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 = (x1 ,x2 ,...,xn ) ∈ Rn+ , and x1/r = (x11/r ,x21/r ,...,
xn1/r ), where r ∈ R, r =0; let En−1 ⊂ Rn−1 be the simplex
En−1 =
u1 , ...,un−1 : ui > 0(1 ≤ i ≤ n − 1),
n
−1
ui ≤ 1 ,
(1.1)
i =1
and let dμ = du1 ,...,dun−1 be the differential of the volume in En−1 .
r with
The weighted arithmetic mean A(x,u) and the power mean Mr (x,u) of order
respect to the numbers x1 ,x2 ,...,xn and the positive weights u1 ,u2 ,...,un with ni=1 ui = 1
1/r
are defined, respectively, as A(x,u) = ni=1 ui xi , Mr (x,u) = ( ni=1 ui xir ) for r =0, and
n ui
Δ
Δ
M0 (x,u) = i=1 xi . For u = (1/n,1/n,...,1/n), we denote A(x,u) = A(x), Mr (x,u) = Mr (x).
The well-known logarithmic mean L(x1 ,x2 ) of two positive numbers x1 and x2 is
L x1 ,x2
⎧
⎪
⎨
x1 − x2
,
ln
x1 − ln x2
=
⎪
⎩x1 ,
x1 =x2 ,
x1 = x2 .
(1.2)
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Journal of Inequalities and Applications
As further generalization of L(x1 ,x2 ), Stolarsky [1] studied the one-parameter mean, that
is,
⎧
1/r
⎪
⎪
x1r+1 − x2r+1
⎪
⎪
,
⎪
⎪
⎪
⎪ (r + 1) x1 − x2
⎪
⎪
⎪
⎪
⎪ x1 − x2
⎪
⎪
⎨ ln x − ln x ,
1
2
Lr x1 ,x2 = ⎪
⎪
⎪ ⎪
1/(x
1 −x 2 )
⎪
x1
⎪
⎪
⎪ 1 x1
⎪
,
⎪
⎪
⎪
e x2x2
⎪
⎪
⎪
⎩
x1 ,
r = − 1,0, x1 =x2 ,
r = −1, x1 =x2 ,
(1.3)
r = 0, x1 =x2 ,
x1 = x 2 .
Alzer [2, 3] obtained another form of one-parameter mean, that is,
⎧
r x1r+1 − x2r+1
⎪
⎪
⎪
·
,
⎪
⎪
r + 1 x1r − x2r
⎪
⎪
⎪
⎪
⎪
⎪
⎪
ln x1 − ln x2
⎪
⎨x1 x2 ·
,
x1 − x2
Fr x1 ,x2 = ⎪
⎪
⎪
⎪
⎪
x1 − x2
⎪
⎪
⎪
,
⎪
⎪
ln x1 − ln x2
⎪
⎪
⎪
⎩
x1 ,
r = − 1,0, x1 =x2 ,
r = −1, x1 =x2 ,
(1.4)
r = 0, x1 =x2 ,
x1 = x 2 .
These two means can be written also as
⎧
1/r
1
⎪
r
⎪
⎪
⎪
,
⎪
⎨ 0 x1 u + x2 (1 − u) du
r =0,
Lr x1 ,x2 = ⎪
1
⎪
⎪
⎪
⎪
⎩exp
ln x1 u + x2 (1 − u) du ,
r = 0,
0
⎧ 1
r
1/r
⎪
⎪
⎪
x u + x2r (1 − u) du,
⎪
⎨ 0 1
Fr x1 ,x2 = ⎪
⎪ 1
⎪
⎪
⎩
0
x1u x21−u du,
(1.5)
r =0,
r = 0.
Correspondingly, Pittenger [4] and Pearce et al. [5] investigated the means above in n
variables, respectively,
⎧
r 1/r
⎪
⎪
⎪
(n
−
1)!
A(x,u) dμ ,
⎪
⎨
En−1
Lr (x) = ⎪ ⎪
⎪
⎪
ln A(x,u)dμ ,
⎩exp (n − 1)!
Fr (x) = (n − 1)!
where un = 1 −
n −1
i =1
ui .
En−1
En−1
Mr (x,u)dμ,
r =0,
r = 0,
(1.6)
Ning-Guo Zheng et al. 3
Expressions (1.3) and (1.4) can be also written by using 2-order determinants, that is,
⎧⎛
⎞1/r
⎪
r+1 ⎪
1
x
1
x
1
⎪
2
2
⎪⎝
⎠ ,
⎪
·
r = − 1,0, x1 =x2 ,
⎪
r+1 ⎪
⎪
r
+
1
1
x
1
x
⎪
1
1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
1
x
1
ln
x
⎪
2
2
,
⎨
r = −1, x1 =x2 ,
Lr x1 ,x2 = ⎪
1 x1 1 ln x1 ⎪
⎪
⎞
⎧⎛
⎫
⎪
⎪
⎪
⎨ ⎬
⎪
1 x2 ln x2 1 x2 ⎪
⎪
⎠ − 1 , r = 0, x1 =x2 ,
⎪
⎪exp ⎩⎝
⎪
⎭
⎪
1 x1 ln x1 1 x1 ⎪
⎪
⎪
⎪
⎩x ,
x1 = x2 ,
1
⎞ ⎛
⎞
⎧⎛ r
⎪
1 x2r+1 ⎪
⎪
⎠ ⎝(r + 1) 1 x2 ⎠ , r = − 1,0, x1 =x2 ,
⎪
⎝r ⎪
⎪
r+1
r
⎪
1 x1 ⎪ 1 x 1 ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
1 ln x2 1 x2 ⎪
,
⎨x1 x2 r = −1, x1 =x2 ,
1 ln x1 1 x1 Fr x1 ,x2 = ⎪
⎪
⎪
⎪
⎪
⎪
⎪
1 x2 1 ln x2 ⎪
⎪
,
⎪
r = 0, x1 =x2 ,
⎪
⎪
⎪
1 x1 1 ln x1 ⎪
⎪
⎪
⎪
⎩x1 ,
x1 = x 2 .
(1.7)
Utilizing higher-order generalized Vandermonde determinants, Xiao et al. [8, 7, 6, 9] gave
the analogous definitions of Lr (x) and Fr (x).
Obviously, Lr (x) and Fr (x) are symmetric with respect to x1 ,x2 ,...,xn , r → Lr (x) and
r → Fr (x) are continuous for any x ∈ Rn+ .
In [4, 5, 10, 11], the authors studied the Schur-convexities of Lr (x1 ,x2 ) and Fr (x1 ,x2 ).
In this paper, we establish the Schur-convexities of two types of one-parameter mean
values Lr (x) and Fr (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 important applications in analytic inequalities. The following definitions can be found in many
references such as [12–17].
Definition 2.1. For u = (u1 ,u2 ,...,un ), v = (v1 ,v2 ,...,vn ) ∈ Rn , without loss of generality, it is assumed that u1 ≥ u2 ≥ · · · ≥ un and v1 ≥ v2 ≥ · · · ≥ vn . Then u is said to be
majorized
by v (in symbols u ≺ v) if ki=1 ui ≤ ki=1 vi for k = 1,2,...,n − 1 and ni=1 ui =
n
i=1 vi .
Definition 2.2. Let Ω ⊂ Rn . A function ϕ : Ω → R is said to be a Schur-convex (Schurconcave) 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 following so-called Schur condition.
Lemma 2.3 [12, page 57]. Suppose that Ω ⊂ Rn is symmetric with respect to permutations
and convexset, and has a nonempty interior set Ω0 . Let ϕ : Ω → R be continuous on Ω and
continuously differentiable in Ω0 . Then, ϕ is a Schur-convex (Schur-concave) function if and
only if it is symmetric and if
u1 − u2
∂ϕ
−
∂u1
∂ϕ
∂u2
≥ (≤ )0
(2.1)
holds for any u = (u1 ,u2 ,...,un ) ∈ Ω0 .
Lemma 2.4. Let m ≥ 1,n ≥ 2, m, n ∈ N, Λ ⊂ Rm , Ω ⊂ Rn , φ : Λ × Ω → R, φ(v,x) be continuous with respect to v ∈ Λ for any x ∈ Ω. Let Δ 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 set of Λ.
Proof. Let l ≥ 1, l ∈ N, vl ∈ Δ, v0 ∈ Λ, vl →v0 if l→ + ∞. According to Definition 2.2,
φ(vl ,y) ≥ (≤ )φ(vl ,z) holds for any y,z ∈ Ω and y z. Let l→ + ∞, then we have φ(v0 ,y) ≥
(≤ )φ(v0 ,z). Hence v0 ∈ Δ, so Δ is a closed set of Λ.
3. Main results
Theorem 3.1. Given r ∈ R, Lr (x) is Schur-convex if r ≥ 1 and Schur-concave if r ≤ 1.
= (u2 ,u1 ,u3 ,... ,un ).
Proof. Denote u
If r =0, owing to the symmetry of Lr (x) with respect to x1 ,x2 ,...,xn , we have
gr (x) r
A(x,u) dμ =
En−1
En−1
A x, u
r
dμ.
(3.1)
Therefore,
∂gr
=r
∂x1
∂gr
=r
∂x2
En−1
u1 A(x,u)
En−1
u1 A x, u
It follows that
∂gr ∂gr
−
=r
∂x1 ∂x2
∂gr ∂gr
−
=r
∂x1 ∂x2
r −1
dμ = r
r −1
dμ = r
u1 A(x,u)
En−1
u2 A x, u
En−1
En−1
u2 A x, u
En−1
r −1
r −1
r −1
u2 A(x,u)
r −1
dμ,
(3.2)
dμ.
r −1 − A x, u
dμ,
− A(x,u)
r −1 (3.3)
dμ.
By combining (3.3) with (3.2), we have
∂gr ∂gr
r
−
=
∂x1 ∂x2 2
En−1
u1 − u2
A(x,u)
r −1
r −1 − A x, u
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) x1 u1 + x2 u2 − x2 u1 − x1 u2
ξ+
n
r −2
ui x i
i =3
= (r − 1) u1 − u2
x1 − x2
ξ+
n
(3.5)
r −2
ui x i
,
i =3
where ξ is between x1 u1 + x2 u2 and x2 u1 + x1 u2 .
From (3.4) and (3.5), we have
x1 − x2
∂gr
∂gr
−
∂x1 ∂x2
2
r(r − 1) x1 − x2 Sr (x),
2
=
(3.6)
where
Sr (x) =
En−1
u1 − u2
2
ξ+
n
r −2
dμ ≥ 0.
ui x i
(3.7)
i=3
Hence, for r =0, we get
x1 − x2
∂Lr
∂Lr
−
∂x1 ∂x2
∂gr
∂gr
1 1−r · x1 − x2
−
= (n − 1)!· · Lr
r
∂x1 ∂x2
2
r − 1 1−r · Lr
· x1 − x2 Sr (x).
= (n − 1)!·
2
(3.8)
From Lemma 2.3, it is clear that Lr 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 → Lr (x), let r →0,1−, or 1+ in Lr (x),
we know that L0 (x) is a Schur-concave function, and L1 (x) is both a Schur-concave func
tion and a Schur-convex function.
Theorem 3.2. Given r ∈ R, Fr (x) is Schur-convex if r ≥ 1 and Schur-concave if r ≤ 1.
= (u2 ,u1 ,u3 ,... ,un ). For r =0,
Proof. Denote u
Fr (x) = (n − 1)!
∂Fr
= (n − 1)!
∂x1
En−1
En−1
Mr (x,u)dμ = (n − 1)!
x1r −1 u1 Mr (x, u)
1−r
En−1
Mr x, u
dμ,
(3.9)
dμ = (n − 1)!
En−1
u1
Mr (x, u)
x1
!1−r
dμ,
(3.10)
∂Fr
= (n − 1)!
∂x2
En−1
x2r −1 u1 Mr x, u
1−r
dμ = (n − 1)!
En−1
u1
Mr x, u
x2
!1−r
dμ.
(3.11)
6
Journal of Inequalities and Applications
Combination of (3.10) with (3.11) yields
∂Fr ∂Fr
−
= (n − 1)!
∂x1 ∂x2
En−1
u1
Mr (x,u)
x1
!1−r
−
Mr x, u
x2
!1−r dμ.
(3.12)
By using the mean value theorem, we find
Mr (x,u)
x1
!1−r
−
= u1 +
u2 x2r +
n
x1r
=
Mr x, u
x2
i=3
!1−r
ui xir
(1−r)/r
− u1 +
u2 x1r +
n
x2r
i =3
ui xir
(1−r)/r
(1−2r)/r
1 − r u2 x2r + ni=3 ui xir u2 x1r + ni=3 ui xir −
u1 + θ 1
r
x1r
x2r
1 − r u2 x22r + x2r
=
·
r
n
i =3
ui xir − u2 x12r − x1r
x1r x2r
n
i =3
(3.13)
(1−2r)/r
ui xir · u1 + θ 1
(1−2r)/r = (1 − r) x2 − x1 u1 + θ 1
T x,u;θ 2 ,
where θ 1 is between (u2 x2r + ni=3 ui xir )/x1r and (u2 x1r + ni=3 ui xir )/x2r , θ 2 is between x1 and
x2 , and T(x,u;θ 2 ) = (2u2 θ 22r −1 + θ 2r −1 ni=3 ui xir )/x1r x2r ≥ 0.
From (3.12) and (3.13), we have
x1 − x2
∂Fr
∂Fr
−
∂x1 ∂x2
2
= (r − 1) x1 − x2 (n − 1)!
En−1
u1 u1 + θ 1
(1−2r)/r (3.14)
T x,u;θ 2 dμ.
It follows that Fr is Schur-convex for r > 1 and Schur-concave for r < 1 and r =0 by
Lemma 2.3.
According to Lemma 2.4 and the continuity of r → Fr (x), let r →0,1−, or 1+ in Fr (x).
We know that F0 (x) is a Schur-concave function, and F1 (x) is both a Schur-concave func
tion and a Schur-convex function.
Theorem 3.3. Lr (x1/r ) and Fr (x1/r ) are Schur-concave functions if r ≥ 1, and Schur-convex
functions if r ≤ 1 and r =0.
Ning-Guo Zheng et al. 7
Proof. We can easily obtain that
Lr x
Fr x
∂Lr x1/r
∂Fr x1/r
x1 − x2
x1 − x2
∂x1
∂x1
1/r
1/r
= (n − 1)!
En−1
= (n − 1)!
−
!1/r
∂Lr x1/r
∂x2
∂Fr x1/r
−
∂x2
M1/r (x,u)dμ
"
En−1
A(x,u)
#1/r
1/r
= F1/r
(x),
(3.15)
dμ = Lr1/r (x),
=
∂F1/r (x) ∂F1/r (x)
1
(1−r)/r
x1 − x2
−
(x),
·F1/r
r
∂x1
∂x2
= r x1 − x2
∂L1/r (x)
∂x1
∂L1/r (x)
r −1
−
(x).
·L1/r
∂x2
(3.16)
From Theorems 3.1 and 3.2, we know that both L1/r (x) and F1/r (x) are Schur-concave
functions if r ≥ 1 and Schur-convex functions if 0<r ≤ 1 or r < 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, the complete
elementary symmetric function
Cr (x) =
x1i1 x2i2 ,...,xnin
(4.1)
i1 +i2 +···+in =r,
i1 ,...,in ≥0 are integers
is Schur-convex.
Proof. If r ≥ 1, r ∈ N, then (see [20, page 164])
Cr (x) =
n−1+r r
Lr (x).
r
(4.2)
By Theorem 3.1 and Lemma 2.3, it is easy to see that Lrr (x) is a Schur-convex function.
Therefore, Cr (x) is a Schur-convex function.
Corollary 4.2. The complete symmetric function of the first degree:
Dr (x) =
x1i1 x2i2 ,...,xnin
1/r
i1 +i2 +···+in =r,
i1 ,...,in ≥0 are integers
(see [6, Theorem 5] and [9]), is Schur-concave for r ≥ 1, r ∈ N.
(4.3)
8
Journal of Inequalities and Applications
Proof. If r ≥ 1, r ∈ N, then we have (see [6, Theorem 5])
n−1+r
F1/r (x).
Dr (x) =
r
(4.4)
By considering Theorem 3.2, we prove the required result.
Corollary 4.3. Let r =0, x,y ∈ Rn+ , xr yr . Then Lr (x) ≤ Lr (y) and Fr (x) ≤ Fr (y) if
r ≥ 1. They are reversed if r ≤ 1 and r =0.
Proof. Suppose r ≥ 1(r ≤ 1,r =0). Lr (x1/r ) is a Schur-concave (Schur-convex) function by
Theorem 3.3. Then
Lr
1/r r
x
≤ (≥ )Lr
1/r r
y
Lr (x) ≤ (≥ )Lr (y).
,
(4.5)
For Fr (x1/r ), the proof is similar; we omit the details.
Corollary 4.4. If r ≥ 1, then
A(x) ≤ Lr (x) ≤ Mr (x),
(4.6)
A(x) ≤ Fr (x) ≤ Mr (x).
Inequalities (4.6) are reversed if r ≤ 1.
Proof. If r ≥ 1, owing to Theorem 3.1 and
x1 ,x2 ,...,xn A(x),A(x),... ,A(x) A(x),
(4.7)
we have
Lr (x) ≥ Lr A(x) = (n − 1)!
En−1
n
r
A(x)ui
1/r
dμ
i=1
= A(x) (n − 1)!
En−1
n
r
ui
(4.8)
1/r
dμ
= A(x).
i =1
Obviously, if r ≤ 1, r =0, inequality (4.8) is reversed by Theorem 3.1. For r = 0, because
of the continuity of r → Lr (x), we have L0 (x) ≤ A(x).
By the same way, we find that Fr (x) ≥ A(x) if r ≥ 1, and Fr (x) ≤ A(x) if r ≤ 1. In addition,
xr = x1r ,x2r ,...,xnr Mrr (x),Mrr (x),...,Mrr (x)
Mr (x),Mr (x),...,Mr (x)
r
r
M r (x) .
(4.9)
Ning-Guo Zheng et al. 9
If r ≥ 1, according to Corollary 4.3, we get
Lr (x) ≤ Lr M r (x) = (n − 1)!
En−1
n
r
Mr (x)ui
1/r
dμ
= Mr (x).
(4.10)
i=1
If r ≤ 1, inequality (4.10) is obviously reversed by Corollary 4.3 again.
Similarly, we have Fr (x) ≤ Mr (x) if r ≥ 1, and Fr (x) ≥ Mr (x) if r ≤ 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 suggestions.
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Ning-Guo Zheng: Huzhou Broadcast and TV University, Huzhou, Zhejiang 313000, China
Email address: yczng@hzvtc.net
Zhi-Hua Zhang: Zixing Educational Research Section, Chenzhou, Hunan 423400, China
Email address: zxzh1234@163.com
Xiao-Ming Zhang: Haining College, Zhejiang Broadcast and TV University, Haining,
Zhejiang 314400, China
Email address: zjzxm79@tom.com