Skip to main content

Schur-convexity of dual form of some symmetric functions

Abstract

By the properties of a Schur-convex function, Schur-convexity of the dual form of some symmetric functions is simply proved.

MSC:26D15, 05E05, 26B25.

1 Introduction

Throughout the article, denotes the set of real numbers, x=( x 1 , x 2 ,, x n ) denotes n-tuple (n-dimensional real vectors), the set of vectors can be written as

R n = { x = ( x 1 , , x n ) : x i R , i = 1 , , n } , R + n = { x = ( x 1 , , x n ) : x i > 0 , i = 1 , , n } .

In particular, the notations and R + denote R 1 and R + 1 , respectively.

For convenience, we introduce some definitions as follows.

Definition 1 [1, 2]

Let x=( x 1 ,, x n ) and y=( y 1 ,, y n ) R n .

  1. (i)

    xy means x i y i for all i=1,2,,n.

  2. (ii)

    Let Ω R n , φ:ΩR is said to be increasing if xy implies φ(x)φ(y). φ is said to be decreasing if and only if −φ is increasing.

Definition 2 [1, 2]

Let x=( x 1 ,, x n ) and y=( y 1 ,, y n ) R n .

  1. (i)

    x is said to be majorized by y (in symbols xy) if i = 1 k x [ i ] i = 1 k y [ i ] for k=1,2,,n1 and i = 1 n x i = i = 1 n y i , where x [ 1 ] x [ n ] and y [ 1 ] y [ n ] are rearrangements of x and y in a descending order.

  2. (ii)

    Let Ω R n , φ:ΩR is said to be a Schur-convex function on Ω if xy on Ω implies φ(x)φ(y). φ is said to be a Schur-concave function on Ω if and only if −φ is Schur-convex function on Ω.

Definition 3 [1, 2]

Let x=( x 1 ,, x n ) and y=( y 1 ,, y n ) R n .

  1. (i)

    Ω R n is said to be a convex set if x,yΩ, 0α1 implies αx+(1α)y=(α x 1 +(1α) y 1 ,,α x n +(1α) y n )Ω.

  2. (ii)

    Let Ω R n be a convex set. A function φ:ΩR is said to be a convex function on Ω if

    φ ( α x + ( 1 α ) y ) αφ(x)+(1α)φ(y)

for all x,yΩ and all α[0,1]. φ is said to be a concave function on Ω if and only if −φ is a convex function on Ω.

  1. (iii)

    Let Ω R n . A function φ:ΩR is said to be a log-convex function on Ω if the function lnφ is convex.

Definition 4 [1]

  1. (i)

    Ω R n is called a symmetric set, if xΩ implies PxΩ for every n×n permutation matrix P.

  2. (ii)

    The function φ:ΩR is called symmetric if for every permutation matrix P, φ(Px)=φ(x) for all xΩ.

Theorem A (Schur-convex function decision theorem [[1], p.84])

Let Ω R n be symmetric and have a nonempty interior convex set. Ω 0 is the interior of Ω. φ:ΩR is continuous on Ω and differentiable in Ω 0 . Then φ is the Schur-convex (Schur-concave) function if and only if φ is symmetric on Ω and

( x 1 x 2 ) ( φ x 1 φ x 2 ) 0(0)
(1)

holds for any x Ω 0 .

The Schur-convex functions were introduced by Schur in 1923 and have important applications in analytic inequalities, elementary quantum mechanics and quantum information theory. See [1].

In recent years, many scholars use the Schur-convex function decision theorem to determine the Schur-convexity of many symmetric functions.

Xia et al. [3] proved that the symmetric function

E k ( x 1 + x ) = 1 i 1 < < i k n j = 1 k x i j 1 + x i j ,k=1,,n,
(2)

is Schur-convex on R + n .

Chu et al. [4] proved that the symmetric function

E k ( x 1 x ) = 1 i 1 < < i k n j = 1 k x i j 1 x i j ,k=1,,n,
(3)

is Schur-convex on [ k 1 2 ( n 1 ) , 1 ) n and Schur-concave on [ 0 , k 1 2 ( n 1 ) ] n .

Xia and Chu [5] proved that the symmetric function

E k ( 1 x x ) = 1 i 1 < < i k n j = 1 k 1 x i j x i j ,k=1,,n,
(4)

is Schur-convex on ( 0 , 2 n k 1 2 ( n 1 ) ] n and Schur-concave on [ 2 n k 1 2 ( n 1 ) , 1 ] n .

Xia and Chu [6] also proved that the symmetric function

E k ( 1 + x 1 x ) = 1 i 1 < < i k n j = 1 k 1 + x i j 1 x i j ,k=1,,n,
(5)

is Schur-convex on ( 0 , 1 ) n .

Mei et al. [7] proved that the symmetric function

E k ( 1 x x ) = 1 i 1 < < i k n j = 1 k ( 1 x i j x i j ) ,k=1,,n,
(6)

is Schur-convex on ( 0 , 1 ) n . More results for Schur convexity of the symmetric functions, we refer the reader to [8].

In this paper, by the properties of a Schur-convex function, we study Schur-convexity of the dual form of the above symmetric functions, and we obtained the following results.

Theorem 1 The symmetric function

E k ( x 1 + x ) = 1 i 1 < < i k n j = 1 k x i j 1 + x i j ,k=1,,n,
(7)

is a Schur-concave function on R + n .

Theorem 2 The symmetric function

E k ( x 1 x ) = 1 i 1 < < i k n j = 1 k x i j 1 x i j ,k=1,,n,
(8)

is a Schur-convex function on [ 1 2 , 1 ) n .

Theorem 3 The symmetric function

E k ( 1 x x ) = 1 i 1 < < i k n j = 1 k 1 x i j x i j ,k=1,,n,
(9)

is a Schur-convex function on ( 0 , 1 2 ] n .

Theorem 4 The symmetric function

E k ( 1 + x 1 x ) = 1 i 1 < < i k n j = 1 k 1 + x i j 1 x i j ,k=1,,n,
(10)

is a Schur-convex function on ( 0 , 1 ) n .

Theorem 5 The symmetric function

E k ( 1 x x ) = 1 i 1 < < i k n j = 1 k ( 1 x i j x i j ) ,k=1,,n,
(11)

is a Schur-convex function on ( 0 , 5 2 ) n .

2 Lemmas

To prove the above three theorems, we need the following lemmas.

Lemma 1 ([[1], p.97], [2])

If φ is symmetric and convex (concave) on a symmetric convex set Ω, then φ is Schur-convex (Schur-concave) on Ω.

Lemma 2 [[2], p.64]

Let Ω R n , φ:Ω R + . Then logφ is Schur-convex (Schur-concave) if and only if φ is Schur-convex (Schur-concave).

Lemma 3 ([[1], p.642], [2])

Let Ω R n be an open convex set, φ:ΩR. For x,yΩ, define one variable function g(t)=φ(tx+(1t)y) on the interval (0,1). Then φ is convex (concave) on Ω if and only if g is convex (concave) on [0,1] for all x,yΩ.

Lemma 4 Let x=( x 1 ,, x m ) and y=( y 1 ,, y m ) R + m . Then the function p(t)=logg(t) is concave on [0,1], where

g(t)= j = 1 m t x j + ( 1 t ) y j 1 + t x j + ( 1 t ) y j .

Proof

p (t)= g ( t ) g ( t ) ,

where

g ( t ) = j = 1 m x j y j ( 1 + t x j + ( 1 t ) y j ) 2 , p ( t ) = g ( t ) g ( t ) ( g ( t ) ) 2 g 2 ( t ) ,

where

g (t)= j = 1 m 2 ( x j y j ) 2 ( 1 + t x j + ( 1 t ) y j ) 3 .

Thus,

g ( t ) g ( t ) ( g ( t ) ) 2 = ( j = 1 m 2 ( x j y j ) 2 ( 1 + t x j + ( 1 t ) y j ) 3 ) ( j = 1 m t x j + ( 1 t ) y j 1 + t x j + ( 1 t ) y j ) ( j = 1 m x j y j ( 1 + t x j + ( 1 t ) y j ) 2 ) 2 0 ,

and then p (t)0, that is, p(t) is concave on [0,1].

The proof of Lemma 4 is completed. □

Lemma 5 Let x=( x 1 ,, x m ) and y=( y 1 ,, y m ) [ 1 2 , 1 ) m . Then the function q(t)=logψ(t) is convex on [0,1], where

ψ(t)= j = 1 m t x j + ( 1 t ) y j 1 t x j ( 1 t ) y j .

Proof

q (t)= ψ ( t ) ψ ( t ) ,

where

ψ ( t ) = j = 1 m x j y j ( 1 t x j ( 1 t ) y j ) 2 , q ( t ) = ψ ( t ) ψ ( t ) ( ψ ( t ) ) 2 ψ 2 ( t ) ,

where

ψ (t)= j = 1 m 2 ( x j y j ) 2 ( 1 t x j ( 1 t ) y j ) 3 .

By the Cauchy inequality, we have

ψ ( t ) ψ ( t ) ( ψ ( t ) ) 2 = ( j = 1 m 2 ( x j y j ) 2 ( 1 t x j ( 1 t ) y j ) 3 ) ( j = 1 m t x j + ( 1 t ) y j 1 t x j ( 1 t ) y j ) ( j = 1 m x j y j ( 1 t x j ( 1 t ) y j ) 2 ) 2 ( j = 1 m 2 | x j y j | ( 1 t x j ( 1 t ) y j ) 3 2 t x j + ( 1 t ) y j 1 t x j ( 1 t ) y j ) 2 ( j = 1 m x j y j ( 1 t x j ( 1 t ) y j ) 2 ) 2 = ( j = 1 m 2 | x j y j | t x j + ( 1 t ) y j ( 1 t x j ( 1 t ) y j ) 2 ) 2 ( j = 1 m x j y j ( 1 t x j ( 1 t ) y j ) 2 ) 2 .

From x j , y j [ 1 2 ,1) it follows that 2 t x j + ( 1 t ) y j 1, hence ψ (t)ψ(t) ( ψ ( t ) ) 2 0, and then q (t)0, that is, q(t) is convex on [0,1].

The proof of Lemma 5 is completed. □

Lemma 6 Let x=( x 1 ,, x m ) and y=( y 1 ,, y m ) ( 0 , 1 2 ] m . Then the function r(t)=logφ(t) is convex on [0,1], where

φ(t)= j = 1 m 1 t x j ( 1 t ) y j t x j + ( 1 t ) y j .

Proof

r (t)= φ ( t ) φ ( t ) ,

where

φ ( t ) = j = 1 m x j y j ( t x j + ( 1 t ) y j ) 2 , r ( t ) = φ ( t ) φ ( t ) ( φ ( t ) ) 2 φ 2 ( t ) ,

where

φ (t)= j = 1 m 2 ( x j y j ) 2 ( t x j + ( 1 t ) y j ) 3 .

By the Cauchy inequality, we have

φ ( t ) φ ( t ) ( φ ( t ) ) 2 = ( j = 1 m 2 ( x j y j ) 2 ( t x j + ( 1 t ) y j ) 3 ) ( j = 1 m 1 t x j ( 1 t ) y j t x j + ( 1 t ) y j ) ( j = 1 m x j y j ( t x j + ( 1 t ) y j ) 2 ) 2 ( j = 1 m 2 | x j y j | ( t x j + ( 1 t ) y j ) 3 2 1 t x j ( 1 t ) y j t x j + ( 1 t ) y j ) 2 ( j = 1 m x j y j ( t x j + ( 1 t ) y j ) 2 ) 2 = ( j = 1 m 2 | x j y j | 1 t x j ( 1 t ) y j ( t x j + ( 1 t ) y j ) 2 ) 2 ( j = 1 m x j y j ( t x j + ( 1 t ) y j ) 2 ) 2 .

From x j , y j (0, 1 2 ] it follows that 2 1 t x j ( 1 t ) y j 1, hence φ (t)φ(t) ( φ ( t ) ) 2 0, and then r (t)0, that is, r(t) is convex on [0,1].

The proof of Lemma 6 is completed. □

Lemma 7 Let x=( x 1 ,, x m ) and y=( y 1 ,, y m ) ( 0 , 1 ) m . Then the function h(t)=logf(t) is convex on [0,1], where

f(t)= j = 1 m 1 + t x j + ( 1 t ) y j 1 t x j ( 1 t ) y j .

Proof

h (t)= f ( t ) f ( t ) ,

where

f ( t ) = j = 1 m 2 ( x j y j ) ( 1 t x j ( 1 t ) y j ) 2 , h ( t ) = f ( t ) f ( t ) ( f ( t ) ) 2 f 2 ( t ) ,

where

f (t)= j = 1 m 4 ( x j y j ) 2 ( 1 t x j ( 1 t ) y j ) 3 .

By the Cauchy inequality, we have

f ( t ) f ( t ) ( f ( t ) ) 2 = ( j = 1 m 4 ( x j y j ) 2 ( 1 t x j ( 1 t ) y j ) 3 ) ( j = 1 m 1 + t x j + ( 1 t ) y j 1 t x j ( 1 t ) y j ) ( j = 1 m 2 ( x j y j ) ( 1 t x j ( 1 t ) y j ) 2 ) 2 ( j = 1 m 2 | x j y j | ( 1 t x j ( 1 t ) y j ) 3 2 1 + t x j + ( 1 t ) y j 1 t x j ( 1 t ) y j ) 2 ( j = 1 m 2 ( x j y j ) ( 1 t x j ( 1 t ) y j ) 2 ) 2 = ( j = 1 m 2 | x j y j | 1 + t x j + ( 1 t ) y j ( 1 t x j ( 1 t ) y j ) 2 ) 2 ( j = 1 m 2 ( x j y j ) ( 1 t x j ( 1 t ) y j ) 2 ) 2 .

From x j , y j (0,1) it follows that 2 1 + t x j + ( 1 t ) y j 1, hence f (t)f(t) ( f ( t ) ) 2 0, and then h (t)0, that is, h(t) is convex on [0,1].

The proof of Lemma 7 is completed. □

Lemma 8 Let x=( x 1 ,, x m ) and y=( y 1 ,, y m ) ( 0 , 5 2 ) m . Then the function s(t)=logw(t) is convex on [0,1], where

w(t)= j = 1 m ( 1 t x j + ( 1 t ) y j ( t x j + ( 1 t ) y j ) ) .

Proof

s (t)= w ( t ) w ( t ) ,

where

w ( t ) = j = 1 m ( x j y j ) ( 1 ( t x j + ( 1 t ) y j ) 2 + 1 ) , s ( t ) = w ( t ) w ( t ) ( w ( t ) ) 2 w 2 ( t ) ,

where

w (t)= j = 1 m 2 ( x j y j ) 2 ( t x j + ( 1 t ) y j ) 3 .

By the Cauchy inequality, we have

w ( t ) w ( t ) ( w ( t ) ) 2 = ( j = 1 m 2 ( x j y j ) 2 ( t x j + ( 1 t ) y j ) 3 ) ( j = 1 m ( 1 t x j + ( 1 t ) y j ( t x j + ( 1 t ) y j ) ) ) ( j = 1 m ( x j y j ) ( 1 ( t x j + ( 1 t ) y j ) 2 + 1 ) ) 2 ( j = 1 m 2 | x j y j | ( t x j + ( 1 t ) y j ) 3 2 1 t x j + ( 1 t ) y j ( t x j + ( 1 t ) y j ) ) 2 ( j = 1 m ( x j y j ) ( 1 ( t x j + ( 1 t ) y j ) 2 + 1 ) ) 2 = ( j = 1 m 2 | x j y j | 1 ( t x j + ( 1 t ) y j ) 2 ( t x j + ( 1 t ) y j ) 2 ) 2 ( j = 1 m ( x j y j ) 1 + ( t x j + ( 1 t ) y j ) 2 ( t x j + ( 1 t ) y j ) 2 ) 2 .

Let u j :=t x j +(1t) y j . From x j , y j (0, 5 2 ) it follows that u j 2 5 2. Since

u j 2 5 2 ( u j 2 + 2 ) 2 5 u j 4 + 4 u j 2 1 0 2 ( 1 u j 2 ) ( 1 + u j 2 ) 2 2 1 u j 2 1 + u j 2 ,

so w (t)w(t) ( w ( t ) ) 2 0, and then s (t)0, that is, s(t) is convex on [0,1].

The proof of Lemma 8 is completed. □

3 Proof of main results

Proof of Theorem 4 For any 1 i 1 << i k n, by Lemma 3 and Lemma 7, it follows that log j = 1 k 1 + x i j 1 x i j is convex on ( 0 , 1 ) k . Obviously, log j = 1 k 1 + x i j 1 x i j is also convex on ( 0 , 1 ) n , and then log E k ( 1 + x 1 x )= 1 i 1 < < i k n log j = 1 k 1 + x i j 1 x i j is convex on ( 0 , 1 ) n . Furthermore, it is clear that log E k ( 1 + x 1 x ) is symmetric on ( 0 , 1 ) n . By Lemma 1, it follows that log E k ( 1 + x 1 x ) is Schur-convex on ( 0 , 1 ) n , and then from Lemma 2 we conclude that E k ( 1 + x 1 x ) is also Schur-convex on ( 0 , 1 ) n .

The proof of Theorem 4 is completed. □

Similar to the proof of Theorem 4, we can use Lemma 4, Lemma 5, Lemma 6 and Lemma 8 respectively to prove Theorem 1, Theorem 2, Theorem 3 and Theorem 5; therefore we omit the details of the proof.

Remark 1 Using the Schur-convex function decision theorem, Liu et al. [9] have proved Theorem 3. Xia and Chu [10] have proved that the symmetric function

E k ( 1 + x x ) = 1 i 1 < < i k n j = 1 k 1 + x i j x i j ,k=1,,n,
(12)

is a Schur-convex function on R + n .

The reader may wish to prove the inequality (12) by the properties of a Schur-convex function.

References

  1. Marshall AW, Olkin I, Arnold BC: Inequalities: Theory of Majorization and Its Application. 2nd edition. Springer, New York; 2011.

    Book  MATH  Google Scholar 

  2. Wang BY: Foundations of Majorization Inequalities. Beijing Normal University Press, Beijing; 1990. (in Chinese)

    Google Scholar 

  3. Xia WF, Wang GD, Chu YM: Schur convexity and inequalities for a class of symmetric functions. Int. J. Pure Appl. Math. 2010, 58(4):435–452.

    MathSciNet  MATH  Google Scholar 

  4. Chu YM, Xia WF, Zhao TH: Schur convexity for a class of symmetric functions. Sci. China Math. 2010, 53(2):465–474. 10.1007/s11425-009-0188-2

    Article  MathSciNet  MATH  Google Scholar 

  5. Xia WF, Chu YM: Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications. Ukr. Math. J. 2009, 61(10):1541–1555. 10.1007/s11253-010-0296-8

    Article  MathSciNet  MATH  Google Scholar 

  6. Xia W-F, Chu Y-M: On Schur-convexity of some symmetric functions. J. Inequal. Appl. 2010., 2010: Article ID 543250. doi:10.1155/2010/543250

    Google Scholar 

  7. Mei H, Bai CL, Man H: Extension of an inequality guess. J. Inn. Mong. Univ. Natl. 2006, 21(2):127–129. (in Chinese)

    Google Scholar 

  8. Shi H-N: Theory of Majorization and Analytic Inequalities. Harbin Institute of Technology Press, Harbin; 2012. (in Chinese)

    Google Scholar 

  9. Liu HQ, Yu Q, Zhang Y: Some properties of a class of symmetric functions and its applications. J. Hengyang Norm. Univ. 2012, 33(6):167–171. (in Chinese)

    MathSciNet  Google Scholar 

  10. Xia W, Chu Y: Schur convexity with respect to a class of symmetric functions and their applications. Bull. Math. Anal. Appl. 2011, 3(3):84–96. http://www.bmathaa.org

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The work was supported by Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHR (IHLB)) (PHR201108407). Thanks for the help.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jing Zhang.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors co-authored this paper together. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Shi, HN., Zhang, J. Schur-convexity of dual form of some symmetric functions. J Inequal Appl 2013, 295 (2013). https://doi.org/10.1186/1029-242X-2013-295

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2013-295

Keywords