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New proofs of Schur-concavity for a class of symmetric functions

Abstract

By properties of the Schur-convex function, Schur-concavity for a class of symmetric functions is simply proved uniform.

2000 Mathematics Subject Classification: Primary 26D15; 05E05; 26B25.

1. Introduction

Throughout the article, denotes the set of real numbers, x= (x1, x2, ..., x n ) denotes n-tuple (n-dimensional real vectors), the set of vectors can be written as

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

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

For convenience, we introduce some definitions as follows.

Definition 1. [1, 2] Let x =(x1, ..., x n ) and y =(y1, ..., y n ) n.

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

(ii) Let Ω n, φ: Ω → is said to be increasing if xy implies φ(x) ≥ φ(y ). φ is said to be decreasing if and only if is increasing.

Definition 2. [1, 2] Let x =(x1, ..., x n ) and y = (y1, ..., y n ) n.

(i) x is said to be majorized by y(in symbols x y) if i = 1 k x [ i ] i = 1 k y [ i ] for k = 1, 2,..., n - 1 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.

(ii) Let Ω n, φ: Ω → is said to be a Schur-convex function on Ω if x y on Ω implies φ (x) ≤ φ (y). φ is said to be a Schur-concave function on Ω if and only if is Schur-convex function on Ω.

Definition 3. [1, 2] Let x = (x1, ..., x n ) and y= (y1, ..., y n ) n.

(i) Ω nis said to be a convex set if x, y Ω, 0 ≤ α ≤ 1 implies α x+ (1- α)y =(αx1 + (1 - α)y1, ...,αx n + (1- α)y n ) Ω.

(ii) Let Ω nbe convex set. A function φ: Ω → is said to be a convex function on Ω if

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

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

Recall that the following so-called Schur's condition is very useful for determining whether or not a given function is Schur-convex or Schur-concave.

Theorem A. [[1], p. 5] Let Ω nis symmetric and has a nonempty interior convex set. Ω0is the interior of Ω. φ: Ω → is continuous on Ω and differentiable in Ω0. 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 Ω0.

In recent years, by using Theorem A, many researchers have studied the Schur-convexity of some of symmetric functions.

Chu et al. [3] defined the following symmetric functions

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

and established the following results by using Theorem A.

Theorem B. For k = 1,..., n, F n (x , k) is an Schur-concave function on + n .

Jiang [4] are discussed the following symmetric functions

H k * ( x ) = 1 i 1 < . . . < i k n j = 1 k x i j 1 / k , k = 1 , . . . , n ,
(3)

and established the following results by using Theorem A.

Theorem C. Fork=1,...,n, H k * ( x ) is an Schur-concave function on + n .

Xia and Chu [5] investigated the following symmetric functions

ϕ n ( x , k ) = 1 i 1 < . . . < i k n j = 1 k x i j 1 + x i j , k = 1 , . . . , n ,
(4)

and established the following results by using Theorem A.

Theorem D. For k = 1,..., n, F n (x , k) is an Schur-concave function on + n .

In this note, by properties of the Schur-convex function, we simply prove Theorems B, C and D uniform.

2. New proofs three theorems

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

Lemma 1. [[1], p. 67], [2]If φ is symmetric and convex (concave) on symmetric convex set Ω, then φ is Schur-convex (Schur-concave) on Ω.

Lemma 2. [[1], p. 73],[2]Let Ω n, φ: Ω → +. Then lnφ is Schur-convex (Schur-concave) if and only if φ is Schur-convex (Schur-concave).

Lemma 3. [[1], p. 446], [2]Let Ω nbe open convex set, φ : Ω → . For x, y Ω, defined one variable function g(t) = φ (t x + (1 - t)y ) on interval (0, 1). Then φ is convex (concave) on Ω if and only if g is convex (concave) on (0, 1) for all x,y Ω.

Lemma 4. Let x= (x1,..., x m ) and y = (y1, ..., y m ) m. Then the following functions are concave on (0,1).

(i) f ( t ) = ln j = 1 m ( t x j + ( 1 - t ) y j ) - ln j = 1 m ( 1 + t x j + ( 1 - t ) y j ) ,

(ii) g ( t ) = ln j = 1 m ( t x j + ( 1 - t ) y j ) 1 / m ,

(iii) h ( t ) = 1 m lnψ ( t ) , where

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

Proof. (i) Directly calculating yields

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

and

f ( t ) = - j = 1 m ( x j - y j ) 2 1 ( t x j + ( 1 - t ) y j ) 2 - 1 ( 1 + t x j + ( 1 - t ) y j ) 2 = - j = 1 m ( x j - y j ) 2 1 + 2 t x j + 2 ( 1 - t ) y j ( t x j + ( 1 - t ) y j ) 2 ( 1 + t x j + ( 1 - t ) y j ) 2 .

Since f''(t) ≤ 0, f(t) is concave on (0,1).

(ii) Directly calculating yields

g ( t ) = 1 m j = 1 m ( x j - y j ) 1 m - 1 j = 1 m ( t x j + ( 1 - t ) y j ) 1 / m

and

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

Since g''(t) ≤ 0, f(t) is concave on (0,1)

(iii) By computing,

h ( t ) = 1 m ψ ( t ) ψ ( t ) , h ( t ) = 1 m ψ ( t ) ψ ( t ) - ( ψ ( t ) ) 2 ψ 2 ( t ) ,

where

ψ ( t ) = j = 1 m x j - y j ( 1 + t x j + ( 1 - t ) y j ) 2

and

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

Thus,

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 0 ,

and then h'' (t) ≤ 0, so f(t) is concave on (0,1).

The proof of Lemma 4 is completed.

Proof of Theorem A: For any 1 ≤ i1 < · · · < i k n, by Lemma 3 and Lemma 4(i), it follows that ln j = 1 k x i j - ln j = 1 k ( 1 + x i j ) is concave on + n , and then ln F n ( x , k ) = 1 i 1 < < i k n ln j = 1 k x i j - ln j = 1 k ( 1 + x i j ) is concave on + n . Furthermore, it is clear that ln F n (x, k) is symmetric on + n , by Lemma 1, it follows that ln F n ( x, k) is concave on + n , and then from Lemma 2 we conclude that F n (x , k) is also concave on + n .

The proof of Theorem A is completed.

Similar to the proof of Theorem A, by Lemma 4 (ii) and Lemma 4 (iii), we can prove Theorems B and C, respectively. Omitted detailed process.

References

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Acknowledgements

Shi was supported in part by the Scientific Research Common Program of Beijing Municipal Commission of Education (KM201111417006). This article was typeset by using A M S - L A T E X .

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Correspondence to Huan-Nan Shi.

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Shi, HN., Zhang, J. & Gu, C. New proofs of Schur-concavity for a class of symmetric functions. J Inequal Appl 2012, 12 (2012). https://doi.org/10.1186/1029-242X-2012-12

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Keywords

  • majorization
  • Schur-concavity
  • inequality
  • symmetric functions
  • concave functions