
xi , and r ∈ {1, 2, . . . , n}, then
c−x
,r
nc/s − 1
for c s,
cx
,r
for c 0,
nc/s 1
s − λx
;r
for 0 λ 1.
Fn x, r Fn
n−λ
3
n
Theorem 3.2. If x x1 , x2 , . . . , xn ∈ 0, 1n , s n
i1
Fn x, r Fn
3.1
xi , r ∈ {1, 2, . . . , n}, and 0 λ 1, then
s λx
;r .
nλ
3.2
Proof. Theorem 3.2 follows from Theorem 2.1 and the fact that
s λx
nλ
s λx1 s λx2
s λxn
,
,...,
nλ
nλ
nλ
≺ x1 , x2 , . . . , xn x.
3.3
If we take r 1 and s 1 in Theorem 3.13 and Theorem 3.2, respectively, then we
get the following.
Corollary 3.3. If x x1 , x2 , . . . , xn ∈ 0, 1n with
1
n
1 xi
i1
2
1 − xi
n
1 xi
i1
1 − xi
i1
xi 1 and 0 λ 1, then
n
n 1 − λ1 xi i1
n
n − 1 − λ1 − xi n
n 1 λ1 xi i1
n − 1 λ1 − xi ,
3.4
.
If we take r n and s 1 in Theorem 3.1(3) and Theorem 3.2, respectively, then one gets the
following.
10
Journal of Inequalities and Applications
n
Corollary 3.4. If x x1 , x2 , . . . , xn ∈ 0, 1n with
1
n
1 xi
i1
2
1 − xi
n
1 xi
i1
1 − xi
xi 1 and 0 λ 1, then
n
n 1 − λ1 xi n − 1 − λ1 − xi i1
i1
n
n 1 λ1 xi n − 1 λ1 − xi i1
,
3.5
.
Remark 3.5. If we take λ 0 in Corollaries 3.3 and 3.4, then we have
n
1 xi
i1
for 0 < xi < 1 and
n
i1
1 − xi
n
1 xi
nn 1
,
n−1
i1
1 − xi
n1
n−1
n
3.6
xi 1.
Theorem 3.6. If x x1 , x2 , . . . , xn ∈ 0, 1n , then
r 1x
ij
1i1 <i2 <···<ir n j1
1 − xij
An 1 x r
n!
.
r!n − r! An 1 − x
3.7
Proof. Theorem 3.6 follows from Theorem 2.1 and 1.7 together with the fact that
An x, An x, . . . , An x ≺ x1 , x2 , . . . , xn x.
3.8
If we take r n in Theorem 3.6, then we have the following.
Corollary 3.7. If x x1 , x2 , . . . , xn ∈ 0, 1n , then
Gn 1 x
Gn 1 − x
.
An 1 x
An 1 − x
3.9
Theorem 3.8. Let A A1 A2 · · · An1 be an n-dimensional simplex in Rn and let P be an arbitrary
point in the interior of A. If Bi is the intersection point of straight line Ai P and hyperplane i A1 A2 · · · Ai−1 Ai1 · · · An1 , i 1, 2, . . . , n 1, then
1
r A B PB
ij ij
ij
1i1 <i2 <···<ir n1 j1
2
P Aij
r A B PA
ij ij
ij
1i1 <i2 <···<ir n1 j1
P Bij
n2 r
n 1!
,
r!n − r 1!
n
n 1!
2n 1r .
r!n − r 1!
3.10
Journal of Inequalities and Applications
Proof. It is easy to see that
imply that
n1
i1
11
P Bi /Ai Bi 1 and
n1
i1
P Ai /Ai Bi n, and these identities
P Bn1
P B 1 P B2
,
,...,
,
A1 B 1 A2 B 2
An1 Bn1
n
n
P A1 P A 2
P An1
n
,
,...,
≺
,
,...,
.
n1 n1
n1
A1 B 1 A2 B 2
An1 Bn1
1
1
1
,
,...,
n1 n1
n1
≺
3.11
Therefore, Theorem 3.8 follows from Theorem 2.1 and 1.7 together with 3.11.
Theorem 3.9. Suppose that A ∈ Mn C n 2 is a complex matrix, and λ1 , λ2 , . . . , λn are the
eigenvalues of A. If A is a positive definite Hermitian matrix, then
1
r tr A λ
ij
1i1 <i2 <···<ir n j1
2
r
n!
tr A − λij
r!n − r!
2
1
λij
1i1 <i2 <···<ir n j1
4
n1 r
n!
,
r!n − r! n − 1
r tr A λ
ij
1i1 <i2 <···<ir n j1
3
tr A − λij
r
1i1 <i2 <···<ir n j1
2
1
λij
n!
r!n − r!
tr A tr A −
√
n
√
n
det A
det A
r
,
r
n
detI A 1
,
n
detI A − 1
3.12
n!
2n r
.
1
r!n − r!
tr A
Proof. We clearly see that λi > 0 i 1, 2, . . . , n, and
1
λ1 λ2
λn
1 1
3.13
, ,...,
,
,...,
≺
,
n n
n
tr A tr A
tr A
√
√
√
n
n
n
λ 1 λ2
λn
det A det A
det A
,
,...,
≺ log
,
,...,
log
,
3.14
tr A
tr A
tr A
tr A tr A
tr A
1
1
1
1
1
1
,
.
.
.
,
≺
log
,
,
,
.
.
.
,
,
log n
n
1 λ1 1 λ2
1 λn
detI A
detI A n detI A
3.15
n tr A
n tr A n tr A
,
,...,
≺ 1 λ1 , 1 λ2 , . . . , 1 λn .
3.16
n
n
n
Therefore, Theorem 3.91 follows from 1.7, 3.13, and the Schur convexity of
Fn x, r, Theorems 3.92 and 3 follow from 3.14 and 3.15 together with the Schur
multiplitively convexity of Fn x, r, and Theorem 3.94 follows from 3.16 and the Schur
harmonic convexity of Fn x, r.
12
Journal of Inequalities and Applications
Acknowledgments
The research was supported by NSF of China no. 60850005 and the NSF of Zhejiang
Province nos. D7080080, Y7080185, and Y607128.
References
1 I. Schur, “Über eine Klasse Von Mittelbildungen mit Anwendungen auf die Determinantentheorie,”
Sitzungsberichte der Berliner Mathematischen Gesellschaft, vol. 22, pp. 9–20, 1923.
2 G. H. Hardy, J. E. Littlewood, and G. Pólya, “Some simple inequalities satisfied by convex functions,”
Messenger of Mathematics, vol. 58, pp. 145–152, 1928/1929.
3 X.-M. Zhang, “Schur-convex functions and isoperimetric inequalities,” Proceedings of the American
Mathematical Society, vol. 126, no. 2, pp. 461–470, 1998.
4 F. Qi, J. Sándor, S. S. Dragomir, and A. Sofo, “Notes on the Schur-convexity of the extended mean
values,” Taiwanese Journal of Mathematics, vol. 9, no. 3, pp. 411–420, 2005.
5 Y. Chu and X. Zhang, “Necessary and sufficient conditions such that extended mean values are Schurconvex or Schur-concave,” Journal of Mathematics of Kyoto University, vol. 48, no. 1, pp. 229–238, 2008.
6 G. M. Constantine, “Schur convex functions on the spectra of graphs,” Discrete Mathematics, vol. 45,
no. 2-3, pp. 181–188, 1983.
7 Y. Chu, X. Zhang, and G. Wang, “The Schur geometrical convexity of the extended mean values,”
Journal of Convex Analysis, vol. 15, no. 4, pp. 707–718, 2008.
8 K. Guan, “A class of symmetric functions for multiplicatively convex function,” Mathematical
Inequalities & Applications, vol. 10, no. 4, pp. 745–753, 2007.
9 W.-D. Jiang, “Some properties of dual form of the Hamy’s symmetric function,” Journal of Mathematical
Inequalities, vol. 1, no. 1, pp. 117–125, 2007.
10 Y. Chu and Y. Lv, “The Schur harmonic convexity of the Hamy symmetric function and its
applications,” Journal of Inequalities and Applications, vol. 2009, Article ID 838529, 10 pages, 2009.
11 A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, vol. 143 of
Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1979.
12 K. Guan, “Schur-convexity of the complete symmetric function,” Mathematical Inequalities &
Applications, vol. 9, no. 4, pp. 567–576, 2006.
13 S. Wu, “Generalization and sharpness of the power means inequality and their applications,” Journal
of Mathematical Analysis and Applications, vol. 312, no. 2, pp. 637–652, 2005.