Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 3 Issue 3(2011), Pages 84-96.
SCHUR CONVEXITY WITH RESPECT TO A CLASS OF
SYMMETRIC FUNCTIONS AND THEIR APPLICATIONS
(COMMUNICATED BY MUHAMMAD ASLAM NOOR)
WEIFENG XIA, YUMING CHU
Abstract. For x = (x1 , x2 , · · · , xn ) ∈ Rn
+ , the symmetric function φn (x, r)
is defined by


r
∏
∑
1 + xij

,
φn (x, r) = φn (x1 , x2 , · · · , xn ; r) =
xij
1≤i <i ···<i ≤n j=1
1
2
r
where r = 1, 2, · · · , n, and i1 , i2 , · · · , in are positive integers.
In this article, the Schur convexity, Schur multiplicative convexity and
Schur harmonic convexity of φn (x, r) are discussed. As applications, some
inequalities, including Weierstrass inequality, are established by use of the
theory of majorization.
1. Introduction
In this paper, we shall adopt the notation and terminology as follows: Rn denotes
the n−dimensional Euclidean space (n ≥ 2), Rn+ = {(x1 , x2 , · · · , xn ) : xi > 0, i =
1, 2, · · · , n}, R = (−∞, +∞), R+ = (0, +∞) and N = {1, 2, · · · , n, · · · }. For x =
(x1 , x2 , · · · , xn ), y = (y1 , y2 , · · · , yn ) ∈ Rn and α ∈ R, we denote by
x + y = (x1 + y1 , x2 + y2 , · · · , xn + yn ),
xy = (x1 y1 , x2 y2 , · · · , xn yn ),
αx = (αx1 , αx2 , · · · , αxn ),
ex = (ex1 , ex2 , · · · , exn ),
α + x = (α + x1 , α + x2 , · · · , α + xn )
and
α − x = (α − x1 , α − x2 , · · · , α − xn ).
2000 Mathematics Subject Classification. 05E05, 26B25.
Key words and phrases. symmetric function; Schur convex; Schur multiplicatively convex;
Schur harmonic convex.
c
⃝2011
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted April 7, 2011. Published June 20, 2011.
This work was supported by the Natural Science Foundation of China (11071069), the Natural
Science Foundation of Zhejiang Province (Y7080185) and the Innovation Team Foundation of the
Department of Education of Zhejiang Province (T200924).
84
A CLASS OF SYMMETRIC FUNCTIONS AND THEIR APPLICATIONS
85
Moreover, we denote by
α
α
xα = (xα
1 , x2 , · · · , xn ),
log x = (log x1 , log x2 , · · · , log xn )
and
1
=
x
(
1 1
1
, ,··· ,
x1 x2
xn
)
for x ∈ Rn+ .
For x = (x1 , x2 , · · · , xn ) ∈ Rn+ , r ∈ N and r ≤ n, the Hamy symmetric function
Hn (x, r) is defined by T. Hara, M. Uchiyama and S. Takahasi [1] as follows:

 r1
r
∑
∏

Hn (x, r) = Hn (x1 , x2 , · · · , xn ; r) =
xij  ,
1≤i1 <i2 <···<ir ≤n
j=1
where i1 , i2 , · · · , in ∈ N.
Corresponding to this is the r−th order Hamy mean
σn (x, r) = σn (x1 , x2 , · · · , xn ; r) =
1
Hn (x, r),
Cnr
n!
where Cnr = (n−r)!r!
. T. Hara, M. Uchiyama and S. Takahasi [1] established the
following refinement of the classical arithmetic and geometric means inequalities:
Gn (x) = σn (x, n) ≤ σn (x, n − 1) ≤ · · · ≤ σn (x, 2) ≤ σn (x, 1) = An (x).
∏n
∑n
1
Here, An (x) = n1 i=1 xi and Gn (x) = ( i=1 xi ) n denote the classical arithmetic
and geometric means of x, respectively. We also denote the harmonic mean of x by
Hn (x) = ∑nn 1 .
i=1 xi
The paper [2] by H. T. Ku, M. C. Ku and X. M. Zhang contains some interesting
1
inequalities including the fact that (σn (x, r)) r is log-concave. More results can be
found in the book [3] by P. S. Bullen.
Recently, the Schur convexity of the Hamy symmetric function Hn (x, r) was
discussed and some analytic inequalities were established by K. Z. Guan [4].
In this article, we define the following new symmetric function:


r
∏
∑
1 + xi j

,
φn (x, r) = φn (x1 , x2 , · · · , xn ; r) =
x
i
j
j=1
(1.1)
1≤i1 <i2 ···<ir ≤n
for x = (x1 , x2 , · · · , xn ) ∈ Rn+ , r ∈ N and r ≤ n. Here, i1 , i2 , · · · , in ∈ N.
The main purpose of this paper is to discuss the Schur convexity, Schur multiplicative convexity and Schur harmonic convexity for the symmetric function
φn (x, r). As applications, some inequalities are established by use of the theory
of majorization.
Schur convex, Schur multiplicatively convex and Schur harmonic convex functions are defined as follows.
86
W. XIA, Y. CHU
Definition 1.1. Let E ⊆ Rn be a set, a real-valued function F on E is called a
Schur convex function if
F (x1 , x2 , · · · , xn ) ≤ F (y1 , y2 , · · · , yn )
for each pair of n-tuples x = (x1 , · · · , xn ) and y = (y1 , · · · , yn ) in E, such that
x ≺ y, i.e.
k
k
∑
∑
x[i] ≤
y[i] , k = 1, 2, · · · , n − 1
i=1
and
i=1
n
∑
i=1
x[i] =
n
∑
y[i] ,
i=1
where x[i] denotes the ith largest component in x. F is called Schur concave if −F
is Schur convex .
Definition 1.2. Let E ⊆ Rn+ be a set, a real-valued function F : E → R+ is called a
Schur multiplicatively convex function on E if F (x1 , x2 , · · · , xn ) ≤ F (y1 , y2 , · · · , yn )
for each pair of n−tuples x = (x1 , x2 , · · · , xn ) and y = (y1 , y2 , · · · , yn ) in E, such
that log x ≺ log y. F is called Schur multiplicatively concave if F1 is Schur multiplicatively convex.
Definition 1.3. Let E ⊆ Rn+ be a set, a real-valued function F on E is called a
Schur harmonic convex function on E if
F (x1 , x2 , · · · , xn ) ≤ F (y1 , y2 , · · · , yn )
(1.2)
for each pair of n−tuples x = (x1 , x2 , · · · , xn ) and y = (y1 , y2 , · · · , yn ) in E, such
that x1 ≺ y1 . F is called a Schur harmonic concave function on E if inequality (1.2)
is reversed.
The Schur convexity was introduced by I. Schur [5] in 1923, G. H. Hardy, J. E.
Littlewood and G. Pólya were also interested in some inequalities that are related
to the Schur convexity [6]. It has many important applications in extended mean
values [7], theory of statistical experiments [8], graphs and matrices [9], combinatorial optimization [10], reliability [11], gamma functions [12], information-theoretic
topics [13], stochastic orderings [14] and other related fields. Recently, the Schur
multiplicative and harmonic convexities were introduced and investigated in paper
[15, 16, 22-24].
2. Lemmas
In this section, we introduce and establish some Lemmas, which are used in the
proof of our main results.
Lemma 2.1. [5] Let f : Rn+ → R+ be a continuous symmetric function. If f is
differentiable in Rn+ , then f is Schur convex in Rn+ if and only if
(
)
∂f
∂f
(xi − xj )
−
≥0
(2.1)
∂xi
∂xj
for all i, j = 1, 2, · · · , n and x = (x1 , · · · , xn ) ∈ Rn+ . And f is Schur concave
in Rn+ if and only if inequality (2.1) is reversed for all i, j = 1, 2, · · · , n and
A CLASS OF SYMMETRIC FUNCTIONS AND THEIR APPLICATIONS
87
x = (x1 , · · · , xn ) ∈ Rn+ . Here f is a symmetric function in Rn+ which means
that f (P x) = f (x) for any x ∈ Rn+ and any n × n permutation matrix P .
Remark 2.1. Since f is symmetric, the Schur’s condition in Lemma 2.1, i.e. (2.1)
can be reduced to
)
(
∂f
∂f
≥ 0.
(x1 − x2 )
−
∂x1
∂x2
Lemma 2.2. [15, 16] Let f : Rn+ → R+ be a continuous symmetric function. If f
is differentiable in Rn+ , then f is Schur multiplicatively convex in Rn+ if and only if
)
(
∂f
∂f
− x2
≥0
(2.2)
(log x1 − log x2 ) x1
∂x1
∂x2
for all x = (x1 , x2 , · · · , xn ) ∈ Rn+ . And f is Schur multiplicatively concave in Rn+
if and only if inequality (2.2) is reversed.
Lemma 2.3. [22] Let f : Rn+ → R+ be a continuous symmetric function. If f is
differentiable in Rn+ , then f is Schur harmonic convex in Rn+ if and only if
)
(
∂f
∂f
− x22
≥0
(2.3)
(x1 − x2 ) x21
∂x1
∂x2
for all x = (x1 , x2 , · · · , xn ) ∈ Rn+ . And f is Schur harmonic concave in Rn+ if and
only if inequality (2.3) is reversed.
∑n
Lemma 2.4. [4, 16, 17] Let x = (x1 , x2 , · · · , xn ) ∈ Rn+ and i=1 xi = s. If c ≥ s,
then
(
)
c−x
c − x1 c − x2
c − xn
=
,
,
·
·
·
,
≺ (x1 , x2 , · · · , xn ) = x.
nc
nc
nc
nc
s −1
s −1 s −1
s −1
∑n
Lemma 2.5. [17] Let x = (x1 , x2 , · · · , xn ) ∈ Rn+ and i=1 xi = s. If c ≥ 0, then
(
)
c+x
c + x1 c + x2
c + xn
=
,
,
·
·
·
,
≺ (x1 , x2 , · · · , xn ) = x.
nc
nc
nc
nc
s +1
s +1 s +1
s +1
∑n
Lemma 2.6. [18] Suppose that x = (x1 , x2 , · · · , xn ) ∈ Rn+ and
i=1 xi = s. If
0 ≤ λ ≤ 1, then
(
)
s − λx
s − λx1 s − λx2
s − λxn
=
,
,··· ,
≺ (x1 , x2 , · · · , xn ) = x.
n−λ
n−λ
n−λ
n−λ
3. Main results
Theorem 3.1. For r ∈ {1, 2, · · · , n}, the symmetric function φn (x, r) is Schur
convex in Rn+ .
Proof. By Lemma 2.1 and Remark 2.1 we only need to prove that
(
)
∂φn (x, r) ∂φn (x, r)
(x1 − x2 )
−
≥ 0.
∂x1
∂x2
for all x = (x1 , x2 , · · · , xn ) ∈ Rn+ and r = 1, 2, · · · , n.
The proof is divided into four cases.
Case 1. If r = 1, then (1.1) leads to
(3.1)
88
W. XIA, Y. CHU
φn (x, 1) = φn (x1 , x2 , · · · , xn ; 1) =
n
∏
1 + xi
i=1
xi
.
(3.2)
From (3.2) and simple computation we get
(
(x1 − x2 )
∂φn (x, 1) ∂φn (x, 1)
−
∂x1
∂x2
)
=
(x1 − x2 )2 (1 + x1 + x2 )
φn (x, 1) ≥ 0.
x1 x2 (1 + x1 )(1 + x2 )
Case 2. If n ≥ 2 and r = n, then (1.1) yields that
φn (x, n) = φn (x1 , x2 , · · · , xn ; n) =
n
∑
1 + xi
i=1
xi
(3.3)
and
(
(x1 − x2 )
∂φn (x, n) ∂φn (x, n)
−
∂x1
∂x2
)
=
(x1 − x2 )2 (x1 + x2 )
≥ 0.
x21 x22
Case 3. If n ≥ 3 and r = 2, then by (1.1) we have
φn (x, 2) = φn (x1 , x2 , · · · , xn ; 2)


(
) n (
)
1 + x1
1 + x2  ∏ 1 + x1
1 + xj 
=
φn−1 (x2 , x3 , · · · , xn ; 2)
+
+
x1
x2
x1
xj
j=3


(
) n (
)
1 + x2
1 + x1  ∏ 1 + x2
1 + xj 
φn−1 (x1 , x3 , · · · , xn ; 2). (3.4)
=
+
+
x2
x1
x2
xj
j=3
Simple computation and (3.4) lead to
(
)
∂φn (x, 2) ∂φn (x, 2)
(x1 − x2 )
−
∂x1
∂x2
[
x1 + x2
= (x1 − x2 )2 φn (x, 2)
x1 x2 (x1 + 2x1 x2 + x2 )
+
n
∑
j=3
≥ 0.

x2j + (x1 + x2 )(1 + 2xj )xj

x1 x2 (x1 + 2x1 xj + xj )(x2 + 2x2 xj + xj )
A CLASS OF SYMMETRIC FUNCTIONS AND THEIR APPLICATIONS
89
Case 4. If n ≥ 4 and 3 ≤ r ≤ n − 1, then from (1.1) we have
φn (x, r) = φn (x1 , x2 , · · · , xn ; r)


r−1
∑
1 + xi j
1
+
x
1


= φn−1 (x2 , x3 , · · · , xn ; r)
+
x1
x
i
j
j=1
3≤i1 <i2 <···<ir−1 ≤n


r−2
∏
∑ 1 + xij
 1 + x1 + 1 + x2 +

×
x1
x2
x
i
j
j=1
3≤i1 <i2 <···<ir−2 ≤n


r−1
∏
∑
1 + xi j
1
+
x
2


+
= φn−1 (x1 , x3 , · · · , xn ; r)
x2
xij
j=1
3≤i1 <i2 <···<ir−1 ≤n


r−2
∏
∑
1 + xij
1
+
x
1
+
x
1
2


×
+
+
(3.5)
x1
x2
xij
j=1
∏
3≤i1 <i2 <···<ir−2 ≤n
and
(
)
∂φn (x, r) ∂φn (x, r)
−
∂x1
∂x2

∑
(x1 − x2 )2

φ
(x,
r)
=
n
x21 x22
(x1 − x2 )
1+x1
3≤i1 <i2 <···<ir−2 ≤n x1
+
x1 + x2
∑r−2
+ j=1
1+x2
x2
1+xij
x ij

∑r−1 1+xi
(1 + x1 + x2 ) + (x1 + x2 ) j=1 xi j
j
+
∑r−1 1+xij 1+x2 ∑r−1 1+xij 
1+x1
3≤i1 <i2 <···<ir−1 ≤n ( x1 +
j=1 xi )( x2 +
j=1 xi )
∑
j
j
≥ 0.
Therefore (3.1) follows from Cases 1-4, and the proof of Theorem 3.1 is completed.
Theorem 3.2. For r ∈ {1, 2, · · · , n}, the symmetric function φn (x, r) is Schur
multiplicatively convex in Rn+ .
Proof. According to Lemma 2.2 we only need to prove that
(
)
∂φn (x, r)
∂φn (x, r)
(log x1 − log x2 ) x1
− x2
≥0
∂x1
∂x2
for all x = (x1 , x2 , · · · , xn ) ∈ Rn+ and r = 1, 2, · · · , n.
The proof is divided into four cases.
Case I. If r = 1, then (3.2) leads to
(
)
∂φn (x, 1)
∂φn (x, 1)
(log x1 − log x2 ) x1
− x2
∂x1
∂x2
(log x1 − log x2 )(x1 − x2 )
φn (x, 1) ≥ 0.
=
(1 + x1 )(1 + x2 )
(3.6)
90
W. XIA, Y. CHU
Case II. If n ≥ 2 and r = n, then (3.3) implies that
(
)
∂φn (x, n)
∂φn (x, n)
− x2
(log x1 − log x2 ) x1
∂x1
∂x2
(x1 − x2 )(log x1 − log x2 )
=
≥ 0.
x1 x2
Case III. If n ≥ 3 and r = 2, then (3.4) leads to
(
)
∂φn (x, 2)
∂φn (x, 2)
(log x1 − log x2 ) x1
− x2
∂x1
∂x2
[
1
= (x1 − x2 )(log x1 − log x2 )φn (x, 2)
x1 + 2x1 x2 + x2

n
∑
(1 + 2xj )xj
 ≥ 0.
+
(x
+
2x
x
+
x
)(x
+
2x
x
+
x
)
1
1
j
j
2
2
j
j
j=3
Case IV. If n ≥ 4 and 3 ≤ r ≤ n − 1, then (3.5) reveals
(
)
∂φn (x, r)
∂φn (x, r)
(log x1 − log x2 ) x1
− x2
∂x1
∂x2

∑
(x1 − x2 )(log x1 − log x2 ) 
=
1+x1
x1 x2
+
3≤i1 <i2 <···<ir−2 ≤n
+
∑
1+x1
3≤i1 <i2 <···<ir−1 ≤n ( x1 +
1+
∑r−1
∑r−1
1+xij
j=1 xij
j=1
1+xij
x ij
2
)( 1+x
x2 +
x1
1
1+x2
x2
∑r−2
+ j=1

∑r−1
1+xij
j=1 xij
)
1+xij
xij
 φn (x, r) ≥ 0.
Therefore (3.6) follows from Cases I-IV, and the proof of Theorem 3.2 is completed.
Theorem 3.3. For r ∈ {1, 2, · · · , n}, the symmetric function φn (x, r) is Schur
harmonic concave in Rn+ .
Proof. According to Lemma 2.3 we only need to prove that
(
)
2 ∂φn (x, r)
2 ∂φn (x, r)
(x1 − x2 ) x1
− x2
≤0
∂x1
∂x2
for all x = (x1 , x2 , · · · , xn ) ∈ Rn+ and r = 1, 2, · · · , n.
The proof is divided into four cases.
Case A. If r = 1, then from (3.2) we have
)
(
∂φn (x, 1)
∂φn (x, 1)
− x22
(x1 − x2 ) x21
∂x1
∂x2
2
(x1 − x2 )
φn (x, 1) ≤ 0.
=−
(1 + x1 )(1 + x2 )
Case B. If n ≥ 2 and r = n, then (3.3) leads to
(
)
2 ∂φn (x, n)
2 ∂φn (x, n)
(x1 − x2 ) x1
− x2
= 0.
∂x1
∂x2
(3.7)
A CLASS OF SYMMETRIC FUNCTIONS AND THEIR APPLICATIONS
91
Case C. If n ≥ 3 and r = 2, then (3.4) shows that
(
)
∂φn (x, 2)
∂φn (x, 2)
(x1 − x2 ) x21
− x22
∂x1
∂x2
n
∑
x2j
= −(x1 − x2 )2 φn (x, 2)
≤ 0.
(x1 + 2x1 xj + xj )(x2 + 2x2 xj + xj )
j=3
Case D. If n ≥ 4 and 3 ≤ r ≤ n − 1, then (3.5) deduces that
(
)
2 ∂φn (x, r)
2 ∂φn (x, r)
(x1 − x2 ) x1
− x2
∂x1
∂x2
2
(x1 − x2 )
=−
φn (x, r)
x1 x2
∑
1
×
∑
∑r−1
1+x
i
r−1
1+x1
1+x2
j
3≤i1 <i2 <···<ir−1 ≤n ( x1 +
j=1 xi )( x2 +
j=1
j
1+xij
x ij
)
≤ 0.
Therefore (3.7) follows from Cases A-D, and the proof of Theorem 3.3 is completed.
4. Applications
In this section, we establish some inequalities by use of Theorems 3.1-3.3 and
the theory of majorization.
∑n
Theorem 4.1. If x = (x1 , x2 , · · · , xn ) ∈ Rn+ , s = i=1 xi and r ∈ {1, 2, · · · , n},
then
c−x
(1) φn (x, r) ≥ φn ( nc
, r) for c ≥ s;
( s −1
)
∑n 1
cHn (x)−1
(2) φn (x, r) ≤ φn
i=1 xi ;
cx−1 x, r for c ≥
c+x
(3) φn (x, r) ≥ φn ( nc
, r) for c ≥ 0;
( s +1
)
n (x)+1
(4) φn (x, r) ≤ φn cHcx+1
x, r for c ≥ 0;
(5) φn (x, r) ≥ φn ( s−λx
, r) for 0 ≤ λ ≤ 1;
(n−λ
)
(6) φn (x, r) ≤ φn ∑n n−λ
for 0 ≤ λ ≤ 1;
1
λ ,r
−
i=1 xi
x
, r) for 0 ≤ λ ≤ 1;
(7) φn (x, r) ≥ φn ( s+λx
(n+λ
)
n+λ
(8) φn (x, r) ≤ φn ∑n 1 + λ , r for 0 ≤ λ ≤ 1.
i=1 xi
x
Proof. Theorem 4.1(1) and (2) follow from Lemma 2.4 , Theorem 3.1 and Theorem
3.3.
Theorem4.1(3) and (4) follow from Lemma 2.5 , Theorem 3.1 and Theorem 3.3.
Theorem 4.1(5) and (6) follow from Lemma 2.6 , Theorem 3.1 and Theorem 3.3.
Theorem 4.1(7) and (8) follow from Theorem 3.1 , Theorem 3.3 together with
the face that
(
)
s + λx
s + λx1 s + λx2
s + λxn
=
,
,··· ,
≺ (x1 , x2 , · · · , xn ) = x.
n+λ
n+λ
n+λ
n+λ
92
W. XIA, Y. CHU
Theorem 4.2. If x = (x1 , x2 , · · · , xn ) ∈ Rn+ and r ∈ {1, 2, · · · , n}, then


] n!
[
r
∏
∑
1 + xij
rAn (1 + x) r!(n−r)!


(i)
≥
;
xij
An (x)
j=1
1≤i1 <i2 <···<ir ≤n


r
∏
∑
n!
 (1 + xij ) ≤ [rAn (1 + x)] r!(n−r)! .
(ii)
1≤i1 <i2 <···<ir ≤n
j=1
Proof. We clearly see that
(An (x), An (x), · · · , An (x)) ≺ (x1 , x2 , · · · , xn ) = x.
(4.1)
Therefore, Theorem 4.2(i) follows from (4.1) and Theorem 3.1 together with
(1.1), and Theorem 4.2 (ii) follows from (4.1) and Theorem 3.3 together with
(1.1).
If we take r = 1 in Theorem 4.2(i) and (ii), and r = n in Theorem 4.2(i),
respectively, then we have
Corollary 4.1. If x = (x1 , x2 , · · · , xn ) ∈ Rn+ , then
(i)
Gn (1 + x)
An (1 + x)
≥
;
Gn (x)
An (x)
(ii)Gn (1 + x) ≤ An (1 + x);
(
)
1+x
An (1 + x)
(iii)An
≥
.
x
An (x)
∑n
Remark 4.1. If we take i=1 xi = 1 in Corollary 4.1(i), then we get the Weierstrass inequality (see [19, p. 260])
)
n (
∏
1
+ 1 ≥ (n + 1)n .
xi
i=1
Theorem 4.3. If x = (x1 , x2 , · · · , xn ) ∈ Rn+ and r ∈ {1, 2, · · · , n}, then


r
∏
∑
n!
 (1 + xij ) ≥ [r(1 + Hn (x))] r!(n−r)! ;
(i)
1≤i1 <i2 <···<ir ≤n
(ii)
∏
1≤i1 <i2 <···<ir ≤n
j=1


r
∑
1 + xij
j=1
xij

n!
[
] r!(n−r)!
 ≤ r(1 + Hn (x))
.
Hn (x)
Proof. We clearly see that
) (
)
(
1
1
1 1
1
1
1
,
,··· ,
≺
, ,··· ,
= .
Hn (x) Hn (x)
Hn (x)
x1 x2
xn
x
(4.2)
Therefore, Theorem 4.3 (i) follows from (4.2), Theorem 3.1 and (1.1), and Theorem 4.3 (ii) follows from (4.2), Theorem 3.3 and (1.1).
If we take r = 1 and r = n in Theorem 4.3, respectively, then we get the following
Corollaries 4.2 and 4.3.
A CLASS OF SYMMETRIC FUNCTIONS AND THEIR APPLICATIONS
93
Corollary 4.2. If x = (x1 , x2 , · · · , xn ) ∈ Rn+ , then
(i)Gn (1 + x) ≥ 1 + Hn (x);
(ii)
Gn (x)
Hn (x)
≥
.
Gn (1 + x)
1 + Hn (x)
Corollary 4.3. If x = (x1 , x2 , · · · , xn ) ∈ Rn+ , then
(i)An (1 + x) ≥ 1 + Hn (x);
(
)
1+x
1
(ii)An
≤1+
.
x
Hn (x)
Theorem 4.4. If x = (x1 , x2 , · · · , xn ) ∈ Rn+ and r ∈ {1, 2, · · · , n}, then


n!
[
] r!(n−r)!
r
∏
∑
1 + xij

 ≥ r · 1 + Gn (x)
.
xij
Gn (x)
j=1
1≤i1 <i2 <···<ir ≤n
Proof. We clearly see that
log (Gn (x), Gn (x), · · · , Gn (x)) ≺ log(x1 , x2 , · · · , xn ).
Therefore, Theorem 4.4 follows from (4.3), Theorem 3.2 and (1.1).
(4.3)
If we take r = 1 and r = n in Theorem 4.4, respectively, then we get
Corollary 4.4. If x = (x1 , x2 , · · · , xn ) ∈ Rn+ , then
(i)Gn (1 + x) ≥ 1 + Gn (x);
(
)
1+x
1 + Gn (x)
(ii)An
≥
.
x
Gn (x)
Theorem 4.5. If x = (x1 , x2 , · · · , xn ) ∈ Rn+ , then


(n−1)!
∑n
[
] (r−1)!(n−r)!
r
∏
∑
(n−1)!
2 + xi j
2 + i=1 xi
r!(n−r−1)!


∑n
(i)
≤ (2r)
+ 2(r − 1)
1 + xi j
1 + i=1 xi
j=1
1≤i1 <i2 ···<ir ≤n
1 ≤ r ≤ n − 1;
for
(ii)
∑n
2 + i=1 xi
∑n
+ 2(n − 1);
≤
1 + xi
1 + i=1 xi


(n−1)!
(
) (r−1)!(n−r)!
r
n
∏
∑
∑
(n−1)!
 (2 + xij ) ≥ (2r) r!(n−r−1)! 2r +
xi
n
∑
2 + xi
i=1
(iii)
1≤i1 <i2 ···<ir ≤n
for
j=1
i=1
1 ≤ r ≤ n − 1.
Proof. Theorem 4.5 follows from Theorem 3.1, Theorem 3.3 and (1.1) together with
the fact that
n
∑
(1 + x1 , 1 + x2 , · · · , 1 + xn ) ≺ (1 +
xi , 1, 1, · · · , 1).
i=1
94
W. XIA, Y. CHU
Theorem 4.6. Let A = A1 A2 · · · An+1 be a n−dimensional simplex in Rn and P be
an arbitrary point in ∑
the interior of A. If Bi is the intersection point of straight line
Ai P and hyperplane i = A1 A2 · · · Ai−1 Ai+1 · · · An+1 , i = 1, 2, · · · , n+1. Then for
r ∈ {1, 2, · · · , n + 1} we have


r
∑
(n+1)!
Aij Bij + P Bij

 ≥ [r(n + 2)] r!(n−r+1)! ;
(i)
P Bij
j=1
1≤i1 <i2 ···<ir ≤n+1


(n+1)!
] r!(n−r+1)!
[
r
∏
∑
Aij Bij + P Bij
r(n
+
2)

≤
(ii)
;
Aij Bij
n+1
j=1
1≤i1 <i2 ···<ir ≤n+1


[
] (n+1)!
r
∏
∑
Aij Bij + P Aij
r(2n + 1) r!(n−r+1)!


(iii)
;
≥
P A ij
n
j=1
1≤i1 <i2 ···<ir ≤n+1


[
] (n+1)!
r
∏
∑
Aij Bij + P Aij
r(2n + 1) r!(n−r+1)!


(iv)
≤
.
Aij Bij
n+1
j=1
∏
1≤i1 <i2 ···<ir ≤n+1
Proof. It is easy to see that
(
∑n+1
P Bi
i=1 Ai Bi
1
1
1
,
,··· ,
n+1 n+1
n+1
)
= 1 and
(
≺
∑n+1
P Ai
i=1 Ai Bi
= n, these imply that
P B1 P B2
P Bn+1
,
,··· ,
A1 B1 A2 B2
An+1 Bn+1
)
(4.4)
and
(
n
n
n
,
,··· ,
n+1 n+1
n+1
)
(
≺
P A1 P A2
P An+1
,
,··· ,
A1 B1 A2 B2
An+1 Bn+1
)
.
(4.5)
Therefore, Theorem 4.6 follows from (4.4), (4.5), Theorem 3.1, Theorem 3.3 and
(1.1).
Remark 4.2. D. S. Mitrinović, J. E. Pec̆airć and V. Volenec [20, p. 473-479]
established a series of inequalities for APiABii and APiBBii , i = 1, 2, · · · , n + 1. Obvious,
our inequalities in Theorem 4.6 are different from theirs.
Theorem 4.7. Suppose that A ∈ Mn (C)(n ≥ 2) is a complex matrix, λ1 , λ2 , · · · , λn
and σ1 , σ2 , · · · , σn are the eigenvalues and singular values of A, respectively. If A
A CLASS OF SYMMETRIC FUNCTIONS AND THEIR APPLICATIONS
95
is a positive definite Hermitian matrix and r ∈ {1, 2, · · · , n}, then


n!
[
] r!(n−r)!
r
∏
∑
1 + λi j
r(n
+
trA)
≥

(i)
;
λi j
trA
j=1
1≤i1 <i2 ···<ir ≤n


n!
[
] r!(n−r)!
r
∑
∏
r(n
+
trA)
 (1 + λij ) ≤
(ii)
;
n
1≤i1 <i2 ···<ir ≤n j=1

 [
n!
] r!(n−r)!
√
r
n
∏
∑
2 + λi j
det(I
+
A))
r(1
+

≥
√
(iii)
;
n
1 + λi j
det(I + A)
j=1
1≤i1 <i2 ···<ir ≤n
 [

n!
] r!(n−r)!
√
r
n
∏
∑
trA + λij
r(trA
+
detA)
≥

√
(iv)
;
n
λi j
detA
j=1
1≤i1 <i2 ···<ir ≤n




r
r
∏
∑
∏
∑
1 + λi j
1 + σij

≤

.
(v)
λi j
σ ij
j=1
j=1
1≤i1 <i2 ···<ir ≤n
1≤i1 <i2 ···<ir ≤n
∑n
We clearly see that λi > 0(i = 1, 2, · · · , n) and i=1 λi = trA,
(
)
trA trA
trA
,
,··· ,
≺ (λ1 , λ2 , · · · , λn ).
(4.6)
n
n
n
Therefore, Theorem 4.7 (i) and (ii) follows from (4.6), Theorem 3.1, Theorem
3.3 and (1.1).
(iii) It∏
is easy to see that 1+λ1 , 1+λ2 , · · · , 1+λn are the eigenvalues of matrix
n
I + A and i=1 (1 + λi ) = det(I + A), these yield that
(√
)
√
√
log n det(I + A), n det(I + A), · · · , n det(I + A)
(4.7)
Proof. (i) − (ii)
these lead to
≺ log(1 + λ1 , 1 + λ2 , · · · , 1 + λn ).
Therefore, Theorem 4.7 (iii) follows from (4.7), Theorem 3.2 and (1.1).
(iv) Theorem 4.7(iv) follows from (1.1), Theorem 3.2 and the fact that
(√
)
√
√
(
)
n
n
detA n detA
detA
λ1 λ2
λn
log
,
,··· ,
≺ log
,
,··· ,
.
trA
trA
trA
trA trA
trA
(v) A result due to H. Weyl [21] (see also [5, p. 231]) gives
log(λ1 , λ2 , · · · , λn ) ≺ log(σ1 , σ2 , · · · , σn ).
Therefore, Theorem 4.7 (v) follows from (4.8), Theorem 3.2 and (1.1).
(4.8)
Acknowledgments. The authors would like to thank the anonymous referee for
his/her comments that helped us improve this article.
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WEIFENG XIA
School of Teacher Education, Huzhou Teachers College, Zhejiang 313000, China
E-mail address: xwf212@hutc.zj.cn
YUMING CHU
Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
E-mail address: chuyuming@hutc.zj.cn