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Schur convexity for the ratios of the Hamy and generalized Hamy symmetric functions

  • Wei-Mao Qian
Open Access
Research

Abstract

In this paper, we present the Schur convexity and monotonicity properties for the ratios of the Hamy and generalized Hamy symmetric functions and establish some analytic inequalities. The achieved results is inspired by the paper of Hara et al. [J. Inequal. Appl. 2, 387-395, (1998)], and the methods from Guan [Math. Inequal. Appl. 9, 797-805, (2006)]. The inequalities we obtained improve the existing corresponding results and, in some sense, are optimal.

2010 Mathematics Subject Classification: Primary 05E05; Secondary 26D20.

Keywords

Hamy symmetric function generalized Hamy symmetric function Schur convex Schur concave 

1 Introduction

Throughout this paper, we denote + n = { x = ( x 1 , x 2 , , x n ) | x i > 0 , i = 1 , 2 , , n } . Open image in new window. For x + n Open image in new window, the Hamy symmetric function [1] is defined as
F n ( x , r ) = F n ( x 1 , x 2 , , x n ; r ) = 1 i 1 < i 2 < < i r n j = 1 r x i j 1 r , Open image in new window
(1.1)

where r is an integer and 1 ≤ rn.

The generalized Hamy symmetric function was introduced by Guan [2] as follows
F n * ( x , r ) = F n * ( x 1 , x 2 , , x n ; r ) = i 1 + i 2 + + i n = r x 1 i 1 x 2 i 2 x n i n 1 r , Open image in new window
(1.2)

where r is a positive integer.

In [2], Guan proved that both F n (x,r) and F n * ( x , r ) Open image in new window are Schur concave in + n Open image in new window. The main of this paper is to investigate the Schur convexity for the functions F n ( x , r ) F n ( x , r - 1 ) Open image in new window and F n * ( x , r ) F n * ( x , r - 1 ) Open image in new window and establish some analytic inequalities by use of the theory of majorization.

For convenience of readers, we recall some definitions as follows, which can be found in many references, such as [3].

Definition 1.1. The n-tuple x is said to be majorized by the n-tuple y (in symbols xy), if
i = 1 k x [ i ] i = 1 k y [ i ] , i = 1 n x [ i ] = i = 1 n y [ i ] , Open image in new window

where 1 ≤ kn - 1, and x[i]denotes the i th largest component of x.

Definition 1.2. Let E n be a set. A real-valued function F : E is said to be Schur convex on E if F(x) ≤ F(y) for each pair of n-tuples x = (x1,..., x n ) and y = (y1,..., y n ) in E, such that xy. F is said to be Schur concave if -F is Schur convex.

The theory of Schur convexity is one of the most important theories in the fields of inequalities. It can be used in combinatorial optimization [4], isoperimetric problems for polytopes [5], theory of statistical experiments [6], graphs and matrices [7], gamma functions [8], reliability and availability [9], optimal designs [10] and other related fields.

Our aim in what follows is to prove the following results.

Theorem 1.1. Let x + n , 2 r n Open image in new window is an integer, then the function ϕ r ( x ) = F n ( x , r ) F n ( x , r - 1 ) Open image in new window is Schur concave in + n Open image in new window and increasing with respect to x i (i= 1,2, ...,n).

Theorem 1.2. Let x + n , 2 r n Open image in new window is an integer, then the function ϕ r * ( x ) = F n * ( x , r ) F n * ( x , r - 1 ) Open image in new window is Schur concave in + n Open image in new window and increasing with respect to x i (i= 1,2,... n).

Corollary 1.1. If x i > 0 , i = 1 , 2 , , n , i = 1 n x i = s Open image in new window and that cs, then
G n ( x ) G n ( c - x ) = F n ( x , n ) F n ( c - x , n ) F n ( x , n - 1 ) F n ( c - x , n - 1 ) F n ( x , 1 ) F n ( c - x , 1 ) = A n ( x ) A n ( c - x ) Open image in new window
and
G n ( x ) G n ( c + x ) = F n ( x , n ) F n ( c + x , n ) F n ( x , n - 1 ) F n ( c + x , n - 1 ) F n ( x , 1 ) F n ( c + x , 1 ) = A n ( x ) A n ( c + x ) , Open image in new window

where A n ( x ) = 1 n i = 1 n x i , G n ( x ) = i = 1 n x i 1 n Open image in new window are the arithmetic and geo-metric means of x, respectively.

Corollary 1.2. If x i > 0 , i = 1 , 2 , , n , i = 1 n x i = s Open image in new window and that cs, then
F n * ( x , r ) F n * ( c - x , r ) F n * ( x , r - 1 ) F n * ( c - x , r - 1 ) F n * ( x , 2 ) F n * ( c - x , 2 ) F n * ( x , 1 ) F n * ( c - x , 1 ) = A n ( x ) A n ( c - x ) Open image in new window
and
F n * ( x , r ) F n * ( c + x , r ) F n * ( x , r - 1 ) F n * ( c + x , r - 1 ) F n * ( x , 2 ) F n * ( c + x , 2 ) F n * ( x , 1 ) F n * ( c + x , 1 ) = A n ( x ) A n ( c + x ) . Open image in new window

2 Lemmas

In order to establish our main results, we need several lemmas, which we present in this section.

Lemma 2.1 (see [3]). Let E n be a symmetric convex set with nonempty interior intE and φ : E be a continuous symmetric function. If φ is differentiable on intE, then φ is Schur convex (or Schur concave, respectively) on E if and only if
( x i - x j ) φ x i - φ x j 0 ( or 0 , respectively ) Open image in new window

for all i,j = 1,2,...,n and x = (x1,...,x n ) ∈ intE.

The r th elementary symmetric function (see [11]) is defined as
E n ( x , r ) = E n ( x 1 , x 2 , , x n ; r ) = 1 i 1 < i 2 < < i r n j = 1 r x i j , Open image in new window
(2.1)

where 1 ≤ rn is a positive integer, and E n (x, 0) = 1.

By (2.1) and simple computations, we have the following lemma.

Lemma 2.2. Let x + n , 1 i n Open image in new window, if
x i ¯ = ( x 1 , x 2 , , x i - 1 , x i + 1 , , x n ) . Open image in new window
Then,
E n ( x 1 , x 2 , , x n ; r ) = x i E n - 1 ( x i ¯ , r - 1 ) + E n - 1 ( x i ¯ , r ) . Open image in new window
(2.2)

Lemma 2.3 (see [11]). Let x + n , r Open image in new window is an integer and 1 ≤ rn - 1.

Then,
( E n ( x , r ) ) 2 > E n ( x , r - 1 ) E n ( x , r + 1 ) . Open image in new window
(2.3)
Another important symmetric function is the complete symmetric function (see [3]), which is defined by
C r ( x ) = C r ( x 1 , x 2 , , x n ) = i 1 + i 2 + + i n = r x 1 i 1 x 2 i 2 x n i n , Open image in new window

where i1, i2,..., i n are non-negative integer, r ∈ {1, 2,...} and C0(x) = 1.

Lemma 2.4 (see [12]). Let x i > 0, i = 1, 2,..., n, and x i ¯ = ( x 1 , x 2 , , x i - 1 , x i + 1 , , x n ) Open image in new window.

Then,
C r ( x ) = x i C r - 1 ( x ) + C r ( x i ¯ ) . Open image in new window
Lemma 2.5 (see [13]). If 0 < r < s , x + n Open image in new window, then
C r ( x ) C s - 1 ( x ) > C r - 1 C s ( x ) . Open image in new window
Lemma 2.6 (see [14]). If x i > 0 , i = 1 , 2 , , n , i = 1 n x i = s Open image in new window and cs, then
  1. (1)

    c - x n c s - 1 = c - x 1 n c s - 1 , c - x 2 n c s - 1 , , c - x n n c s - 1 ( x 1 , x 2 , , x n ) = x , Open image in new window,

     

(2) c + x s + n c = c + x 1 s + n c , c + x 2 s + n c , , c + x n s + n c x 1 s , x 2 s , , x n s = x s Open image in new window.

3 Proof of Theorems

Proof of Theorem 1.1. It is obvious that ϕ r (x) is symmetric and has continuous partial derivatives in + n Open image in new window. By Lemma 2.1, we only need to prove that
( x 1 - x 2 ) ϕ r ( x ) x 1 - ϕ r ( x ) x 2 0 . Open image in new window
(3.1)
For any fixed 2 ≤ rn, let u i = x i r , i = 1 , 2 , , n Open image in new window and u = ( u 1 , u 2 , , u n ) + n Open image in new window, we have
ϕ r ( x ) = F n ( x , r ) F n ( x , r - 1 ) = E n ( u , r ) E n ( u , r - 1 ) . Open image in new window
Differentiating ϕ r (x) with respect to x1 yields
ϕ r ( x ) x 1 = 1 E n 2 ( u , r - 1 ) E n ( u , r - 1 ) E n ( u , r ) u 1 u 1 x 1 - E n ( u , r ) E n ( u , r - 1 ) u 1 u 1 x 1 . Open image in new window
(3.2)
Using Lemma 2.2 repeatedly, we get
E n ( u , r ) = u 1 u 2 E n - 2 ( u 3 , , u n ; r - 2 ) + ( u 1 + u 2 ) E n - 2 ( u 3 , , u n ; r - 1 ) + E n - 2 ( u 3 , , u n ; r ) . Open image in new window
(3.3)
Equations (3.2) and (3.3) lead to
ϕ r ( x ) x 1 = 1 r E n 2 ( u , r - 1 ) ( u 1 1 - r u 2 A + u 1 1 - r B ) , Open image in new window
(3.4)
where
A = E n ( u , r - 1 ) E n - 2 ( u 3 , , u n ; r - 2 ) - E n ( u , r ) E n - 2 ( u 3 , , u n ; r - 3 ) Open image in new window
and
B = E n ( u , r - 1 ) E n - 2 ( u 3 , , u n ; r - 1 ) - E n ( u , r ) E n - 2 ( u 3 , , u n ; r - 2 ) . Open image in new window
Similarly, we can deduce that
ϕ r ( x ) x 2 = 1 r E n 2 ( u , r - 1 ) ( u 1 u 2 1 - r A + u 2 1 - r B ) . Open image in new window
(3.5)
From (3.4) and (3.5), one has
( x 1 - x 2 ) ϕ r ( x ) x 1 - ϕ r ( x ) x 2 = x 1 - x 2 r E n 2 ( u , r - 1 ) x 1 1 r x 2 1 r ( x 1 - 1 - x 2 - 1 ) A + ( x 1 1 r - 1 - x 2 1 r - 1 ) B . Open image in new window
(3.6)
It follows from (3.3) and Lemma 2.3 that
A = ( u 1 + u 2 ) [ E n 2 2 ( u 3 , , u n ; r 2 ) E n 2 ( u 3 , , u n ; r 1 ) × E n 2 ( u 3 , , u n ; r 3 ) ] + E n 2 ( u 3 , , u n ; r 1 ) E n 2 ( u 3 , , u n ; r 2 ) E n 2 ( u 3 , , u n ; r ) E n 2 ( u 3 , , u n ; r 3 ) > 0. Open image in new window

Similarly, we can get B > 0.

It follows from the function x k - r r ( k = 0 , 1 ) Open image in new window is decreasing in (0, +∞) that
( x 1 - x 2 ) x 1 k - r r - x 2 k - r r 0 , ( k = 0 , 1 ) . Open image in new window
(3.7)

Therefore, inequality (3.1) follows from (3.6) and (3.7) together with A > 0 and B > 0.

Next, we prove that ϕ r ( x ) = F n ( x , r ) F n ( x , r - 1 ) Open image in new window is increasing with respect to x i (i= 1,2,...,n).

By the symmetry of ϕ r (x) with respect to x i (i = 1, 2,..., n), we only need to prove that
ϕ r ( x ) x 1 0 , Open image in new window

which can be derived directly from A > 0 and B > 0 together with Equation (3.4).

Proof of Theorem 1.2. It is obvious that ϕ r * ( x ) Open image in new window is symmetric and has continuous partial derivatives in + n Open image in new window. By Lemma 2.1, we only need to prove that
( x 1 - x 2 ) ϕ r * ( x ) x 1 - ϕ r * ( x ) x 2 0 . Open image in new window
(3.8)
For any fixed 2 ≤ rn, let u i = x i r , i = 1 , 2 , , n Open image in new window and u = ( u 1 , u 2 , , u n ) + n Open image in new window. Then,
ϕ r * ( x ) = F n * ( x , r ) F n * ( x , r - 1 ) = C r ( u ) C r - 1 ( u ) . Open image in new window
(3.9)
Differentiating ϕ r * ( x ) Open image in new window with respect to x1, we have
ϕ r * ( x ) x 1 = 1 C r - 1 2 ( u ) C r - 1 ( u ) C r ( u ) u 1 u 1 x 1 - C r ( u ) C r - 1 ( u ) u 1 u 1 x 1 . Open image in new window
(3.10)
It follows from Lemma 2.4 that
C r ( u ) u 1 = C r - 1 ( u ) + u 1 C r - 1 ( u ) u 1 = C r - 1 ( u ) + u 1 C r - 2 ( u ) + u 1 C r - 2 ( u ) u 1 = C r - 1 ( u ) + u 1 C r - 2 ( u ) + u 1 2 C r - 2 ( u ) u 1 = = C r - 1 ( u ) + u 1 C r - 2 ( u ) + u 1 2 C r - 3 ( u ) + + u 1 r - 2 C 1 ( u ) + u 1 r - 1 . Open image in new window
(3.11)
Equations (3.10) and (3.11) lead to
ϕ r * ( x ) x 1 = 1 C r 1 2 ( u ) { [ C r 1 2 ( u ) C r ( u ) C r 2 ( u ) ] + u 1 [ C r 1 ( u ) C r 2 ( u ) C r ( u ) C r 3 ( u ) ] + + u 1 r 2 [ C r 1 ( u ) C 1 ( u ) C r ( u ) C 0 ( u ) ] + C r 1 ( u ) u 1 r 1 } 1 r u 1 1 r . Open image in new window
(3.12)
Similarly, we have
ϕ r * ( x ) x 2 = 1 C r 1 2 ( u ) { [ C r 1 2 ( u ) C r ( u ) C r 2 ( u ) ] + u 2 [ C r 1 ( u ) C r 2 ( u ) C r ( u ) C r 3 ( u ) ] + + u 2 r 2 [ C r 1 ( u ) C 1 ( u ) C r ( u ) C 0 ( u ) ] + C r 1 ( u ) u 2 r 1 } 1 r u 2 1 r . Open image in new window
(3.13)
From (3.12) and (3.13), one has
( x 1 x 2 ) ( ϕ r * ( x ) x 1 ϕ r * ( x ) x 2 ) = x 1 x 2 r C r 1 2 ( u ) { [ C r 1 2 ( u ) C r ( u ) C r 2 ( u ) ] ( x 1 1 r r x 2 1 r r ) + [ C r 1 ( u ) C r 2 ( u ) C r ( u ) C r 3 ( u ) ] ( x 1 2 r r x 2 2 r r ) + + [ C r 1 ( u ) C 1 ( u ) C r ( u ) C 0 ( u ) ] × ( x 1 ( r 1 ) r r x 2 ( r 1 ) r r ) } . Open image in new window
(3.14)
By Lemma 2.5, we know that
C r - 1 2 ( u ) - C r ( u ) C r - 2 ( u ) > 0 , C r - 1 ( u ) C r - 2 ( u ) - C r ( u ) C r - 3 ( u ) > 0 , , C r - 1 ( u ) C 1 ( u ) - C r ( u ) C 0 ( u ) > 0 . Open image in new window
(3.15)
The monotonicity of the function x j - r r ( 1 j r - 1 ) Open image in new window in (0, +∞) leads to the conclusion that
( x 1 - x 2 ) ( x 1 j - r r - x 2 j - r r ) 0 . Open image in new window
(3.16)

Therefore, inequality (3.8) follows from (3.14)-(3.16).

Next, we prove that ϕ r * ( x ) = F n * ( x , r ) F n * ( x , r - 1 ) Open image in new window is increasing with respect to x i (i= 1,2,...,n).

From (3.12) and (3.15), we clearly see that
ϕ r * ( x ) x 1 0 . Open image in new window
(3.17)

Inequality (3.17) implies that ϕ r * ( x ) Open image in new window is increasing with respect to x1, then from the symmetry of ϕ r * ( x ) Open image in new window with respect to x i (i = 1, 2,..., n) we know that ϕ r * ( x ) Open image in new window is increasing with respect to each x i (i = 1, 2,..., n).

Proof of Corollary 1.1. By Theorem 1.1 and Lemma 2.6, we have ϕ r c - x n c s - 1 ϕ r ( x ) Open image in new window and ϕ r c + x s + n c ϕ r x s Open image in new window which imply Corollary 1.1.

Remark 1. Let 0 < x i 1 2 , i = 1 , 2 , , n Open image in new window, then
G n ( x ) G n ( 1 - x ) A n ( x ) A n ( 1 - x ) , Open image in new window
(3.18)

where (1 - x) = (1 - x1, 1 - x2,... , 1 - x n ), commonly referred to as Ky Fan inequality (see [15]), which has attracted the attention of a considerable number of mathematicians (see [16, 17, 18, 19, 20]).

Letting i = 1 n x i 1 Open image in new window and taking c = 1 in Corollary 1.1, we get
G n ( x ) G n ( 1 - x ) = F n ( x , n ) F n ( 1 - x , n ) F n ( x , n - 1 ) F n ( 1 - x , n - 1 ) F n ( x , 1 ) F n ( 1 - x , 1 ) = A n ( x ) A n ( 1 - x ) . Open image in new window
(3.19)

It is obvious that inequality (3.19) can be called Ky Fan-type inequality.

Remark 2. Let x i > 0, i = 1, 2,..., n, the following inequalities
i = 1 n ( x i - 1 - 1 ) ( n - 1 ) n Open image in new window
and
i = 1 n ( x i - 1 + 1 ) ( n + 1 ) n Open image in new window

are the well-known Weierstrass inequalities (see [11]).

Taking c = s = 1 in Corollary 1.1, one has
i = 1 n ( x i - 1 - 1 ) F n ( 1 - x , n - 1 ) F n ( x , n - 1 ) n F n ( 1 - x , 2 ) F n ( x , 2 ) n ( n - 1 ) n Open image in new window
and
i = 1 n ( x i - 1 + 1 ) F n ( 1 + x , n - 1 ) F n ( x , n - 1 ) n F n ( 1 + x , 2 ) F n ( x , 2 ) n ( n + 1 ) n . Open image in new window

It is obvious that our inequalities can be called Weierstrass-type inequalities.

Proof of Corollary 1.2. By Theorem 1.2 and Lemma 2.6, we have ϕ r * c - x n c s - 1 ϕ r * ( x ) Open image in new window and ϕ r * c + x s + n c ϕ r * x s Open image in new window, which imply Corollary 1.2.

Notes

Acknowledgements

This work was supported by NSF of China under grant No. 11071069.

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© Qian; licensee Springer. 2011

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

Authors and Affiliations

  1. 1.Huzhou Broadcast and TV UniversityHuzhouChina

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