Báo cáo hóa học: " Research Article Schur-Convexity for a Class of Symmetric Functions and Its Applications" pptx

1.1 The notion of Schur convexity was first introduced by Schur in 1923 1.. The following definition for Schur convex or concave can be found in 1, 3, 7 and the references therein.. F is

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

1 School of Teacher Education, Huzhou Teachers College, Huzhou 313000, China

2 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

Forx  x1, x2, , xn  ∈ R n

, the symmetric functionφn x, r is defined by φ n x, r  φ n x1,

x2, , xn;r 1≤i1<i2···<i r ≤nr

j1 x i j /1xi j1/r, wherer  1, 2, , n and i1, i2, , inare positive integers In this article, the Schur convexity, Schur multiplicative convexity and Schur harmonic convexity ofφn x, r are discussed As applications, some inequalities are established by use of the

theory of majorization

Copyrightq 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 useR nto denote then-dimensional Euclidean space over the field

of real numbers andR n  {x1, x2, , xn  : x i > 0, i  1, 2, , n} In particular, we use R to

denoteR1.

For the sake of convenience, we use the following notation system

Forx  x1, x2, , xn , y  y1, y2, , yn  ∈ R n, andα > 0, let

x  y x1 y1, x2 y2, , x n  y n,

xy x1y1, x2y2, , x n y n,

αx  αx1, αx2, , αxn ,

x αx α

1, x α

2, , x α n



,

1

 1

x1, x1

2, , x1 n



,

logx logx1, log x2, , log xn,

e x  e x1, e x2, , e x n .

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 optimization 8, isoperimetric problem for polytopes 9, linear regression 10, graphs and matrices 11, gamma and digamma functions12, 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

convex function if

for each pair ofn-tuples x  x1, , xn  and y  y1, , yn  on E, such that x is majorized

k



i1

x i≤k

i1

y i , k  1, 2, , n − 1, n



i1

x in

i1

y i ,

1.3

wherex idenotes theith 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

called a Schur multiplicatively convex function onE if

for each pair of n-tuples x  x1, x2, , xn  and y  y1, y2, , yn  on E, such that x is

logarithmically majorized byy in symbols log x ≺ log y, that is,

k

i1

x i≤ k

i1

y i , k  1, 2, , n − 1, n

i1

x i n

i1

y i

1.5

HoweverF is called Schur multiplicatively concave if 1/F is Schur multiplicatively convex.

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

 n ≥ 2 be a set A real-valued function F on E is called a Schur

harmonic convex function if

for each pair ofn-tuples x  x1, x2, , xn  and y  y1, y2, , yn  on E, such that 1/x ≺ 1/y.

The main purpose of this paper is to discuss the Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity of the following symmetric function:

φ n x, r  φ n x1, x2, , x n;r 

1≤i 1<i2···<i r ≤n

⎛

⎝r

j1

xi j

1 x i j

⎞

⎠

1/r

wherex  x1, x2, , xn  ∈ R n 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

function If f is differentiable in R n , then f is Schur convex if and only if



xi − x j∂f

∂f

∂xj



for all i, j  1, 2, , n and x  x1, , xn  ∈ R n Also f is Schur concave if and only if 2.1 is

reversed for all i, j  1, 2, , n and x  x1, , xn  ∈ R n

 Here, f is a symmetric function in R n



reduced to

x1− x2 ∂f

∂x1 −∂x ∂f

2



Lemma 2.3 see 7,16 Let f : R n

 → Rbe a continuous symmtric function If f is differentiable

in R n , then f is Schur multiplicatively convex if and only if

 logx1− log x2



x1∂f

∂x1 − x2∂f

∂x2



for all x  x1, x2, , x n  ∈ R n Also f is Schur multiplicatively concave if and only if 2.3 is

reversed.

f is Schur harmonic convex if and only if

x1− x2



x2 1

∂f

∂x1 − x2 2

∂f

∂x2



for all x  x1, x2, , x n  ∈ R n Also f is Schur harmonic concave if and only if 2.4 is reversed.

convex if and only ifFx  1/f1/x : R n → Ris Schur concave

This fact,Lemma 2.1andRemark 2.2together with elementary calculation imply that Lemma 2.4is true

i1 xi  s If c ≥ s, then

c − x

1

nc/s − 1 ,

c − x2

nc/s − 1 , ,

c − xn nc/s − 1



≺ x1, x2, , xn   x. 2.5

andn

i1 xi  s If c ≥ 0, then

c  x

1

nc/s  1 ,

c  x2

nc/s  1 , ,

c  xn nc/s  1



≺ x1, x2, , xn   x. 2.6

i1 x i  s If 0 ≤ λ ≤ 1, then

s − λx

1

s − λx2

n − λ , ,

s − λxn

n − λ



3 Main Results

.

x1− x2∂φ n x, r

∂x1 − ∂φn x, r

∂x2



for allx  x1, x2, , x n  ∈ R nandr  1, 2, , n.

The proof is divided into four cases.

Case 1 If r  1, then 1.7 leads to

φ n x, 1  φ n x1, x2, , x n; 1  n

i1

x i

However3.2 and elementary computation lead to

x1− x2∂φ n x, 1

∂x1 −∂φ n ∂x x, 1

2



 −x1− x221  x1 x2

x1x21  x11  x2 φn x, 1 ≤ 0. 3.3

Case 2 If n ≥ 2 and r  n, then 1.7 yields

φ n x, n  φ n x1, x2, , x n;n 

n

i1

xi

1 x i

1/n

From3.4 and elementary computation, we have

x1− x2∂φ n x, n

∂x1 −∂φ n ∂x x, n

2



 −x1− x222  x1 x2

n1  x121  x22

 n



i1

x i

1 x i

1/n−1

≤ 0. 3.5

φn x, 2  φ n x1, x2, · · · , xn; 2



 x

1

1 x1  x2

1 x2

1/2⎡

⎣ n

j3



x1

1 x1  x j

1 x j

1/2⎤

⎦φ n−1 x2, x3, , xn; 2



 x

2

1 x2  x1

1 x1

1/2⎡

⎣ n

j3



x2

1 x2  x j

1 x j

1/2⎤

⎦φ n−1 x1, x3, , x n; 2

3.6

Elementary computation and3.6 yield

x1− x2∂φ n x, 2

∂x1 −∂φn ∂x x, 2

2



 − x1− x22

1  x11  x2

φn x, 2

2

×

⎡

⎣ 2  x1 x2

x1 x2 2x1x2 n

j3



1  x1 x2  3  2x1 2x2x j1 x j



x1 x j  2x1xjx2 x j  2x2xj

⎤

⎦ ≤ 0.

3.7

φ n x, r  φ n x1, x2, , x n;r

 φ n−1 x2, x3, , x n;r

3≤i 1<i2<···<i r−1 ≤n

⎛

⎝ x1

1 x1 r−1

j1

xi j

1 x i j

⎞

⎠

1/r

3≤i 1<i2<···<i r−2 ≤n

⎛

⎝ x1

1 x1  x2

1 x2 r−2

j1

x i j

1 x i j

⎞

⎠

1/r

 φ n−1 x1, x3, , x n;r

3≤i 1<i2<···<i r−1 ≤n

⎛

⎝ x2

1 x2 r−1

j1

x i j

1 x i j

⎞

⎠

1/r

3≤i 1<i2<···<i r−2 ≤n

⎛

⎝ x1

1 x1  x2

1 x2 r−2

j1

xi j

1 x i j

⎞

⎠

1/r ,

3.8

x1− x2∂φ n x, r

∂x1 − ∂φn ∂x x, r

2



 − x1− x22

1  x121  x22

×

⎡

⎢

3≤i 1<i2<···<i r−2 ≤n

2x1x2

x1/1x1  x2/1x2 r−2 j1xi j /1x i j



3≤i 1<i2<···<i r−1 ≤n

1x1x22x1x2r−1 j1xi j /1xi j



x1/1x1 r−1

j1



x i j /1x i jx2/1x2 r−1

j1



x i j /1 x i j

⎤

⎥

×φ n x, r r ≤ 0.

3.9 Therefore,3.1 follows from Cases1 4and the proof ofTheorem 3.1is completed

For the Schur multiplicative convexity or concavity ofφn x, r, we have the following

theorem

 logx1− log x2



x1

∂φ n x, r

∂φ n x, r

∂x2



for allx  x1, x2, , n ∈ 1, ∞ nandr  1, 2, , n Then proof is divided into four cases Case 1 If r  1, then 3.2 leads to



logx1− log x2



x1

∂φ n x, 1

∂φ n x, 1

∂x2



 −

 logx1− log x2



x1− x2

1  x11  x2 φn x, 1 ≤ 0.

3.11

Case 2 If r  n, n ≥ 2, then 3.4 yields

 logx1− log x2



x1∂φn x, n

∂x1 − x2∂φn x, n

∂x2





 logx1− log x2



x1− x2

n1  x121  x22 1 − x1x2

n

i1

xi

1 x i

1/n−1

≤ 0.

3.12

Case 3 If n ≥ 3 and r  2, then 3.6 implies



logx1− log x2



x1∂φn x, 2

∂x1 − x2∂φn x, 2

∂x2



 φ n x, 2

2

 logx1− log x2



x1− x2

1  x121  x22

×

⎡

⎣ 1− x1x2

x1/1  x1  x2/1  x2

n

j3

−x1x2 1 − x1x2x j /1 x j



x1/1  x1  x j/1 x j

x2/1  x2  x j/1 x j

⎤

⎦ ≤ 0.

3.13

Case 4 If n ≥ 4 and 3 ≤ r ≤ n − 1, then from 3.8 we have



logx1− log x2



x1∂φn x, r

∂x1 − x2∂φn x, r

∂x2







logx1− log x2



x1− x2

1  x121  x22

×

⎡

3≤i 1<i2<···<i r−2 ≤n

1− x1x2

x1/1  x1  x2/1  x2 r−2 j1xi j /1 x i j



3≤i 1<i2<···<i r−1 ≤n

−x1x21 − x1x2r−1 j1xi j /1 x i j





x1/1x1 r−1 j1x i j /1 x i jx2/1x2 r−1 j1x i j /1 x i j

⎤

⎥

⎦

× φn x, r r ≤ 0.

3.14 Therefore,Theorem 3.2follows from3.10 and Cases1 4

in0, ∞ nandφ n x, n is Schur multiplicatively convex in 0, 1 n

R n

x1− x2



x2 1

∂φ n x, r

2

∂φ n x, r

∂x2



for allx  x1, x2, , xn  ∈ R nandr  1, 2, , n.

The proof is divided into four cases

Case 1 If r  1, then from 3.2 we have

x1− x2



x2 1

∂φn x, 1

2

∂φn x, 1

∂x2



 x1− x22

1  x11  x2φn x, 1 ≥ 0. 3.16

Case 2 If n ≥ 2 and r  n, then 3.4 leads to

x1− x2



x2

1

∂φn x, n

2

∂φn x, n

∂x2



 x1− x22x1 x2 2x1x2

n1  x121  x22

 n



i1

x i

1 x i

1/n−1

≥ 0.

3.17

Case 3 If n ≥ 3 and r  2, then 3.6 yields

x1− x2



x2 1

∂φn x, 2

2

∂φn x, 2

∂x2



 x1− x22

1  x11  x2

φn x, 2

2

×

⎡

⎣1 n

j3



x1x2 x1x j  x2x j  3x1x2x j1 x j



x1 x j  2x1xjx2 x j  2x2xj

⎤

⎦

≥ 0.

3.18

Case 4 If n ≥ 4 and 3 ≤ r ≤ n − 1, then 3.8 implies

x1− x2



x2

1

∂φ n x, r

2

∂φ n x, r

∂x2



 x1− x22

1  x121  x22

φn x, r

r

×

⎡

⎢

3≤i 1<i2<···<i r−2 ≤n

x1 x2 2x1x2

x1/1  x1  x2/1  x2 r−2

j1



x i j /1 x i j

3≤i 1<i2<···<i r−1 ≤n

x1x2x1x22x1x2r−2

j1



x i j /1x i j



x1/1x1r−1 j1xi j /1 x i j



x2/1  x2r−1 j1xi j /1 xi j



⎤

⎥

≥ 0.

3.19 Therefore,3.15 follows 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

i1 x i ,H n x  n/n

i1 1/x i , and r ∈ {1, 2, , n}, then

1 φ n x, r ≤ φ n c − x/nc/s − 1, r for c ≥ s;

2 φ n x, r ≥ φ n cH n x − 1/cx − 1x, r for c ≥n

i1 1/x i ;

3 φ n x, r ≤ φ n c  x/nc/s  1, r for c ≥ 0;

4 φ n x, r ≥ φ n cH n x  1/cx  1x, r for c ≥ 0;

5 φ n x, r ≤ φ n s − λx/n − λ, r for 0 ≤ λ ≤ 1;

6 φ n x, r ≥ φ n n − λ/n i1 1/x i − λ/x , r for 0 ≤ λ ≤ 1;

7 φ n x, r ≤ φ n s  λx/n  λ, r for 0 ≤ λ ≤ 1;

8 φ n x, r ≥ φ n n  λ/n i1 1/x i  λ/x , r for 0 ≤ λ ≤ 1.

the fact that

s  λx

1

s  λx2

n  λ , ,

s  λxn

n  λ



1≤i 1<i2<···<i r ≤n

⎛

⎝r

j1

x i j

1 x i j

⎞

⎠

1/r

≤



r A An x

n 1  x

n!/r·r!n−r!

;

1≤i 1<i2<···<i r ≤n

⎛

⎝r

j1

1

1 x i j

⎞

⎠

1/r

≥r 1

An 1  x

n!/r·r!n−r!

.

4.2

Proof We clearly see that

A n x, A n x, , A n x ≺ x1, x2, , xn   x. 4.3

Therefore,Theorem 4.2i follows from 4.3 andTheorem 3.1together with1.7, and Theorem 4.2ii follows from 4.3 andTheorem 3.4together with1.7

If we taker  1 inTheorem 4.2i and r  n inTheorem 4.2, respectively, then we have the following corollary

i G n x

Gn 1  x ≤

A n x

An 1  x;

ii A n  x

1 x



≤ A n x

An 1  x;

iii A n

 1

1 x



≥ A 1

n 1  x .

4.4

i1 x i  1 in Corollary 4.3i, then we obtain the Weierstrass inequality:see 20, page 260

n

i1

 1

xi  1



Theorem 4.5 If x  x1, x2, , xn  ∈ R n

and r ∈ {1, 2, , n}, then

1≤i 1<i2<···<i r ≤n

⎛

⎝r

j1

xi j

1 x i j

⎞

⎠

1/r

≥



1 H n x

n!/r·r!n−r!

;

1≤i 1<i2<···<i r ≤n

⎛

⎝r

j1

1

1 x i j

⎞

⎠

1/r

≤



1 H n x

n!/r·r!n−r!

.

4.6

Proof We clearly see that

 1

H n x ,

1

H n x , ,

1

H n x



≺

 1

x1, x1

2, , x1 n



Therefore,Theorem 4.5i follows from 4.7 andTheorem 3.4together with1.7, and Theorem 4.5ii follows from 4.7 andTheorem 3.1together with1.7

If we take r  1 and r  n inTheorem 4.5, respectively, then we get the following corollary

i G Gn x

n 1  x ≥

1 H n x;

ii G n 1  x ≥ 1  H n x;

iii A n  x

1 x



≥ Hn x

1 H n x;

iv A n

 1

1 x



1 H n x .

4.8

1≤i 1<i2<···<i r ≤n

⎛

⎝r

j1

x i j

1 x i j

⎞

⎠

1/r

≤



1 G n x

n!/r·r!n−r!

Proof We clearly see that

logGn x, G n x, , G n x ≺ logx1, x2, , x n . 4.10 Therefore,Theorem 4.7follows from4.10,Theorem 3.2, and1.7

If we take r  1 and r  n inTheorem 4.7, respectively, then we get the following corollary

i A n  x

1 x



≤ G n x

1 G n x;

ii G n 1  x ≥ 1  G n x.

4.11

Corollary 4.8i is reversed for x ∈ 0, 1 nand inequality inCorollary 4.8ii is true for x ∈ R n



1≤i 1<i2···<i r ≤n

⎛

⎝r

j1

1 x i j

2 x i j

⎞

⎠

1/r

≥

1n

i1 xi

2n i1 xi 

r − 1

2

n−1!/r!n−r!

× r

2

n−1!/r·r!n−r−1!

for 1 ≤ r ≤ n − 1;

ii n

i1

1 x i

2 x i ≥ 1

n

i1 xi

2n i1 x i 

n − 1

2 ;

1≤i 1<i2···<i r ≤n

⎛

⎝r

j1

1

2 x i j

⎞

⎠

1/r

2n i1 x i 

r − 1

2

n−1!/r−1!n−r!

× r

2

n−1!/r·r!n−r−1!

for 1 ≤ r ≤ n − 1;

iv n

i1

1

2 x i ≤

1

2n i1 x i 

n − 1

2 .

4.12

1  x1, 1  x2, , 1  x n ≺



1n

i1

x i , 1, 1, , 1



point in the interior of A If B i is the intersection point of straight line A i P and hyperplane i 

A1A2· · · A i−1Ai1 · · · A n1, i  1, 2, , n  1, then for r ∈ {1, 2, , n  1} one has

1≤i<i ···<i ≤n1

⎛

⎝r

j1

PBi j

Ai j Bi j  PB i j

⎞

⎠

1/r

≤

 r

n  2

n1!/r·r!n−r1!

;

ii

1≤i 1<i2···<i r ≤n1

⎛

⎝r

j1

A i j B i j

Ai j Bi j  PB i j

⎞

⎠

1/r

≥



r

n  2

n1!/r·r!n−r1!

;

1≤i 1<i2···<i r ≤n1

⎛

⎝r

j1

PAi j

A i j B i j  PA i j

⎞

⎠

1/r

≤



r

2n  1

n1!/r·r!n−r1!

;

1≤i 1<i2···<i r ≤n1

⎛

⎝r

j1

A i j B i j

Ai j Bi j  PA i j

⎞

⎠

1/r

≥



r

2n  1

n1!/r·r!n−r1!

.

4.14

i1 PB i /A i B i  1 and n1 i1 PA i /A i B i   n, these identical

equations imply

 1

n  1 ,

1

n  1 , ,

1

n  1



≺

 PB

1

A1B1, PB2

A2B2, , A PBn1

n1 B n1



,

 n

n  1 ,

n

n  1 , ,

n

n  1



≺

PA

1

A1B1, PA2

A2B2, , An1Bn1 PA n1



.

4.15

Therefore,Theorem 4.11follows from4.15, Theorems3.1,3.4, and1.7

different from theirs

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

1≤i 1<i2···<i r ≤n

⎛

⎝r

j1

λ i j

1 λ i j

⎞

⎠

1/r

≤



r



trA

n  trA

n!/r·r!n−r!

;

1≤i 1<i2···<i r ≤n

⎛

⎝r

j1

1

1 λ i j

⎞

⎠

1/r

≥



r

n  trA

n!/r·r!n−r!

;

1≤i 1<i2···<i r ≤n

⎛

⎝r

j1

1 λ i j

2 λ i j

⎞

⎠

1/r

≤

r

 n detI  A

1n

detI  A

n!/r·r!n−r!

;

1≤i<i ···<i ≤n

⎛

⎝r

j1

1

trA  λi j

⎞

⎠

1/r

≤



r



1

trA √n

detA

n!/r·r!n−r!

;