Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 493759, 15 pages
doi:10.1155/2009/493759
Research Article
Schur-Convexity for a Class of Symmetric
Functions and Its Applications
Wei-Feng Xia1 and Yu-Ming Chu2
1
2
School of Teacher 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 16 May 2009; Accepted 14 September 2009
Recommended by Jozef Banas
For x x1 , x2 , . . . , xn ∈ Rn , the symmetric function φn x, r is defined by φn x, r φn x1 ,
x2 , . . . , xn ; r 1≤i1 <i2 ···<ir ≤n rj1 xij /1xij 1/r , where r 1, 2, . . . , n and i1 , i2 , . . . , in are positive
integers. In this article, the Schur convexity, Schur multiplicative convexity and Schur harmonic
convexity of φn x, r are discussed. As applications, some inequalities are established by use of the
theory of majorization.
Copyright q 2009 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.
1. Introduction
Throughout this paper we use Rn to denote the n-dimensional Euclidean space over the field
of real numbers and Rn {x1 , x2 , . . . , xn : xi > 0, i 1, 2, . . . , n}. In particular, we use R to
denote R1 .
For the sake of convenience, we use the following notation system.
For x x1 , x2 , . . . , xn , y y1 , y2 , . . . , yn ∈ Rn , and α > 0, let
x y x1 y1 , x2 y2 , . . . , xn yn ,
xy x1 y1 , x2 y2 , . . . , xn yn ,
αx αx1 , αx2 , . . . , αxn ,
xα x1α , x2α , . . . , xnα ,
1
x
1 1
1
, ,...,
,
x1 x2
xn
2
Journal of Inequalities and Applications
log x log x1 , log x2 , . . . , log xn ,
ex ex1 , ex2 , . . . , exn .
1.1
The notion of Schur convexity was first introduced by Schur in 1923 1. It has
many important applications in analytic inequalities 2–7, combinatorial optimization 8,
isoperimetric problem for polytopes 9, linear regression 10, graphs and matrices 11,
gamma and digamma functions 12, reliability and availability 13, and other related fields.
The following definition for Schur convex or concave can be found in 1, 3, 7 and the
references therein.
Definition 1.1. Let E ⊆ Rn n ≥ 2 be a set, a real-valued function F on E is called a Schur
convex function if
Fx1 , x1 , . . . , xn ≤ F y1 , y2 , . . . , yn
1.2
for each pair of n-tuples x x1 , . . . , xn and y y1 , . . . , yn on E, such that x is majorized
by y in symbols x ≺ y, that is,
k
k
xi ≤
yi ,
i1
k 1, 2, . . . , n − 1,
i1
n
xi i1
n
1.3
yi ,
i1
where xi denotes the ith largest component in x. F is called Schur concave if −F is Schur
convex.
The notation of multiplicative convexity was first introduced by Montel 14. The
Schur multiplicative convexity was investigated by Niculescu 15, Guan 7, and Chu et
al. 16.
Definition 1.2 see 7, 16. Let E ⊆ Rn n ≥ 2 be a set, a real-valued function F : E → R is
called a Schur multiplicatively convex function on E if
Fx1 , x2 , . . . , xn ≤ F y1 , y2 , . . . , yn
1.4
for each pair of n-tuples x x1 , x2 , . . . , xn and y y1 , y2 , . . . , yn on E, such that x is
logarithmically majorized by y in symbols log x ≺ log y, that is,
k
k
xi ≤
yi ,
i1
k 1, 2, . . . , n − 1,
i1
n
i1
xi n
1.5
yi .
i1
However F is called Schur multiplicatively concave if 1/F is Schur multiplicatively convex.
Journal of Inequalities and Applications
3
In paper 17, Anderson et al. discussed an attractive class of inequalities, which
arise from the notion of harmonic convex functions. Here, we introduce the notion of Schur
harmonic convexity.
Definition 1.3. Let E ⊆ Rn n ≥ 2 be a set. A real-valued function F on E is called a Schur
harmonic convex function if
Fx1 , x2 , . . . , xn ≤ F y1 , y2 , . . . , yn
1.6
for each pair of n-tuples x x1 , x2 , . . . , xn and y y1 , y2 , . . . , yn on E, such that 1/x ≺ 1/y.
F is called a Schur harmonic concave function on E if 1.6 is reversed.
The main purpose of this paper is to discuss the Schur convexity, Schur multiplicative
convexity, and Schur harmonic convexity of the following symmetric function:
⎛
⎞1/r
r
xij
⎝
⎠ ,
φn x, r φn x1 , x2 , . . . , xn ; r 1 xij
j1
1≤i1 <i2 ···<ir ≤n
1.7
where x x1 , x2 , . . . , xn ∈ Rn n ≥ 2, r 1, 2, . . . , n, and i1 , i2 , . . . , ir are positive integers. As
applications, some inequalities are established by use of the theory of majorization.
2. Lemmas
In order to establish our main results we need several lemmas, which we present in this
section.
The following lemma is so-called Schur’s condition which is very useful for
determining whether or not a given function is Schur convex or Schur concave.
Lemma 2.1 see 6, 7, 18. Let f : Rn 0, ∞n → R 0, ∞ be a continuous symmetric
function. If f is differentiable in Rn , then f is Schur convex if and only if
xi − xj
∂f
∂f
−
∂xi ∂xj
≥0
2.1
for all i, j 1, 2, . . . , n and x x1 , . . . , xn ∈ Rn . Also f is Schur concave if and only if 2.1 is
reversed for all i, j 1, 2, . . . , n and x x1 , . . . , xn ∈ Rn . Here, f is a symmetric function in Rn
meaning that fP x fx for any x ∈ Rn and any n × n permutation matrix P .
Remark 2.2. Since f is symmetric, the Schur’s condition in Lemma 2.1, that is, 2.1 can be
reduced to
∂f
∂f
−
x1 − x2 ∂x1 ∂x2
≥ 0.
2.2
4
Journal of Inequalities and Applications
Lemma 2.3 see 7, 16. Let f : Rn → R be a continuous symmtric function. If f is differentiable
in Rn , then f is Schur multiplicatively convex if and only if
log x1 − log x2
x1
∂f
∂f
− x2
∂x1
∂x2
2.3
≥0
for all x x1 , x2 , . . . , xn ∈ Rn . Also f is Schur multiplicatively concave if and only if 2.3 is
reversed.
Lemma 2.4. Let f : Rn → R be a continuous symmetric function. If f is differentiable in Rn , then
f is Schur harmonic convex if and only if
∂f
∂f
− x22
≥0
x1 − x2 x12
∂x1
∂x2
2.4
for all x x1 , x2 , . . . , xn ∈ Rn . Also f is Schur harmonic concave if and only if 2.4 is reversed.
Proof. From Definitions 1.1 and 1.3, we clearly see the fact that f : Rn → R is Schur harmonic
convex if and only if Fx 1/f1/x : Rn → R is Schur concave.
This fact, Lemma 2.1 and Remark 2.2 together with elementary calculation imply that
Lemma 2.4 is true.
Lemma 2.5 see 5, 6. Let x x1 , x2 , . . . , xn ∈ Rn and
c−x
nc/s − 1
i1
c − x1
c − x2
c − xn
,
,...,
nc/s − 1 nc/s − 1
nc/s − 1
Lemma 2.6 see 6. Let x x1 , x2 , . . . , xn ∈ Rn and
cx
nc/s 1
n
n
≺ x1 , x2 , . . . , xn x.
i1
c x1
c x2
c xn
,
,...,
nc/s 1 nc/s 1
nc/s 1
xi s. If c ≥ s, then
xi s. If c ≥ 0, then
≺ x1 , x2 , . . . , xn x.
Lemma 2.7 see 19. Suppose that x x1 , x2 , . . . , xn ∈ Rn and
s − λx
n−λ
s − λx1 s − λx2
s − λxn
,
,...,
n−λ
n−λ
n−λ
2.5
n
i1
2.6
xi s. If 0 ≤ λ ≤ 1, then
≺ x1 , x2 , . . . , xn x.
2.7
3. Main Results
Theorem 3.1. For r ∈ {1, 2, . . . , n}, the symmetric function φn x, r is Schur concave in Rn .
Proof. By Lemma 2.1 and Remark 2.2, we only need to prove that
∂φn x, r ∂φn x, r
−
≤0
x1 − x2 ∂x1
∂x2
for all x x1 , x2 , . . . , xn ∈ Rn and r 1, 2, . . . , n.
3.1
Journal of Inequalities and Applications
5
The proof is divided into four cases.
Case 1. If r 1, then 1.7 leads to
φn x, 1 φn x1 , x2 , . . . , xn ; 1 n
xi
.
1 xi
i1
3.2
However 3.2 and elementary computation lead to
∂φn x, 1 ∂φn x, 1
x1 − x2 2 1 x1 x2 φn x, 1 ≤ 0.
−
−
x1 − x2 ∂x1
∂x2
x1 x2 1 x1 1 x2 3.3
Case 2. If n ≥ 2 and r n, then 1.7 yields
1/n
n
xi
.
φn x, n φn x1 , x2 , . . . , xn ; n 1 xi
i1
3.4
From 3.4 and elementary computation, we have
∂φn x, n ∂φn x, n
−
x1 − x2 ∂x1
∂x2
−
x1 − x2 2 2 x1 x2 n1 x1 2 1 x2 2
1/n−1
n
xi
≤ 0.
1 xi
i1
3.5
Case 3. If n ≥ 3 and r 2, then by 1.7 we have
φn x, 2 φn x1 , x2 , · · · , xn ; 2
x1
x2
1 x1 1 x2
⎡ 1/2 ⎤
1/2 n
x
x
j
1
⎣
⎦φn−1 x2 , x3 , . . . , xn ; 2
1
x
1
xj
1
j3
⎡ 1/2 ⎤
1/2 n
x
x2
x2
x1
j
⎣
⎦φn−1 x1 , x3 , . . . , xn ; 2.
1 x2 1 x1
1
x
1
xj
2
j3
3.6
6
Journal of Inequalities and Applications
Elementary computation and 3.6 yield
∂φn x, 2 ∂φn x, 2
−
x1 − x2 ∂x1
∂x2
φn x, 2
x1 − x2 2
2
1 x1 1 x2 ⎡
⎤
n 1 x x 3 2x 2x x
x
2
x
1
2
1
2 j 1 xj
1
2
⎦ ≤ 0.
×⎣
x1 x2 2x1 x2 j3
x1 xj 2x1 xj x2 xj 2x2 xj
−
3.7
Case 4. If n ≥ 4 and 3 ≤ r ≤ n − 1, then from 1.7, we have
φn x, r φn x1 , x2 , . . . , xn ; r
⎞1/r
r−1 x
ij
x
1
⎝
⎠
φn−1 x2 , x3 , . . . , xn ; r
1 x1 j1 1 xij
3≤i1 <i2 <···<ir−1 ≤n
⎛
⎞1/r
r−2 x
ij
x
x
1
2
⎝
⎠
×
1 x1 1 x2 j1 1 xij
3≤i1 <i2 <···<ir−2 ≤n
⎛
φn−1 x1 , x3 , . . . , xn ; r
⎛
3.8
⎞1/r
r−1 x
ij
⎝ x2 ⎠
1 x2 j1 1 xij
3≤i1 <i2 <···<ir−1 ≤n
⎞1/r
r−2 x
ij
x
x
1
2
⎝
⎠ ,
×
1 x1 1 x2 j1 1 xij
3≤i1 <i2 <···<ir−2 ≤n
x1 − x2 ⎛
∂φn x, r ∂φn x, r
−
∂x1
∂x2
x1 − x2 2
−
1 x1 2 1 x2 2
⎡
2x1 x2
⎢
×⎣
r−2 x
/
1x
3≤i1 <i2 <···<ir−2 ≤n x1 /1x1 x2 /1x2 i
i
j
j
j1
⎤
x
1x1 x2 2x1 x2 r−1
/1x
i
i
j
j
j1
⎥
⎦
r−1 r−1 x2 /1x2 j1 xij / 1 xij
3≤i1 <i2 <···<ir−1 ≤n x1 /1x1 j1 xij / 1xij
×
φn x, r
≤ 0.
r
3.9
Therefore, 3.1 follows from Cases 1–4 and the proof of Theorem 3.1 is completed.
Journal of Inequalities and Applications
7
For the Schur multiplicative convexity or concavity of φn x, r, we have the following
theorem
Theorem 3.2. It holds that φn x, r is Schur multiplicatively concave in 1, ∞n .
Proof. According to Lemma 2.3 we only need to prove that
log x1 − log x2
∂φn x, r
∂φn x, r
− x2
x1
∂x1
∂x2
≤0
3.10
for all x x1 , x2 , . . . , n ∈ 1, ∞n and r 1, 2, . . . , n. Then proof is divided into four cases.
Case 1. If r 1, then 3.2 leads to
log x1 − log x2 x1 − x2 ∂φn x, 1
∂φn x, 1
φn x, 1 ≤ 0.
− x2
−
log x1 − log x2 x1
∂x1
∂x2
1 x1 1 x2 3.11
Case 2. If r n, n ≥ 2, then 3.4 yields
∂φn x, n
∂φn x, n
log x1 − log x2 x1
− x2
∂x1
∂x2
1/n−1
n
log x1 − log x2 x1 − x2 xi
≤ 0.
1 − x1 x2 1 xi
n1 x1 2 1 x2 2
i1
3.12
Case 3. If n ≥ 3 and r 2, then 3.6 implies
∂φn x, 2
∂φn x, 2
− x2
∂x1
∂x2
φn x, 2 log x1 − log x2 x1 − x2 2
1 x1 2 1 x2 2
⎡
1 − x1 x2
×⎣
x1 /1 x1 x2 /1 x2 log x1 − log x2
x1
⎤
−x1 x2 1 − x1 x2 xj / 1 xj
⎦ ≤ 0.
x2 /1 x2 xj / 1 xj
j3 x1 /1 x1 xj / 1 xj
n
3.13
8
Journal of Inequalities and Applications
Case 4. If n ≥ 4 and 3 ≤ r ≤ n − 1, then from 3.8 we have
∂φn x, r
∂φn x, r
− x2
log x1 − log x2 x1
∂x1
∂x2
log x1 − log x2 x1 − x2 1 x1 2 1 x2 2
⎡
1 − x1 x2
⎢
×⎣
r−2 3≤i1 <i2 <···<ir−2 ≤n x1 /1 x1 x2 /1 x2 j1 xij / 1 xij
⎤
−x1 x2 1 − x1 x2 r−1
j1 xij / 1 xij
⎥
⎦
r−1 r−1 x2 /1x2 j1 xij / 1 xij
3≤i1 <i2 <···<ir−1 ≤n x1 /1x1 j1 xij / 1 xij
×
φn x, r
≤ 0.
r
3.14
Therefore, Theorem 3.2 follows from 3.10 and Cases 1–4.
Remark 3.3. From 3.11 and 3.12 we know that φn x, 1 is Schur multiplicatively concave
in 0, ∞n and φn x, n is Schur multiplicatively convex in 0, 1n .
Theorem 3.4. For r ∈ {1, 2, . . . , n}, the symmetric function φn x, r is Schur harmonic convex in
Rn .
Proof. According to Lemma 2.4 we only need to prove that
2 ∂φn x, r
2 ∂φn x, r
− x2
≥0
x1 − x2 x1
∂x1
∂x2
3.15
for all x x1 , x2 , . . . , xn ∈ Rn and r 1, 2, . . . , n.
The proof is divided into four cases.
Case 1. If r 1, then from 3.2 we have
∂φn x, 1
∂φn x, 1
x1 − x2 2
φn x, 1 ≥ 0.
− x22
x1 − x2 x12
∂x1
∂x2
1 x1 1 x2 3.16
Case 2. If n ≥ 2 and r n, then 3.4 leads to
x1 − x2 ∂φn x, n
x12
∂x1
−
∂φn x, n
x22
∂x2
x1 − x2 2 x1 x2 2x1 x2 n1 x1 2 1 x2 2
1/n−1
n
xi
≥ 0.
1 xi
i1
3.17
Journal of Inequalities and Applications
9
Case 3. If n ≥ 3 and r 2, then 3.6 yields
∂φn x, 2
∂φn x, 2
− x22
x1 − x2 x12
∂x1
∂x2
φn x, 2
x1 − x2 2
2
1 x1 1 x2 ⎡
⎤
n x x x x x x 3x x x
1 xj
1 2
1 j
2 j
1 2 j
× ⎣1 ⎦
x1 xj 2x1 xj x2 xj 2x2 xj
j3
3.18
≥ 0.
Case 4. If n ≥ 4 and 3 ≤ r ≤ n − 1, then 3.8 implies
x1 − x2 ∂φn x, r
x12
∂x1
−
∂φn x, r
x22
∂x2
φn x, r
r
1 x1 1 x2 ⎡
x1 x2 2x1 x2
⎢
×⎣
r−2 3≤i1 <i2 <···<ir−2 ≤n x1 /1 x1 x2 /1 x2 j1 xij / 1 xij
x1 − x2 2
2
2
⎤
x1 x2 x1 x2 2x1 x2 r−2
j1 xij / 1xij
⎥
⎦
r−1 r−1 x2 /1 x2 j1 xij / 1 xij
3≤i1 <i2 <···<ir−1 ≤n x1 /1x1 j1 xij / 1 xij
≥ 0.
3.19
Therefore, 3.15 follows from Cases 1–4 and the proof of Theorem 3.4 is completed.
4. Applications
In this section, we establish some inequalities by use of Theorems 3.1, 3.2 and 3.4 and the
theory of majorization.
Theorem 4.1. If x x1 , x2 , · · · , xn ∈ Rn , s {1, 2, . . . , n}, then
n
i1
xi ,Hn x n/
1 φn x, r ≤ φn c − x/nc/s − 1, r for c ≥ s;
2 φn x, r ≥ φn cHn x − 1/cx − 1x, r for c ≥
n
3 φn x, r ≤ φn c x/nc/s 1, r for c ≥ 0;
4 φn x, r ≥ φn cHn x 1/cx 1x, r for c ≥ 0;
5 φn x, r ≤ φn s − λx/n − λ, r for 0 ≤ λ ≤ 1;
i1 1/xi ;
n
i1 1/xi ,
and r ∈
10
Journal of Inequalities and Applications
6 φn x, r ≥ φn n − λ/
n
i1
1/xi − λ/x , r for 0 ≤ λ ≤ 1;
7 φn x, r ≤ φn s λx/n λ, r for 0 ≤ λ ≤ 1;
8 φn x, r ≥ φn n λ/ ni1 1/xi λ/x , r for 0 ≤ λ ≤ 1.
Proof. Theorem 4.1 follows from Theorem 3.1, Theorem 3.4 and Lemmas 2.5–2.7 together with
the fact that
s λx
nλ
s λx1 s λx2
s λxn
,
,...,
nλ
nλ
nλ
≺ x1 , x2 , . . . , xn x.
Theorem 4.2. If x x1 , x2 , · · · , xn ∈ Rn , An x 1/n
i
⎛
⎝
j1
1≤i1 <i2 <···<ir ≤n
ii
⎛
⎝
1≤i1 <i2 <···<ir ≤n
⎞1/r
r
xij
1 xij
⎠
⎞1/r
r
1 ⎠
1
xij
j1
n
i1
4.1
xi , and r ∈ {1, 2, . . . , n}, then
An x n!/r·r!n−r!
≤ r
;
An 1 x
≥ r
1
An 1 x
n!/r·r!n−r!
4.2
.
Proof. We clearly see that
An x, An x, . . . , An x ≺ x1 , x2 , . . . , xn x.
4.3
Therefore, Theorem 4.2i follows from 4.3 and Theorem 3.1 together with 1.7, and
Theorem 4.2ii follows from 4.3 and Theorem 3.4 together with 1.7.
If we take r 1 in Theorem 4.2i and r n in Theorem 4.2, respectively, then we have
the following corollary.
Corollary 4.3. If x x1 , x2 , . . . , xn ∈ Rn and Gn x n
i1 xi Gn x
An x
≤
;
Gn 1 x An 1 x
x An x
≤
;
ii An
1x
An 1 x
1
1
.
≥
iii An
1x
An 1 x
1/n
, then
i
4.4
Remark 4.4. If we take ni1 xi 1 in Corollary 4.3i, then we obtain the Weierstrass
inequality: see 20, page 260
n 1
i1
xi
1 ≥ n 1n .
4.5
Journal of Inequalities and Applications
11
Theorem 4.5. If x x1 , x2 , . . . , xn ∈ Rn and r ∈ {1, 2, . . . , n}, then
⎛
i
⎝
j1
1≤i1 <i2 <···<ir ≤n
⎛
ii
⎞1/r
r
xij
⎝
1 xij
⎞1/r
r
1≤i1 <i2 <···<ir ≤n
Hn x n!/r·r!n−r!
≥ r
;
1 Hn x
⎠
1 ⎠
1
xij
j1
≤ r
1
1 Hn x
n!/r·r!n−r!
4.6
.
Proof. We clearly see that
1
1
1
,
,...,
Hn x Hn x
Hn x
≺
1 1
1
, ,...,
x1 x2
xn
1
.
x
4.7
Therefore, Theorem 4.5i follows from 4.7 and Theorem 3.4 together with 1.7, and
Theorem 4.5ii follows from 4.7 and Theorem 3.1 together with 1.7.
If we take r 1 and r n in Theorem 4.5, respectively, then we get the following
corollary.
Corollary 4.6. If x x1 , x2 , . . . , xn ∈ Rn , then
i
Gn x
Hn x
≥
;
Gn 1 x 1 Hn x
ii
Gn 1 x ≥ 1 Hn x;
iii
iv
x Hn x
≥
;
1x
1 Hn x
1
1
.
An
≤
1x
1 Hn x
An
4.8
Theorem 4.7. If x x1 , x2 , . . . , xn ∈ 1, ∞n and r ∈ {1, 2, . . . , n}, then
⎛
⎝
1≤i1 <i2 <···<ir ≤n
r
xij
j1
1 xij
⎞1/r
⎠
Gn x n!/r·r!n−r!
≤ r
.
1 Gn x
4.9
Proof. We clearly see that
logGn x, Gn x, . . . , Gn x ≺ logx1 , x2 , . . . , xn .
Therefore, Theorem 4.7 follows from 4.10, Theorem 3.2, and 1.7.
4.10
12
Journal of Inequalities and Applications
If we take r 1 and r n in Theorem 4.7, respectively, then we get the following
corollary.
Corollary 4.8. If x x1 , x2 , . . . , xn ∈ 1, ∞n , then
x Gn x
≤
;
1x
1 Gn x
i
An
ii
Gn 1 x ≥ 1 Gn x.
4.11
Remark 4.9. From Remark 3.3 and 4.10 together with 1.7 we clearly see that inequality in
Corollary 4.8i is reversed for x ∈ 0, 1n and inequality in Corollary 4.8ii is true for x ∈ Rn .
Theorem 4.10. If x x1 , x2 , · · · , xn ∈ Rn , then
i
⎛
⎝
r 1x
ij
j1
1≤i1 <i2 ···<ir ≤n
2 xij
⎞1/r
⎠
≥
n−1!/r!n−r! n−1!/r·r!n−r−1!
1 ni1 xi r − 1
r
×
n
2
2
2 i1 xi
for 1 ≤ r ≤ n − 1;
ii
n
1 xi
i1
2 xi
iii
1 ni1 xi n − 1
;
2
2 ni1 xi
≥
⎛
r
⎝
1≤i1 <i2 ···<ir ≤n
⎞1/r
1 ⎠
r − 1 n−1!/r−1!n−r! r n−1!/r·r!n−r−1!
1
≤
×
n
2 xij
2
2
2 i1 xi
j1
for 1 ≤ r ≤ n − 1;
iv
n
1
1
n−1
.
≤
n
2 xi 2 i1 xi
2
i1
4.12
Proof. Theorem 4.10 follows from Theorems 3.1, 3.4, and 1.7 together with the fact that
n
1 xi , 1, 1, . . . , 1 .
1 x1 , 1 x2 , . . . , 1 xn ≺
4.13
i1
Theorem 4.11. 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 for r ∈ {1, 2, . . . , n 1} one has
i
⎛
⎝
1≤i1 <i2 ···<ir ≤n1
r
P Bij
j1
Aij Bij P Bij
⎞1/r
⎠
≤
r
n2
n1!/r·r!n−r1!
;
Journal of Inequalities and Applications
⎛
ii
⎝
1≤i1 <i2 ···<ir ≤n1
iii
Aij Bij
j1
Aij Bij P Bij
⎝
1≤i1 <i2 ···<ir ≤n1
iv
n 1 n1!/r·r!n−r1!
≥ r
;
n2
⎞1/r
P Aij
j1
Aij Bij P Aij
j1
1≤i1 <i2 ···<ir ≤n1
⎠
r
⎛
r
⎝
⎞1/r
r
⎛
13
⎠
⎞1/r
Aij Bij
Aij Bij P Aij
⎠
≤ r
n
2n 1
n1!/r·r!n−r1!
;
n1!/r·r!n−r1!
n1
≥ r
.
2n 1
4.14
Proof. It is easy to see that
equations imply
n1
i1
P Bi /Ai Bi 1 and
n1
i1
P Ai /Ai Bi n, these identical
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
≺
4.15
Therefore, Theorem 4.11 follows from 4.15, Theorems 3.1, 3.4, and 1.7.
Remark 4.12. Mitrinovic’ et al. 21, pages 473–479 established a series of inequalities for
P Ai /Ai Bi and P Bi /Ai Bi , i 1, 2, . . . , n 1. Obvious, our inequalities in Theorem 4.11 are
different from theirs.
Theorem 4.13. Suppose that A ∈ Mn C n ≥ 2 is a complex matrix, λ1 , λ2 , . . . , λn , and
σ1 , σ2 , . . . , σn are the eigenvalues and singular values of A, respectively. If A is a positive definite
Hermitian matrix and r ∈ {1, 2, . . . , n}, then
i
⎛
⎝
j1
1≤i1 <i2 ···<ir ≤n
ii
iii
r
λij
n!/r·r!n−r!
trA
≤ r
;
n trA
r
⎛
⎝
1≤i1 <i2 ···<ir ≤n
iv
⎠
⎞1/r
n!/r·r!n−r!
1
n
⎝
⎠ ≥ r
;
1 λij
n trA
j1
1≤i1 <i2 ···<ir ≤n
⎛
1 λij
⎞1/r
⎛
⎝
1≤i1 <i2 ···<ir ≤n
r 1λ
ij
j1
r
2 λij
⎞1/r
⎠
n!/r·r!n−r!
n
detI A
≤ r
;
1 n detI A
⎞1/r
1
⎠
trA
λij
j1
≤ r
trA 1
√
n
n!/r·r!n−r!
detA
;
14
v
Journal of Inequalities and Applications
⎞1/r
⎛
⎞1/r
r
1
1
⎝
⎠ ≥
⎝
⎠ .
n
n
n
n
λ
λ
σ
σ
λ
σ
i
i
i
i
i
i
j
j
i1
i1
i1
i1
j1
j1
1≤i1 <i2 ···<ir ≤n
1≤i1 <i2 ···<ir ≥n
⎛
r
4.16
Proof. i–ii We clearly see that λi > 0 i 1, 2, . . . , n and
trA
trA trA
,
,...,
n
n
n
n
i1
λi trA, then we have
≺ λ1 , λ2 , . . . , λn .
4.17
Therefore, Theorem 4.13i and ii follows from 4.17, Theorems 3.1, 3.4, and 1.7.
iii It is easy to see that 1 λ1 , 1 λ2 , . . . , 1 λn are the eigenvalues of matrix I A and
n
i1 1 λi detI A, then we get
log
n
detI A,
n
detI A, . . . , n detI A ≺ log1 λ1 , 1 λ2 , . . . , 1 λn 1 λi ≥ 1,
4.18
i 1, 2, . . . , n.
Therefore, Theorem 4.13iii follows from 4.18, Theorem 3.2, and 1.7.
iv It is not difficult to verify that
trA
trA
trA
trA
trA trA
,
,
.
.
.
,
log √
,
,
.
.
.
,
,
≺
log
√
√
n
n
n
λ1 λ2
λn
det A det A
det A
trA
≥ 1,
λi
4.19
i 1, 2, . . . , n.
Therefore, Theorem 4.13iv follows from 4.19, and Theorem 3.2 together with 1.7.
v A result due to Weyl 22 gives
logλ1 , λ2 , . . . , λn ≺ logσ1 , σ2 , . . . , σn .
4.20
From 4.20, we clearly see that
n
log
λi λ1
n
i1
n
i1
σi
,
n
i1
λi λ2
n
n
i1
σi
,...,
n
i1
λi λn
n
n
i1
σi
λi ni1 σi
λi ni1 σi
λi ni1 σi
,
, i1
, . . . , i1
σ1
σ2
σn
n
n
n
λi ni1 σi
i1 λi i1 σi
, i1
≥ 1, i 1, 2, . . . , n.
λi
σi
≺ log
i1
Therefore, Theorem 4.13v follows from 4.21, Theorem 3.2, and 1.7.
4.21
Journal of Inequalities and Applications
15
Acknowledgment
This work was supported by the National Science Foundation of China no. 60850005 and
the Natural Science Foundation of Zhejiang Province no. D7080080, Y607128, Y7080185.
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