Open Access

Some new judgement theorems of Schur geometric and Schur harmonic convexities for a class of symmetric functions

Journal of Inequalities and Applications20132013:527

https://doi.org/10.1186/1029-242X-2013-527

Received: 22 July 2013

Accepted: 18 October 2013

Published: 11 November 2013

Abstract

The judgement theorems of Schur geometric and Schur harmonic convexities for a class of symmetric functions are given. As their application, some analytic inequalities are established.

MSC:26D15, 05E05, 26B25.

Keywords

Schur geometric convexitySchur harmonic convexityinequalitysymmetric function

1 Introduction

Throughout this paper, 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.

Let π = ( π ( 1 ) , , π ( n ) ) be a permutation of ( 1 , , n ) , all permutations are totally n!. The following conclusion is proved in [[1], pp.127-129].

Theorem A Let A R k be a symmetric convex set, and let φ be a Schur-convex function defined on A with the property that for each fixed x 2 , , x k , φ ( z , x 2 , , x k ) is convex in z on { z : ( z , x 2 , , x k ) A } . Then, for any n > k ,
ψ ( x 1 , , x n ) = π φ ( x π ( 1 ) , , x π ( k ) )
(1)
is Schur-convex on
B = { ( x 1 , , x n ) : ( x π ( 1 ) , , x π ( k ) ) A  for all permutations  π } .
Furthermore, the symmetric function
ψ ¯ ( x ) = 1 i 1 < < i k n φ ( x i 1 , , x i k )
(2)

is also Schur-convex on B.

Theorem A is very effective for judgement of the Schur-convexity of the symmetric functions of the form (2), see the references [1] and [2].

The Schur geometrically convex functions were proposed by Zhang [3] in 2004. Further, the Schur harmonically convex functions were proposed by Chu and Lü [4] in 2009. The theory of majorization was enriched and expanded by using these concepts [515]. Regarding Schur geometrically convex functions and Schur harmonically convex functions, the aim of this paper is to establish the following judgement theorems which are similar to Theorem A.

Theorem 1 Let A R k be a symmetric geometrically convex set, and let φ be a Schur geometrically convex (concave) function defined on A with the property that for each fixed x 2 , , x k , φ ( z , x 2 , , x k ) is GA convex (concave) in z on { z : ( z , x 2 , , x k ) A } . Then, for any n > k ,
ψ ( x 1 , , x n ) = π φ ( x π ( 1 ) , , x π ( k ) )
is Schur geometrically convex (concave) on
B = { ( x 1 , , x n ) : ( x π ( 1 ) , , x π ( k ) ) A  for all permutations  π } .
Furthermore, the symmetric function
ψ ¯ ( x ) = 1 i 1 < < i k n φ ( x i 1 , , x i k )

is also Schur geometrically convex (concave) on B.

Theorem 2 Let A R k be a symmetric harmonically convex set, and let φ be a Schur harmonically convex (concave) function defined on A with the property that for each fixed x 2 , , x k , φ ( z , x 2 , , x k ) is HA convex (concave) in z on { z : ( z , x 2 , , x k ) A } . Then, for any n > k ,
ψ ( x 1 , , x n ) = π φ ( x π ( 1 ) , , x π ( k ) )
is Schur harmonically convex (concave) on
B = { ( x 1 , , x n ) : ( x π ( 1 ) , , x π ( k ) ) A  for all permutations  π } .
Furthermore, the symmetric function
ψ ¯ ( x ) = 1 i 1 < < i k n φ ( x i 1 , , x i k )

is also Schur harmonically convex (concave) on B.

2 Definitions and lemmas

In order to prove some further results, in this section we recall useful definitions and lemmas.

Definition 1 [1, 16]

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

    We say y majorizes x (x is said to be majorized by y), denoted by 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.

     
  2. (ii)

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

     

Definition 2 [1, 16]

Let x = ( x 1 , , x n ) and y = ( y 1 , , y n ) R n , 0 α 1 . A set Ω R n is said to be a convex set if x , y Ω implies α x + ( 1 α ) y = ( α x 1 + ( 1 α ) y 1 , , α x n + ( 1 α ) y n ) Ω .

Definition 3 [1, 16]

  1. (i)

    A set Ω R n is called a symmetric set if x Ω implies x P Ω for every n × n permutation matrix P.

     
  2. (ii)

    A function φ : Ω R is called symmetric if for every permutation matrix P, φ ( x P ) = φ ( x ) for all x Ω .

     
Definition 4 Let Ω R + n , x = ( x 1 , , x n ) Ω and y = ( y 1 , , y n ) Ω .
  1. (i)

    [[3], p.64] A set Ω is called a geometrically convex set if ( x 1 α y 1 β , , x n α y n β ) Ω for all x , y Ω and α , β [ 0 , 1 ] such that α + β = 1 .

     
  2. (ii)

    [[3], p.107] A function φ : Ω R + is said to be a Schur geometrically convex function on Ω if ( log x 1 , , log x n ) ( log y 1 , , log y n ) on Ω implies φ ( x ) φ ( y ) . A function φ is said to be a Schur geometrically concave function on Ω if and only if −φ is a Schur geometrically convex function.

     

Definition 5 [17]

Let Ω R + n .
  1. (i)

    A set Ω is said to be a harmonically convex set if xy λ x + ( 1 λ ) y Ω for every x , y Ω and λ [ 0 , 1 ] , where xy = i = 1 n x i y i and 1 x = ( 1 x 1 , , 1 x n ) .

     
  2. (ii)

    A function φ : Ω R + is said to be a Schur harmonically convex function on Ω if 1 x 1 y implies φ ( x ) φ ( y ) . A function φ is said to be a Schur harmonically concave function on Ω if and only if −φ is a Schur harmonically convex function.

     

Definition 6 [18]

Let I R + , φ : I R + be continuous.
  1. (i)
    A function φ is said to be a GA convex (concave) function on I if
    φ ( x y ) ( ) φ ( x ) + φ ( y ) 2

    for all x , y I .

     
  2. (ii)
    A function φ is said to be a HA convex (concave) function on I if
    φ ( 2 x y x + y ) ( ) φ ( x ) + φ ( y ) 2
     

for all x , y I .

Lemma 1 [[16], p.57]

Let Ω R n be a symmetric convex set with a nonempty interior Ω 0 . φ : Ω R is continuous on Ω and differentiable on Ω 0 . Then φ is a 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 )
(3)

holds for any x = ( x 1 , , x n ) Ω 0 .

Lemma 2 [[3], p.108]

Let Ω R + n be a symmetric geometrically convex set with a nonempty interior Ω 0 . Let φ : Ω R + be continuous on Ω and differentiable on Ω 0 . Then φ is a Schur geometrically convex (Schur geometrically concave) function if and only if φ is symmetric on Ω and
( x 1 x 2 ) ( x 1 φ x 1 x 2 φ x 2 ) 0 ( 0 )
(4)

holds for any x = ( x 1 , , x n ) Ω 0 .

Lemma 3 [17, 19]

Let Ω R + n be a symmetric harmonically convex set with a nonempty interior Ω 0 . Let φ : Ω R + be continuous on Ω and differentiable on Ω 0 . Then φ is a Schur harmonically convex (Schur harmonically concave) function if and only if φ is symmetric on Ω and
( x 1 x 2 ) ( x 1 2 φ x 1 x 2 2 φ x 2 ) 0 ( 0 )
(5)

holds for any x = ( x 1 , , x n ) Ω 0 .

Lemma 4 [18]

Let I R + be an open subinterval, and let φ : I R + be differentiable.
  1. (i)

    φ is GA-convex (concave) if and only if x φ ( x ) is increasing (decreasing).

     
  2. (ii)

    φ is HA-convex (concave) if and only if x 2 φ ( x ) is increasing (decreasing).

     

3 Proofs of main results

Proof of Theorem 1 To verify condition (4) of Lemma 2, denote by π ( i , j ) the summation over all permutations π such that π ( i ) = 1 , π ( j ) = 2 . Because φ is symmetric,
ψ ( x 1 , , x n ) = i , j k i j π ( i , j ) φ ( x 1 , x 2 , x π ( 1 ) , , x π ( i 1 ) , x π ( i + 1 ) , , x π ( j 1 ) , x π ( j + 1 ) , , x π ( k ) ) + i k < j π ( i , j ) φ ( x 1 , x π ( 1 ) , , x π ( i 1 ) , x π ( i + 1 ) , , x π ( k ) ) + j k < i π ( i , j ) φ ( x 2 , x π ( 1 ) , , x π ( j 1 ) , x π ( j + 1 ) , , x π ( k ) ) + k < i , j i j π ( i , j ) φ ( x π ( 1 ) , , x π ( k ) ) .
Then
Δ 1 : = ( x 1 ψ x 1 x 2 ψ x 2 ) ( x 1 x 2 ) = i , j k i j π ( i , j ) [ x 1 φ ( 1 ) ( x 1 , x 2 , x π ( 1 ) , , x π ( i 1 ) , x π ( i + 1 ) , , x π ( j 1 ) , x π ( j + 1 ) , , x π ( k ) ) x 2 φ ( 2 ) ( x 1 , x 2 , x π ( 1 ) , , x π ( i 1 ) , x π ( i + 1 ) , , x π ( j 1 ) , x π ( j + 1 ) , , x π ( k ) ) ] ( x 1 x 2 ) + i k < j π ( i , j ) [ x 1 φ ( 1 ) ( x 1 , x π ( 1 ) , , x π ( i 1 ) , x π ( i + 1 ) , , x π ( k ) ) x 2 φ ( 1 ) ( x 2 , x π ( 1 ) , , x π ( i 1 ) , x π ( i + 1 ) , , x π ( k ) ) ] ( x 1 x 2 ) .
Here,
( x 1 φ ( 1 ) x 2 φ ( 2 ) ) ( x 1 x 2 ) 0 ( 0 )
because φ is Schur geometrically convex (concave), and
[ x 1 φ ( 1 ) ( x 1 , z ) x 2 φ ( 1 ) ( x 2 , z ) ] ( x 1 x 2 ) 0 ( 0 )
because φ ( z , x 2 , , x k ) is GA convex (concave) in its first argument on { z : ( z , x 2 , , x k ) A } . Accordingly, Δ 1 0 (≤0). This shows that ψ is Schur geometrically convex (concave) on
B = { ( x 1 , , x n ) : ( x π ( 1 ) , , x π ( k ) ) A  for all permutations  π } .
Notice that
ψ ¯ ( x ) = ψ ( x ) / k ! ( n k ) ! .

Of course, ψ ¯ is Schur geometrically convex (concave) whenever ψ is Schur geometrically convex (concave).

The proof of Theorem 1 is completed. □

Proof of Theorem 2 We only need to verify condition (5) of Lemma 3, the proof is similar to that of Theorem 1 and is omitted. □

Remark 1 In most applications, A has the form I k for some interval I R and in this case B = I n . Notice that the convexity of φ in its first argument also implies that φ is convex in each argument, the other arguments being fixed, because φ is symmetric.

4 Applications

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

In 2011, Guan and Guan [20] proved the following theorem through Lemma 2.

Theorem 3 The symmetric function E k ( x 1 x ) , k = 1 , , n , is Schur geometrically convex on ( 0 , 1 ) n .

Now, we give a new proof of Theorem 3 by using Theorem 1. Furthermore, we prove the following theorem through Theorem 2.

Theorem 4 The symmetric function E k ( x 1 x ) , k = 1 , , n , is Schur harmonically convex on ( 0 , 1 ) n .

Proof of Theorem 3 Let φ ( z ) = i = 1 k [ z i / ( 1 z i ) ] . Then
log φ ( z ) = i = 1 k [ log z i log ( 1 z i ) ]
and
φ ( z ) z 1 = φ ( z ) ( 1 z 1 + 1 1 z 1 ) , φ ( z ) z 2 = φ ( z ) ( 1 z 2 + 1 1 z 2 ) , Δ : = ( z 1 z 2 ) ( z 1 φ ( z ) z 1 z 2 φ ( z ) z 2 ) Δ = ( z 1 z 2 ) φ ( z ) ( z 1 1 z 1 z 2 1 z 2 ) Δ = ( z 1 z 2 ) 2 φ ( z ) 1 ( 1 z 2 ) ( 1 z 1 ) .
(7)

This shows that Δ 0 when 0 < z i < 1 , i = 1 , , k . According to Lemma 2, φ is Schur geometrically convex on A = { z : z ( 0 , 1 ) k } . Let g ( t ) = t 1 t , then h ( t ) : = t g ( t ) = t ( 1 t ) 2 . From t ( 0 , 1 ) , it follows that h ( t ) = 1 + t ( 1 t ) 3 0 . According to Lemma 4(i), φ is GA convex in its single variable on ( 0 , 1 ) . So E k ( x 1 x ) is Schur geometrically convex on ( 0 , 1 ) n from Theorem 1. The proof of Theorem 3 is completed. □

Proof of Theorem 4 Let φ ( z ) = i = 1 k ( z i / 1 z i ) , then
log φ ( z ) = i = 1 k [ log z i log ( 1 z i ) ] .
From (7), we get
Δ 1 : = ( z 1 z 2 ) ( z 1 2 φ ( z ) z 1 z 2 2 φ ( z ) z 2 ) = ( z 1 z 2 ) φ ( z ) ( z 1 z 2 + z 1 2 1 z 1 z 2 2 1 z 2 ) = ( z 1 z 2 ) 2 φ ( z ) [ 1 + z 1 + z 2 z 1 z 2 ( 1 z 2 ) ( 1 z 1 ) ] .

This shows that Δ 1 0 when 0 < z i < 1 , i = 1 , , k . According to Lemma 3, φ is Schur harmonically convex on A = { z : z ( 0 , 1 ) k } . Let g ( t ) = t 1 t , then p ( t ) : = t 2 g ( t ) = t 2 ( 1 t ) 2 . From t ( 0 , 1 ) , it follows that p ( t ) = 2 t ( 1 t ) 3 0 . According to Lemma 4(ii), φ is HA convex in its single variable on ( 0 , 1 ) . So E k ( x 1 x ) is Schur harmonically convex on ( 0 , 1 ) n from Theorem 2. The proof of Theorem 4 is completed. □

By using Theorem A, the following conclusion is proved in [[1], p.129].

The symmetric function
ψ ¯ ( x ) = 1 i 1 < < i k n x i 1 + + x i k x i 1 x i k
(8)

is Schur-convex on R + n .

Now we use Theorem 1 and Theorem 2, respectively, to study Schur geometric convexity and Schur harmonic convexity of ψ ¯ ( x ) .

Theorem 5 The symmetric function ψ ¯ ( x ) is Schur geometrically convex and Schur harmonically concave on R + n .

Proof Let φ ( y ) = i = 1 k y i / i = 1 k y i , then log φ ( y ) = log ( i = 1 k y i ) i = 1 k log y i . Thus,
φ ( y ) y 1 = φ ( y ) ( 1 i = 1 k y i 1 y 1 ) , φ ( y ) y 2 = φ ( y ) ( 1 i = 1 k y i 1 y 2 ) , Δ : = ( y 1 y 2 ) ( y 1 φ ( y ) y 1 y 2 φ ( y ) y 2 ) Δ = ( y 1 y 2 ) φ ( y ) ( y 1 y 2 i = 1 k y i ) Δ = ( y 1 y 2 ) 2 i = 1 k y i 0 .

According to Lemma 2, φ ( y ) is Schur geometrically convex on R + k . Let g ( z ) = φ ( z , x 2 , , x k ) = z + a b z = 1 b + a b z , where a = i = 2 k x i , b = i = 2 k x i , then h ( z ) : = z g ( z ) = a b z . From z R + , it follows that h ( z ) = a b z 2 0 . According to Lemma 4(i), φ is GA convex in its single variable on R + . So ψ ¯ ( x ) is Schur geometrically convex on R + from Theorem 1.

It is easy to check that
Δ 1 : = ( y 1 y 2 ) ( y 1 2 φ ( y ) y 1 y 2 2 φ ( y ) y 2 ) = ( y 1 y 2 ) 2 ( y 1 + y 2 i = 1 k y i ) i = 1 k y i 0 .

According to Lemma 3, φ ( y ) is Schur harmonically concave on R + k . Let h ( z ) : = z 2 g ( z ) = a b . h ( z ) = 0 when z R + . According to Lemma 4(ii), φ is HA concave in its single variable on R + . So ψ ¯ ( x ) is Schur harmonically concave on R + n from Theorem 2. □

Remark 2 Let
H = n i = 1 n 1 x i , G = ( i = 1 n x i ) 1 n ,
where x i > 0 , i = 1 , , n . Then
( log G , , log G ) ( log x 1 , , log x n ) ,
(9)
( 1 H , , 1 H ) ( 1 x 1 , , 1 x n ) .
(10)
From Theorem 5, it follows that
k C n k H k 1 1 i 1 < < i k n x i 1 + + x i k x i 1 x i k k C n k G k 1 .
(11)

By using Theorem A, the following conclusion is proved in [[1], p.129].

The symmetric function
ψ ( x ) = 1 i 1 < < i k n x i 1 x i k x i 1 + + x i k

is Schur-concave on R + n .

By applying Theorem 2, we further obtain the following result.

Theorem 6 The symmetric function ψ ( x ) is Schur harmonically convex on R + n .

Proof Let λ ( y ) = i = 1 k y i / i = 1 k y i . According to the proof of Theorem 5, φ ( y ) is Schur harmonically concave on R + k . Let λ ( y ) = 1 φ ( y ) . From the definition of Schur harmonically convex, it follows that λ ( y ) is Schur harmonically convex on R + k . Let g ( z ) = λ ( z , x 2 , , x k ) = b z z + a , where a = i = 2 k x i , b = i = 2 k x i . Then h ( z ) : = z 2 g ( z ) = z 2 a b ( z + a ) 2 . With the fact that h ( z ) = 2 z a 2 b ( z + a ) 3 0 for z R + , it follows that φ is HA convex in its single variable on R + . So, from Theorem 2, ψ ( x ) is Schur harmonically convex on R + n . □

Remark 3 From Theorem 6 and (10), it follows that
1 i 1 < < i k n x i 1 x i k x i 1 + + x i k H k 1 C n k k ,
(12)

where x i > 0 , i = 1 , , n .

Remark 4 It needs further discussion that ψ ( x ) is Schur geometrically convex on R + n .

Declarations

Acknowledgements

The work was supported by the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHR (IHLB)) (PHR201108407) and the National Natural Science Foundation of China (Grant No. 11101034).

Authors’ Affiliations

(1)
Department of Electronic Information, Teacher’s College, Beijing Union University
(2)
Basic Courses Department, Beijing Union University

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© Shi and Zhang; licensee Springer. 2013

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