Volume 2009, Article ID 493759,15pages doi:10.1155/2009/493759
Research Article
Schur-Convexity for a Class of Symmetric
Functions and Its Applications
Wei-Feng Xia
1and Yu-Ming Chu
21School of Teacher Education, Huzhou Teachers College, Huzhou 313000, China 2Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
Correspondence should be addressed to Yu-Ming Chu,[email protected]
Received 16 May 2009; Accepted 14 September 2009
Recommended by Jozef Banas
Forx x1, x2, . . . , xn ∈ Rn, the symmetric functionφnx, ris defined by φnx, r φnx1, x2, . . . , xn;r 1≤i1<i2···<ir≤n
r
j1xij/1xij
1/r, wherer1,2, . . . , nandi
1, i2, . . . , inare positive
integers. In this article, the Schur convexity, Schur multiplicative convexity and Schur harmonic convexity ofφnx, rare discussed. As applications, some inequalities are established by use of the theory of majorization.
Copyrightq2009 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 useRnto denote then-dimensional Euclidean space over the field of real numbers andRn {x1, x2, . . . , xn:xi >0, i 1,2, . . . , n}. In particular, we useRto
denoteR1.
For the sake of convenience, we use the following notation system. Forx x1, x2, . . . , xn, y y1, y2, . . . , yn∈Rn, andα >0, let
xyx1y1, x2y2, . . . , xnyn,
xyx1y1, x2y2, . . . , xnyn,
αx αx1, αx2, . . . , αxn,
xαxα
1, x2α, . . . , xαn
,
1
x
1
x1,
1
x2, . . . ,
1
xn
logxlogx1,logx2, . . . ,logxn, 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 inequalities2–7, combinatorial optimization8, isoperimetric problem for polytopes 9, linear regression 10, graphs and matrices 11, gamma and digamma functions12, reliability and availability13, 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. LetE ⊆ Rn n ≥ 2be a set, a real-valued functionF onEis called a Schur convex function if
Fx1, x1, . . . , xn≤Fy1, y2, . . . , yn 1.2
for each pair ofn-tuplesx x1, . . . , xnandy y1, . . . , ynonE, such thatxis majorized
byyin symbolsx≺y, that is,
k
i1
xi≤ k
i1
yi, k1,2, . . . , n−1,
n
i1
xi
n
i1
yi,
1.3
wherexi denotes theith largest component inx.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 1.2see7,16. LetE⊆Rn n≥2be a set, a real-valued functionF :E → Ris called a Schur multiplicatively convex function onEif
Fx1, x2, . . . , xn≤Fy1, y2, . . . , yn 1.4
for each pair of n-tuples x x1, x2, . . . , xn and y y1, y2, . . . , ynon E, such that x is
logarithmically majorized byyin symbols logx≺logy, that is,
k
i1
xi≤ k
i1
yi, k1,2, . . . , n−1,
n
i1
xi
n
i1
yi.
1.5
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. LetE ⊆ Rn n ≥ 2be a set. A real-valued functionF onEis called a Schur harmonic convex function if
Fx1, x2, . . . , xn≤F
y1, y2, . . . , yn
1.6
for each pair ofn-tuplesx x1, x2, . . . , xnandy y1, y2, . . . , ynonE, such that 1/x≺1/y. Fis called a Schur harmonic concave function onEif1.6is 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:
φnx, r φnx1, x2, . . . , xn;r
1≤i1<i2···<ir≤n ⎛ ⎝r
j1
xij
1xij ⎞ ⎠
1/r
, 1.7
wherex x1, x2, . . . , xn∈Rn n≥2,r1,2, . . . , n, andi1, 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 2.1see6,7,18. Letf : Rn 0,∞n → R 0,∞be a continuous symmetric
function. Iffis differentiable inRn, thenfis Schur convex if and only if
xi−xj
∂f ∂xi −
∂f ∂xj
≥0 2.1
for alli, j 1,2, . . . , nand x x1, . . . , xn ∈ Rn. Alsof is Schur concave if and only if 2.1is
reversed for alli, j 1,2, . . . , nandx x1, . . . , xn ∈ Rn. Here,f is a symmetric function inRn
meaning thatfPx fxfor anyx∈Rnand anyn×npermutation matrixP.
Remark 2.2. Sincef is symmetric, the Schur’s condition in Lemma 2.1, that is,2.1can be reduced to
x1−x2
∂f
∂x1 −
∂f ∂x2
Lemma 2.3see7,16. Letf:Rn → Rbe a continuous symmtric function. Iffis differentiable
inRn, thenfis Schur multiplicatively convex if and only if
logx1−logx2
x1∂x∂f 1 −x2
∂f ∂x2
≥0 2.3
for allx x1, x2, . . . , xn ∈ Rn. Also f is Schur multiplicatively concave if and only if 2.3 is
reversed.
Lemma 2.4. Letf:Rn → Rbe a continuous symmetric function. Iffis differentiable inRn, then fis Schur harmonic convex if and only if
x1−x2
x2 1
∂f ∂x1 −x
2 2
∂f ∂x2
≥0 2.4
for allx x1, x2, . . . , xn∈Rn. Alsofis Schur harmonic concave if and only if 2.4is reversed.
Proof. From Definitions1.1and1.3, we clearly see the fact thatf:Rn → Ris Schur harmonic convex if and only ifFx 1/f1/x:Rn → Ris Schur concave.
This fact,Lemma 2.1andRemark 2.2together with elementary calculation imply that
Lemma 2.4is true.
Lemma 2.5see5,6. Letx x1, x2, . . . , xn∈Rnandni1xis. Ifc≥s, then
c−x
nc/s−1
c−x
1
nc/s−1,
c−x2
nc/s−1, . . . ,
c−xn nc/s−1
≺x1, x2, . . . , xn x. 2.5
Lemma 2.6see6. Letx x1, x2, . . . , xn∈Rnandni1xis. Ifc≥0, then
cx
nc/s1
cx
1
nc/s1,
cx2
nc/s1, . . . ,
cxn
nc/s1
≺x1, x2, . . . , xn x. 2.6
Lemma 2.7see19. Suppose thatx x1, x2, . . . , xn∈Rnandni1xis.If0≤λ≤1,then
s−λx
n−λ
s−λx
1
n−λ , s−λx2
n−λ , . . . , s−λxn
n−λ
≺x1, x2, . . . , xn x. 2.7
3. Main Results
Theorem 3.1. Forr∈ {1,2, . . . , n},the symmetric functionφnx, ris Schur concave inRn.
Proof. ByLemma 2.1andRemark 2.2, we only need to prove that
x1−x2
∂φ
nx, r
∂x1 −
∂φnx, r ∂x2
≤0 3.1
The proof is divided into four cases.
Case 1. Ifr1, then1.7leads to
φnx,1 φnx1, x2, . . . , xn; 1 n
i1
xi
1xi. 3.2
However3.2and elementary computation lead to
x1−x2
∂φ
nx,1
∂x1 −
∂φnx,1 ∂x2
−x1−x221x1x2
x1x21x11x2 φnx,
1≤0. 3.3
Case 2. Ifn≥2 andrn, then1.7yields
φnx, n φnx1, x2, . . . , xn;n
n
i1
xi
1xi
1/n
. 3.4
From3.4and elementary computation, we have
x1−x2
∂φ
nx, n
∂x1 −
∂φnx, n ∂x2
−x1−x222x1x2
n1x121x22
n
i1
xi
1xi 1/n−1
≤0. 3.5
Case 3. Ifn≥3 andr2, then by1.7we have
φnx,2 φnx1, x2,· · · , xn; 2
x
1
1x1
x2
1x2
1/2⎡
⎣ n j3
x1
1x1
xj
1xj
1/2⎤
⎦φn−1x2, x3, . . . , xn; 2
x
2
1x2
x1
1x1
1/2⎡
⎣ n j3
x2
1x2
xj
1xj 1/2⎤
⎦φn−1x1, x3, . . . , xn; 2.
Elementary computation and3.6yield
x1−x2
∂φ
nx,2
∂x1 −
∂φnx,2 ∂x2
− x1−x22 1x11x2
φnx,2
2
×
⎡
⎣ 2x1x2
x1x22x1x2
n
j3
1x1x2 32x12x2xj1xj x1xj2x1xjx2xj2x2xj
⎤ ⎦≤0.
3.7
Case 4. Ifn≥4 and 3≤r≤n−1, then from1.7, we have
φnx, r φnx1, x2, . . . , xn;r
φn−1x2, x3, . . . , xn;r
3≤i1<i2<···<ir−1≤n ⎛ ⎝ x1
1x1
r−1
j1
xij
1xij ⎞ ⎠
1/r
×
3≤i1<i2<···<ir−2≤n ⎛ ⎝ x1
1x1
x2
1x2
r−2
j1
xij
1xij ⎞ ⎠
1/r
φn−1x1, x3, . . . , xn;r
3≤i1<i2<···<ir−1≤n ⎛ ⎝ x2
1x2
r−1
j1
xij
1xij
⎞ ⎠
1/r
×
3≤i1<i2<···<ir−2≤n ⎛ ⎝ x1
1x1
x2
1x2
r−2
j1
xij
1xij ⎞ ⎠
1/r
,
3.8
x1−x2
∂φ
nx, r
∂x1 −
∂φnx, r ∂x2
− x1−x22 1x121x22
×
⎡ ⎢
⎣
3≤i1<i2<···<ir−2≤n
2x1x2 x1/1x1 x2/1x2
r−2 j1
xij/
1xij
3≤i1<i2<···<ir−1≤n
1x1x22x1x2
r−1 j1
xij/1xij
x1/1x1 rj−11
xij/
1xij
x2/1x2 rj−11
xij/
1xij
⎤ ⎥ ⎦
×φnx, rr ≤0.
3.9
For the Schur multiplicative convexity or concavity ofφnx, r, we have the following
theorem
Theorem 3.2. It holds thatφnx, ris Schur multiplicatively concave in1,∞n.
Proof. According toLemma 2.3we only need to prove that
logx1−logx2
x1
∂φnx, r ∂x1 −x2
∂φnx, r ∂x2
≤0 3.10
for allx x1, x2, . . . , n∈1,∞nandr1,2, . . . , n.Then proof is divided into four cases.
Case 1. Ifr1, then3.2leads to
logx1−logx2
x1
∂φnx,1 ∂x1 −x2
∂φnx,1 ∂x2
−
logx1−logx2
x1−x2 1x11x2 φnx,
1≤0.
3.11
Case 2. Ifrn, n≥2, then3.4yields
logx1−logx2
x1∂φn∂xx, n 1 −x2
∂φnx, n ∂x2
logx1−logx2
x1−x2
n1x121x22
1−x1x2
n
i1
xi
1xi
1/n−1
≤0.
3.12
Case 3. Ifn≥3 andr2, then3.6implies
logx1−logx2
x1∂φnx,
2
∂x1 −x2
∂φnx,2 ∂x2
φnx,2
2
logx1−logx2
x1−x2 1x121x22
×
⎡
⎣ 1−x1x2
x1/1x1 x2/1x2
n
j3
−x1x2 1−x1x2
xj/1xj
x1/1x1 xj/1xjx2/1x2 xj/1xj ⎤ ⎦≤0.
Case 4. Ifn≥4 and 3≤r≤n−1, then from3.8we have
logx1−logx2
x1∂φn∂xx, r 1 −x2
∂φnx, r ∂x2
logx1−logx2
x1−x2 1x121x22
×
⎡ ⎢
⎣
3≤i1<i2<···<ir−2≤n
1−x1x2
x1/1x1 x2/1x2 rj−12
xij/
1xij
3≤i1<i2<···<ir−1≤n
−x1x21−x1x2rj−11
xij/
1xij
x1/1x1
r−1 j1
xij/
1xijx2/1x2
r−1 j1
xij/
1xij
⎤ ⎥ ⎦
× φnrx, r ≤0.
3.14
Therefore,Theorem 3.2follows from3.10and Cases1–4.
Remark 3.3. From3.11and3.12we know thatφnx,1is Schur multiplicatively concave
in0,∞nandφnx, nis Schur multiplicatively convex in0,1n.
Theorem 3.4. Forr ∈ {1,2, . . . , n},the symmetric functionφnx, ris Schur harmonic convex in Rn
.
Proof. According toLemma 2.4we only need to prove that
x1−x2
x2 1
∂φnx, r
∂x1 −x
2 2
∂φnx, r ∂x2
≥0 3.15
for allx x1, x2, . . . , xn∈Rnandr1,2, . . . , n.
The proof is divided into four cases.
Case 1. Ifr1, then from3.2we have
x1−x2
x2 1
∂φnx,1
∂x1 −x
2 2
∂φnx,1 ∂x2
x1−x22
1x11x2φnx,
1≥0. 3.16
Case 2. Ifn≥2 andrn, then3.4leads to
x1−x2
x2 1
∂φnx, n
∂x1 −x
2 2
∂φnx, n ∂x2
x1−x22x1x22x1x2
n1x121x22
n
i1
xi
1xi 1/n−1
≥0.
Case 3. Ifn≥3 andr2, then3.6yields
x1−x2
x2 1
∂φnx,2
∂x1 −x
2 2
∂φnx,2 ∂x2
x1−x22 1x11x2
φnx,2
2
×
⎡ ⎣1
n
j3
x1x2x1xjx2xj3x1x2xj1xj x1xj2x1xjx2xj2x2xj
⎤ ⎦
≥0.
3.18
Case 4. Ifn≥4 and 3≤r≤n−1, then3.8implies
x1−x2
x2 1
∂φnx, r
∂x1 −x
2 2
∂φnx, r ∂x2
x1−x22 1x121x22
φnx, r r
×
⎡ ⎢
⎣
3≤i1<i2<···<ir−2≤n
x1x22x1x2
x1/1x1 x2/1x2
r−2 j1
xij/
1xij
3≤i1<i2<···<ir−1≤n
x1x2x1x22x1x2
r−2 j1
xij/
1xij
x1/1x1rj−11
xij/
1xij
x2/1x2rj−11
xij/
1xij
⎤ ⎥ ⎦
≥0.
3.19
Therefore,3.15follows from Cases1–4and the proof ofTheorem 3.4is completed.
4. Applications
In this section, we establish some inequalities by use of Theorems3.1,3.2and3.4and the theory of majorization.
Theorem 4.1. If x x1, x2,· · ·, xn ∈ Rn, s ni1xi,Hnx n/
n
i11/xi, and r ∈
{1,2, . . . , n}, then
1φnx, r≤φnc−x/nc/s−1, rforc≥s;
2φnx, r≥φncHnx−1/cx−1x, rforc≥ni11/xi; 3φnx, r≤φncx/nc/s1, rforc≥0;
4φnx, r≥φncHnx 1/cx1x, rforc≥0;
6φnx, r≥φnn−λ/ni1 1/xi−λ/x, rfor0≤λ≤1; 7φnx, r≤φnsλx/nλ, rfor0≤λ≤1;
8φnx, r≥φnnλ/ni11/xiλ/x , rfor0≤λ≤1.
Proof. Theorem 4.1follows fromTheorem 3.1,Theorem 3.4and Lemmas2.5–2.7together with the fact that
sλx
nλ
sλx
1
nλ , sλx2
nλ , . . . , sλxn
nλ
≺x1, x2, . . . , xn x. 4.1
Theorem 4.2. Ifx x1, x2,· · ·, xn∈Rn,Anx 1/nni1xi, andr∈ {1,2, . . . , n}, then
i
1≤i1<i2<···<ir≤n ⎛ ⎝r
j1
xij
1xij
⎞ ⎠
1/r
≤
rAAnx n1x
n!/r·r!n−r!
;
ii
1≤i1<i2<···<ir≤n ⎛ ⎝r
j1
1 1xij
⎞ ⎠
1/r
≥r 1
An1x
n!/r·r!n−r! .
4.2
Proof. We clearly see that
Anx, Anx, . . . , Anx≺x1, x2, . . . , xn x. 4.3
Therefore,Theorem 4.2ifollows from4.3andTheorem 3.1together with1.7, and
Theorem 4.2iifollows from4.3andTheorem 3.4together with1.7.
If we taker1 inTheorem 4.2iandrninTheorem 4.2, respectively, then we have the following corollary.
Corollary 4.3. Ifx x1, x2, . . . , xn∈RnandGnx ni1xi1/n, then
i Gnx
Gn1x ≤
Anx An1x;
ii An
x
1x
≤ Anx
An1x;
iii An
1 1x
≥ A 1
n1x.
4.4
Remark 4.4. If we take ni1xi 1 in Corollary 4.3i, then we obtain the Weierstrass inequality:see20, page 260
n
i1
1
xi 1
Theorem 4.5. Ifx x1, x2, . . . , xn∈Rnandr ∈ {1,2, . . . , n}, then
i
1≤i1<i2<···<ir≤n ⎛ ⎝r
j1
xij
1xij ⎞ ⎠
1/r
≥
r Hnx
1Hnx
n!/r·r!n−r!
;
ii
1≤i1<i2<···<ir≤n ⎛ ⎝r
j1
1 1xij
⎞ ⎠
1/r
≤
r 1
1Hnx
n!/r·r!n−r! .
4.6
Proof. We clearly see that
1
Hnx,
1
Hnx, . . . ,
1
Hnx
≺
1
x1,
1
x2, . . . ,
1
xn
x1. 4.7
Therefore,Theorem 4.5ifollows from4.7andTheorem 3.4together with1.7, and
Theorem 4.5iifollows from4.7andTheorem 3.1together with1.7.
If we take r 1 and r ninTheorem 4.5, respectively, then we get the following corollary.
Corollary 4.6. Ifx x1, x2, . . . , xn∈Rn, then
i Gnx
Gn1x ≥
Hnx
1Hnx;
ii Gn1x≥1Hnx;
iii An
x
1x
≥ Hnx
1Hnx;
iv An
1 1x
≤ 1
1Hnx.
4.8
Theorem 4.7. Ifx x1, x2, . . . , xn∈1,∞nandr ∈ {1,2, . . . , n}, then
1≤i1<i2<···<ir≤n ⎛ ⎝r
j1
xij
1xij ⎞ ⎠
1/r
≤
r Gnx
1Gnx
n!/r·r!n−r!
. 4.9
Proof. We clearly see that
logGnx, Gnx, . . . , Gnx≺logx1, x2, . . . , xn. 4.10
If we take r 1 and r ninTheorem 4.7, respectively, then we get the following corollary.
Corollary 4.8. Ifx x1, x2, . . . , xn∈1,∞n, then
i An
x
1x
≤ Gnx
1Gnx;
ii Gn1x≥1Gnx.
4.11
Remark 4.9. FromRemark 3.3and4.10together with1.7we clearly see that inequality in
Corollary 4.8iis reversed forx∈0,1nand inequality inCorollary 4.8iiis true forx∈Rn.
Theorem 4.10. Ifx x1, x2,· · ·, xn∈Rn,then
i
1≤i1<i2···<ir≤n ⎛ ⎝r
j1
1xij 2xij
⎞ ⎠
1/r
≥
1n i1xi
2ni1xi r−1
2
n−1!/r!n−r!
×r
2
n−1!/r·r!n−r−1!
for1≤r≤n−1;
ii
n
i1
1xi 2xi ≥
1ni1xi
2ni1xi
n−1 2 ;
iii
1≤i1<i2···<ir≤n ⎛ ⎝r
j1
1 2xij
⎞ ⎠
1/r
≤ 1
2ni1xi r−1
2
n−1!/r−1!n−r!
×r
2
n−1!/r·r!n−r−1!
for1≤r≤n−1;
iv
n
i1
1 2xi ≤
1 2ni1xi
n−1 2 .
4.12
Proof. Theorem 4.10follows from Theorems3.1,3.4, and1.7together with the fact that
1x1,1x2, . . . ,1xn≺
1
n
i1
xi,1,1, . . . ,1
. 4.13
Theorem 4.11. LetAA1A2· · ·An1be ann-dimensional simplex inRnand letPbe an arbitrary point in the interior ofA. IfBi is the intersection point of straight lineAiP and hyperplanei
A1A2· · ·Ai−1Ai1· · ·An1, i1,2, . . . , n1, then forr ∈ {1,2, . . . , n1}one has
i
1≤i1<i2···<ir≤n1
⎛ ⎝r
j1
PBij
AijBijPBij ⎞ ⎠
1/r
≤
r
n2
n1!/r·r!n−r1!
ii
1≤i1<i2···<ir≤n1
⎛ ⎝r
j1
AijBij
AijBijPBij ⎞ ⎠ 1/r ≥ r n 1
n2
n1!/r·r!n−r1!
;
iii
1≤i1<i2···<ir≤n1
⎛ ⎝r
j1
PAij
AijBijPAij ⎞ ⎠ 1/r ≤ r n
2n1
n1!/r·r!n−r1!
;
iv
1≤i1<i2···<ir≤n1
⎛ ⎝r
j1
AijBij
AijBijPAij ⎞ ⎠ 1/r ≥ r n 1 2n1
n1!/r·r!n−r1!
.
4.14
Proof. It is easy to see thatni11PBi/AiBi 1 and ni11PAi/AiBi n, these identical equations imply
1
n1, 1
n1, . . . , 1
n1
≺
PB
1
A1B1,
PB2
A2B2, . . . ,
PBn1
An1Bn1
,
n
n1,
n n1, . . . ,
n n1
≺
PA
1
A1B1,
PA2
A2B2, . . . ,
PAn1
An1Bn1
.
4.15
Therefore,Theorem 4.11follows from4.15, Theorems3.1,3.4, and1.7.
Remark 4.12. Mitrinovic’ et al. 21, pages 473–479 established a series of inequalities for
PAi/AiBi and PBi/AiBi,i 1,2, . . . , n1. Obvious, our inequalities in Theorem 4.11are
different from theirs.
Theorem 4.13. Suppose that A ∈ MnC n ≥ 2 is a complex matrix, λ1, λ2, . . . , λn, and σ1, σ2, . . . , σn are the eigenvalues and singular values ofA, respectively. IfA is a positive definite Hermitian matrix andr ∈ {1,2, . . . , n},then
i
1≤i1<i2···<ir≤n ⎛ ⎝r
j1
λij
1λij
⎞ ⎠ 1/r ≤ r trA
ntrA
n!/r·r!n−r!
;
ii
1≤i1<i2···<ir≤n ⎛ ⎝r
j1
1 1λij
⎞ ⎠ 1/r ≥ r n
ntrA
n!/r·r!n−r!
;
iii
1≤i1<i2···<ir≤n ⎛ ⎝r
j1
1λij 2λij
⎞ ⎠ 1/r ≤ r n
detIA 1n detIA
n!/r·r!n−r!
;
iv
1≤i1<i2···<ir≤n ⎛ ⎝r
j1
1 trAλij
⎞ ⎠ 1/r ≤ r 1 trA√ndetA
n!/r·r!n−r!
v
1≤i1<i2···<ir≤n ⎛ ⎝r
j1
1
λij n
i1λi
n i1σi
⎞ ⎠
1/r
≥
1≤i1<i2···<ir≥n ⎛ ⎝r
j1
1
σij n
i1λi
n i1σi
⎞ ⎠
1/r
.
4.16
Proof. i–iiWe clearly see thatλi>0 i1,2, . . . , nandni1λitrA,then we have
trA
n ,
trA
n , . . . ,
trA
n
≺λ1, λ2, . . . , λn. 4.17
Therefore,Theorem 4.13iandiifollows from4.17, Theorems3.1,3.4, and1.7.
iiiIt is easy to see that 1λ1,1λ2, . . . ,1λnare the eigenvalues of matrixIAand
n
i11λi detIA, then we get
log
n
detIA, n
detIA, . . . , n
detIA
≺log1λ1,1λ2, . . . ,1λn
1λi ≥1, i1,2, . . . , n.
4.18
Therefore,Theorem 4.13iiifollows from4.18,Theorem 3.2, and1.7.
ivIt is not difficult to verify that
log
trA
n
√
detA, trA
n
√
detA, . . . , trA
n
√
detA
≺log
trA
λ1 ,
trA
λ2 , . . . ,
trA
λn
,
trA
λi ≥1, i1,2, . . . , n.
4.19
Therefore,Theorem 4.13ivfollows from4.19, andTheorem 3.2together with1.7.
vA result due to Weyl22gives
logλ1, λ2, . . . , λn≺logσ1, σ2, . . . , σn. 4.20
From4.20, we clearly see that
log
n
i1λi
n i1σi
λ1 ,
n i1λi
n i1σi
λ2 , . . . ,
n i1λi
n i1σi
λn
≺log
n
i1λi
n i1σi
σ1 ,
n i1λi
n i1σi
σ2 , . . . ,
n i1λi
n i1σi
σn
n i1λi
n i1σi
λi ,
n i1λi
n i1σi
σi ≥1, i1,2, . . . , n.
, 4.21
Acknowledgment
This work was supported by the National Science Foundation of Chinano. 60850005and the Natural Science Foundation of Zhejiang Provinceno. D7080080, Y607128, Y7080185.
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