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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2007, Article ID 78175, 10 pages
doi:10.1155/2007/78175
Research Article
Schur-Convexity of Two Types of One-Parameter
Mean Values in n Variables
Ning-Guo Zheng, Zhi-Hua Zhang, and Xiao-Ming Zhang
Received 10 July 2007; Revised 9 October 2007; Accepted 9 November 2007
Recommended by Simeon Reich
We establish Schur-convexities of two types of one-parameter mean values in n variables.
As applications, Schur-convexities of some well-known functions involving the complete
elementary symmetric functions are obtained.
Copyright © 2007 Ning-Guo Zheng et al. 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 the paper,
R denotes the set of real numbers and R
+
denotes the set of strictly
positive real numbers. Let n
≥ 2, n ∈ N, x = (x
1
,x
2
, ,x
n
) ∈ R
n
+
,andx
1/r
= (x
1/r
1
,x
1/r
2
, ,
x
1/r
n
), where r ∈ R, r=0; let E
n−1
⊂ R
n−1
be the simplex
E
n−1
=
u
1
, ,u
n−1
: u
i
> 0(1 ≤ i ≤ n − 1),
n−1
i=1
u
i
≤ 1
, (1.1)
and let dμ
= du
1
, ,du
n−1
be the differential of the volume in E
n−1
.
The weighted arithmetic mean A(x,u) and the power mean M
r
(x,u)oforderr with
respect to the numbers x
1
,x
2
, ,x
n
and the positive weights u
1
,u
2
, ,u
n
with
n
i
=1
u
i
= 1
are defined, respectively, as A(x,u)
=
n
i
=1
u
i
x
i
, M
r
(x,u) = (
n
i
=1
u
i
x
r
i
)
1/r
for r=0, and
M
0
(x,u)=
n
i
=1
x
u
i
i
. For u=(1/n,1/n, ,1/n), we denote A(x,u)
Δ
=A(x), M
r
(x,u)
Δ
=M
r
(x).
The well-known logarithmic mean L(x
1
,x
2
) of two positive numbers x
1
and x
2
is
L
x
1
,x
2
=
⎧
⎪
⎨
⎪
⎩
x
1
− x
2
ln x
1
− ln x
2
, x
1
=x
2
,
x
1
, x
1
= x
2
.
(1.2)
2 Journal of Inequalities and Applications
As further generalization of L(x
1
,x
2
), Stolarsky [1] studied the one-parameter mean, that
is,
L
r
x
1
,x
2
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
x
r+1
1
− x
r+1
2
(r +1)
x
1
− x
2
1/r
, r=−1,0, x
1
=x
2
,
x
1
− x
2
ln x
1
− ln x
2
, r =−1, x
1
=x
2
,
1
e
x
x
1
1
x
x
2
2
1/(x
1
−x
2
)
, r = 0, x
1
=x
2
,
x
1
, x
1
= x
2
.
(1.3)
Alzer [2, 3] obtained another form of one-parameter mean, that is,
F
r
x
1
,x
2
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
r
r +1
·
x
r+1
1
− x
r+1
2
x
r
1
− x
r
2
, r=−1,0, x
1
=x
2
,
x
1
x
2
·
ln x
1
− ln x
2
x
1
− x
2
, r =−1, x
1
=x
2
,
x
1
− x
2
ln x
1
− ln x
2
, r = 0, x
1
=x
2
,
x
1
, x
1
= x
2
.
(1.4)
These two means can be written also as
L
r
x
1
,x
2
=
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
1
0
x
1
u +x
2
(1 − u)
r
du
1/r
, r=0,
exp
1
0
ln
x
1
u +x
2
(1 − u)
du
, r = 0,
F
r
x
1
,x
2
=
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
1
0
x
r
1
u +x
r
2
(1 − u)
1/r
du, r=0,
1
0
x
u
1
x
1−u
2
du, r = 0.
(1.5)
Correspondingly, Pittenger [4] and Pearce et al. [5] investigated the means above in n
variables, respectively,
L
r
(x) =
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
(n − 1)!
E
n−1
A(x,u)
r
dμ
1/r
, r=0,
exp
(n − 1)!
E
n−1
ln A(x,u)dμ
, r = 0,
F
r
(x) = (n − 1)!
E
n−1
M
r
(x,u)dμ,
(1.6)
where u
n
= 1 −
n−1
i
=1
u
i
.
Ning-Guo Zheng et al. 3
Expressions (1.3)and(1.4) can be also written by using 2-order determinants, that is,
L
r
x
1
,x
2
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎛
⎝
1
r +1
·
1 x
r+1
2
1 x
r+1
1
1 x
2
1 x
1
⎞
⎠
1/r
, r=−1,0, x
1
=x
2
,
1 x
2
1 x
1
1lnx
2
1lnx
1
, r =−1, x
1
=x
2
,
exp
⎧
⎨
⎩
⎛
⎝
1 x
2
ln x
2
1 x
1
ln x
1
1 x
2
1 x
1
⎞
⎠
−
1
⎫
⎬
⎭
, r = 0, x
1
=x
2
,
x
1
, x
1
= x
2
,
F
r
x
1
,x
2
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎛
⎝
r
1 x
r+1
2
1 x
r+1
1
⎞
⎠
⎛
⎝
(r +1)
1 x
r
2
1 x
r
1
⎞
⎠
, r=−1,0, x
1
=x
2
,
x
1
x
2
1lnx
2
1lnx
1
1 x
2
1 x
1
, r =−1, x
1
=x
2
,
1 x
2
1 x
1
1lnx
2
1lnx
1
, r = 0, x
1
=x
2
,
x
1
, x
1
= x
2
.
(1.7)
Utilizing higher-order generalized Vandermonde determinants, Xiao et al. [8, 7, 6, 9]gave
the analogous definitions of L
r
(x)andF
r
(x).
Obviously, L
r
(x)andF
r
(x) are symmetric with respect to x
1
,x
2
, ,x
n
, r → L
r
(x)and
r
→ F
r
(x) are continuous for any x ∈ R
n
+
.
In [4, 5, 10, 11], the authors studied the Schur-convexities of L
r
(x
1
,x
2
)andF
r
(x
1
,x
2
).
In this paper, we establish the Schur-convexities of two types of one-parameter mean
values L
r
(x)andF
r
(x) for several positive numbers. As applications, Schur-convexities of
some well-known functions involving the complete elementary symmetric functions are
obtained.
2. Some definitions and lemmas
The Schur-convex function was introduced by Schur [12] in 1923, and has many impor-
tant applications in analytic inequalities. The following definitions can be found in many
references such as [12–17].
Definit ion 2.1. For u
= (u
1
,u
2
, ,u
n
), v = (v
1
,v
2
, ,v
n
) ∈ R
n
, without loss of general-
ity, it is assumed that u
1
≥ u
2
≥ ··· ≥ u
n
and v
1
≥ v
2
≥ ··· ≥ v
n
.Thenu is said to be
majorized by v (in symbols u
≺ v)if
k
i
=1
u
i
≤
k
i
=1
v
i
for k = 1,2, ,n − 1and
n
i
=1
u
i
=
n
i
=1
v
i
.
Definit ion 2.2. Let Ω
⊂ R
n
. A function ϕ : Ω → R is said to be a Schur-convex (Schur-
concave) function if u
≺ v implies ϕ(u) ≤ (≥ )ϕ(v).
4 Journal of Inequalities and Applications
Every Schur-convex function is a symmetric function [18]. But it is not hard to see that
not every symmetric function can be a Schur-convex function [15, page 258]. However,
we have the follow ing so-called Schur condition.
Lemma 2.3 [12, page 57]. Suppose that Ω
⊂ R
n
is symmetric w ith respect to permutations
and convexset, and has a nonempty interior set Ω
0
.Letϕ : Ω → R be continuous on Ω and
continuously differentiable in Ω
0
.Then,ϕ is a Schur-convex (Schur-concave) function if and
only if it is symmetric and if
u
1
− u
2
∂ϕ
∂u
1
−
∂ϕ
∂u
2
≥
(≤ )0 (2.1)
holds for any u
= (u
1
,u
2
, ,u
n
) ∈ Ω
0
.
Lemma 2.4. Let m
≥ 1,n ≥ 2, m, n ∈ N, Λ ⊂ R
m
, Ω ⊂ R
n
, φ : Λ × Ω → R, φ(v,x) be con-
tinuous with respect to v
∈ Λ for any x ∈ Ω.LetΔ be a set of all v ∈ Λ such that the function
x
→ φ(v,x) is a Schur-convex (Schur-concave) function. Then Δ is a closed s et of Λ.
Proof. Let l
≥ 1, l ∈ N, v
l
∈ Δ, v
0
∈ Λ, v
l
→v
0
if l→ + ∞.AccordingtoDefinition 2.2,
φ(v
l
,y) ≥ (≤ )φ(v
l
,z)holdsforanyy,z ∈ Ω and y z.Letl→ + ∞,thenwehaveφ(v
0
,y) ≥
(≤ )φ(v
0
,z). Hence v
0
∈ Δ,soΔ is a closed set of Λ.
3. Main results
Theorem 3.1. Given r
∈ R, L
r
(x) is Schur-convex if r ≥ 1 and Schur-concave if r ≤ 1.
Proof. Denote
u = (u
2
,u
1
,u
3
, ,u
n
).
If r
=0, owing to the symmetry of L
r
(x) with respect to x
1
,x
2
, ,x
n
,wehave
g
r
(x)
E
n−1
A(x,u)
r
dμ =
E
n−1
A
x, u
r
dμ. (3.1)
Therefore,
∂g
r
∂x
1
= r
E
n−1
u
1
A(x,u)
r−1
dμ = r
E
n−1
u
2
A
x, u
r−1
dμ,
∂g
r
∂x
2
= r
E
n−1
u
1
A
x, u
r−1
dμ = r
E
n−1
u
2
A(x,u)
r−1
dμ.
(3.2)
It follows that
∂g
r
∂x
1
−
∂g
r
∂x
2
= r
E
n−1
u
1
A(x,u)
r−1
−
A
x, u
r−1
dμ,
∂g
r
∂x
1
−
∂g
r
∂x
2
= r
E
n−1
u
2
A
x, u
r−1
−
A(x,u)
r−1
dμ.
(3.3)
By combining (3.3) with (3.2), we have
∂g
r
∂x
1
−
∂g
r
∂x
2
=
r
2
E
n−1
u
1
− u
2
A(x,u)
r−1
−
A
x, u
r−1
dμ. (3.4)
Ning-Guo Zheng et al. 5
By Lagrange’s mean value theorem, we find that
A(x,u)
r−1
−
A
x, u
r−1
= (r − 1)
x
1
u
1
+ x
2
u
2
− x
2
u
1
− x
1
u
2
ξ +
n
i=3
u
i
x
i
r−2
= (r − 1)
u
1
− u
2
x
1
− x
2
ξ +
n
i=3
u
i
x
i
r−2
,
(3.5)
where ξ is between x
1
u
1
+ x
2
u
2
and x
2
u
1
+ x
1
u
2
.
From (3.4)and(3.5), we have
x
1
− x
2
∂g
r
∂x
1
−
∂g
r
∂x
2
=
r(r − 1)
2
x
1
− x
2
2
S
r
(x), (3.6)
where
S
r
(x) =
E
n−1
u
1
− u
2
2
ξ +
n
i=3
u
i
x
i
r−2
dμ ≥ 0. (3.7)
Hence, for r
=0, we get
x
1
− x
2
∂L
r
∂x
1
−
∂L
r
∂x
2
=
(n − 1)!·
1
r
·
L
r
1−r
·
x
1
− x
2
∂g
r
∂x
1
−
∂g
r
∂x
2
=
(n − 1)!·
r − 1
2
·
L
r
1−r
·
x
1
− x
2
2
S
r
(x).
(3.8)
From Lemma 2.3, it is clear that L
r
is Schur-convex for r>1 and Schur-concave for r<1
and r
=0.
According to Lemma 2.4 and the continuity of r
→ L
r
(x), let r→0,1−,or1+inL
r
(x),
we know that L
0
(x) is a Schur-concave function, and L
1
(x) is both a Schur-concave func-
tion and a Schur-convex function.
Theorem 3.2. Given r ∈ R, F
r
(x) is Schur-convex if r ≥ 1 and Schur-concave if r ≤ 1.
Proof. Denote
u = (u
2
,u
1
,u
3
, ,u
n
). For r=0,
F
r
(x) = (n − 1)!
E
n−1
M
r
(x,u)dμ = (n − 1)!
E
n−1
M
r
x, u
dμ,
(3.9)
∂F
r
∂x
1
= (n − 1)!
E
n−1
x
r−1
1
u
1
M
r
(x,
u
)
1−r
dμ = (n − 1)!
E
n−1
u
1
M
r
(x,
u
)
x
1
1−r
dμ,
(3.10)
∂F
r
∂x
2
= (n − 1)!
E
n−1
x
r−1
2
u
1
M
r
x, u
1−r
dμ = (n − 1)!
E
n−1
u
1
M
r
x, u
x
2
1−r
dμ.
(3.11)
6 Journal of Inequalities and Applications
Combination of (3.10)with(3.11)yields
∂F
r
∂x
1
−
∂F
r
∂x
2
= (n − 1)!
E
n−1
u
1
M
r
(x,u)
x
1
1−r
−
M
r
x, u
x
2
1−r
dμ. (3.12)
By using the mean value theorem, we find
M
r
(x,u)
x
1
1−r
−
M
r
x, u
x
2
1−r
=
u
1
+
u
2
x
r
2
+
n
i
=3
u
i
x
r
i
x
r
1
(1−r)/r
−
u
1
+
u
2
x
r
1
+
n
i
=3
u
i
x
r
i
x
r
2
(1−r)/r
=
1 − r
r
u
2
x
r
2
+
n
i
=3
u
i
x
r
i
x
r
1
−
u
2
x
r
1
+
n
i
=3
u
i
x
r
i
x
r
2
u
1
+ θ
1
(1−2r)/r
=
1 − r
r
·
u
2
x
2r
2
+ x
r
2
n
i
=3
u
i
x
r
i
− u
2
x
2r
1
− x
r
1
n
i
=3
u
i
x
r
i
x
r
1
x
r
2
·
u
1
+ θ
1
(1−2r)/r
= (1 − r)
x
2
− x
1
u
1
+ θ
1
(1−2r)/r
T
x,u;θ
2
,
(3.13)
where θ
1
is between (u
2
x
r
2
+
n
i
=3
u
i
x
r
i
)/x
r
1
and (u
2
x
r
1
+
n
i
=3
u
i
x
r
i
)/x
r
2
, θ
2
is between x
1
and
x
2
,andT(x,u;θ
2
) = (2u
2
θ
2r−1
2
+ θ
r−1
2
n
i
=3
u
i
x
r
i
)/x
r
1
x
r
2
≥ 0.
From (3.12)and(3.13), we have
x
1
− x
2
∂F
r
∂x
1
−
∂F
r
∂x
2
=
(r − 1)
x
1
− x
2
2
(n − 1)!
E
n−1
u
1
u
1
+ θ
1
(1−2r)/r
T
x,u;θ
2
dμ.
(3.14)
It follows that F
r
is Schur-convex for r>1 and Schur-concave for r<1andr=0by
Lemma 2.3.
According to Lemma 2.4 and the continuity of r
→ F
r
(x), let r→0,1−,or1+inF
r
(x).
We know that F
0
(x) is a Schur-concave function, and F
1
(x) is both a Schur-concave func-
tion and a Schur-convex function.
Theorem 3.3. L
r
(x
1/r
) and F
r
(x
1/r
) are Schur-concave functions if r ≥ 1,andSchur-convex
functions if r
≤ 1 and r=0.
Ning-Guo Zheng et al. 7
Proof. We can easily obtain that
L
r
x
1/r
=
(n − 1)!
E
n−1
M
1/r
(x,u)dμ
1/r
= F
1/r
1/r
(x),
F
r
x
1/r
=
(n − 1)!
E
n−1
A(x,u)
1/r
dμ = L
r
1/r
(x),
(3.15)
x
1
− x
2
∂L
r
x
1/r
∂x
1
−
∂L
r
x
1/r
∂x
2
=
1
r
x
1
− x
2
∂F
1/r
(x)
∂x
1
−
∂F
1/r
(x)
∂x
2
·
F
(1−r)/r
1/r
(x),
x
1
− x
2
∂F
r
x
1/r
∂x
1
−
∂F
r
x
1/r
∂x
2
=
r
x
1
− x
2
∂L
1/r
(x)
∂x
1
−
∂L
1/r
(x)
∂x
2
·
L
r−1
1/r
(x).
(3.16)
From Theorems 3.1 and 3.2, we know that both L
1/r
(x)andF
1/r
(x)areSchur-concave
functions if r
≥1 and Schur-convex functions if 0 <r≤1orr<0. According to Lemma 2.3
and (3.16), the required result of Theorem 3.3 is proved.
4. Applications
As applications of the theorems above, we have the following corollaries.
Corollary 4.1 (See [19, Theorem 3.1] and [12, page 82]). For r
≥ 1, r ∈ N,thecomplete
elementary symmetric function
C
r
(x) =
i
1
+i
2
+···+i
n
=r,
i
1
, ,i
n
≥0 are integers
x
i
1
1
x
i
2
2
, ,x
i
n
n
(4.1)
is Schur-convex.
Proof. If r
≥ 1, r ∈ N, then (see [20, page 164])
C
r
(x) =
n − 1+r
r
L
r
r
(x). (4.2)
By Theorem 3.1 and Lemma 2.3, it is easy to see that L
r
r
(x) is a Schur-convex function.
Therefore, C
r
(x) is a Schur-convex function.
Corollary 4.2. The complete symmetric function of the first degree:
D
r
(x) =
i
1
+i
2
+···+i
n
=r,
i
1
, ,i
n
≥0 are integers
x
i
1
1
x
i
2
2
, ,x
i
n
n
1/r
(4.3)
(see [6,Theorem5]and[9]), is Schur-concave for r
≥ 1, r ∈ N.
8 Journal of Inequalities and Applications
Proof. If r
≥ 1, r ∈ N, then we have (see [6,Theorem5])
D
r
(x) =
n − 1+r
r
F
1/r
(x). (4.4)
By considering Theorem 3.2, we prove the required result.
Corollary 4.3. Let r=0, x,y ∈ R
n
+
, x
r
y
r
. Then L
r
(x) ≤ L
r
(y) and F
r
(x) ≤ F
r
(y) if
r
≥ 1. They are reversed if r ≤ 1 and r=0.
Proof. Suppose r
≥ 1(r ≤ 1,r=0). L
r
(x
1/r
) is a Schur-concave (Schur-convex) function by
Theorem 3.3.Then
L
r
x
r
1/r
≤
(≥ )L
r
y
r
1/r
, L
r
(x) ≤ (≥ )L
r
(y). (4.5)
For F
r
(x
1/r
), the proof is similar; we omit the details.
Corollary 4.4. If r ≥ 1, then
A(x)
≤ L
r
(x) ≤ M
r
(x),
A(x)
≤ F
r
(x) ≤ M
r
(x).
(4.6)
Inequalities (4.6) are reversed if r
≤ 1.
Proof. If r
≥ 1, owing to Theorem 3.1 and
x
1
,x
2
, ,x
n
A(x),A(x), ,A(x)
A(x), (4.7)
we have
L
r
(x) ≥ L
r
A(x)
=
(n − 1)!
E
n−1
n
i=1
A(x)u
i
r
dμ
1/r
= A(x)
(n − 1)!
E
n−1
n
i=1
u
i
r
dμ
1/r
= A(x).
(4.8)
Obviously, if r
≤ 1, r=0, inequality (4.8)isreversedbyTheorem 3.1.Forr = 0, because
of the continuity of r
→ L
r
(x), we have L
0
(x) ≤ A(x).
By the same way, we find that F
r
(x) ≥ A(x)ifr ≥ 1, and F
r
(x) ≤ A(x)ifr ≤ 1. In addi-
tion,
x
r
=
x
r
1
,x
r
2
, ,x
r
n
M
r
r
(x),M
r
r
(x), ,M
r
r
(x)
M
r
(x),M
r
(x), ,M
r
(x)
r
M
r
(x)
r
.
(4.9)
Ning-Guo Zheng et al. 9
If r
≥ 1, according to Corollary 4.3,weget
L
r
(x) ≤ L
r
M
r
(x)
=
(n − 1)!
E
n−1
n
i=1
M
r
(x)u
i
r
dμ
1/r
= M
r
(x). (4.10)
If r
≤ 1, inequality (4.10)isobviouslyreversedbyCorollary 4.3 again.
Similarly, we have F
r
(x) ≤ M
r
(x)ifr ≥ 1, and F
r
(x) ≥ M
r
(x)ifr ≤ 1.
Acknowledgments
This work was supported by the NSF of Zhejiang Broadcast and TV University under
Grant no. XKT-07G19. The authors are grateful to the referees for their valuable sugges-
tions.
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Ning-Guo Zheng: Huzhou Broadcast and TV University, Huzhou, Zhejiang 313000, China
Email address:
Zhi-Hua Zhang: Zixing Educational Research Section, Chenzhou, Hunan 423400, China
Email address:
Xiao-Ming Zhang: Haining College, Zhejiang Broadcast and TV University, Haining,
Zhejiang 314400, China
Email address:
