Journal of Inequalities in Pure and
Applied Mathematics
A GENERALIZATION OF AN INEQUALITY INVOLVING THE
GENERALIZED ELEMENTARY SYMMETRIC MEAN
volume 5, issue 4, article 108,
2004.
ZHI-HUA ZHANG AND ZHEN-GANG XIAO
Zixing Educational Research Section
Chenzhou, Hunan 423400, China.
EMail: zxzh1234@163.com
Hunan Institute of Science and Technology
Yueyang, Hunan 414006, China.
EMail: xiaozg@163.com
Received 23 May, 2003;
accepted 09 October, 2004.
Communicated by: F. Qi
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
070-03
Quit
Abstract
A generalization of an inequality involving the generalized elementary symmetric mean and its elementary proof are given.
2000 Mathematics Subject Classification: Primary 26D15
Key words: Generalized elementary symmetric mean, Inequality.
The authors would like to thank Professor Feng Qi and the anonymous referee for
some valuable suggestions which have improved the final version of this paper.
A Generalization of an
Inequality Involving the
Generalized Elementary
Symmetric Mean
Contents
Zhi-Hua Zhang and
Zhen-Gang Xiao
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
5
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 13
J. Ineq. Pure and Appl. Math. 5(4) Art. 108, 2004
1.
Introduction
Let a = (a1 , a2 , · · · , an ) and r be a nonnegative integer, where ai for 1 ≤ i ≤ n
are nonnegative real numbers. Then
En[r]
(1.1)
=
En[r] (a)
n
Y
X
=
i1 +i2 +···+in =r,
i1 ,i2 ,··· ,in ≥0 are integers
[0]
[0]
aikk
k=1
[r]
with En = En (a) = 1 for n ≥ 1 and En = 0 for r < 0 or n ≤ 0 is called the
rth generalized elementary symmetric function of a.
The rth generalized elementary symmetric mean of a is defined by
[r]
X
(1.2)
=
[r]
X
n
[r]
En (a)
(a) = n+r−1 .
In 1934, I. Schur [5, p. 182] obtained the following
!r
Z
Z X
[r]
n
X
ai xi dx1 · · · dxn−1 ,
(1.3)
(a) = (n − 1)! · · ·
i=1
where xn = 1 − (x1 + x2 + · · · + xn−1 ) and the integral is taken over xk ≥ 0
for k = 1, 2, . . . , n − 1. By using (1.3) and Cauchy integral inequality, he also
proved that

2
[r−1]
[r+1]
[r]
X
X
X
(1.4)
(a)
(a) ≥  (a) .
n
n
Zhi-Hua Zhang and
Zhen-Gang Xiao
r
n
n
A Generalization of an
Inequality Involving the
Generalized Elementary
Symmetric Mean
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 13
n
J. Ineq. Pure and Appl. Math. 5(4) Art. 108, 2004
In 1968, K.V. Menon [2] proved that for n = 2 or n ≥ 3 and r = 1, 2, 3,
inequality (1.4) is valid.
In [1], the generalized symmetric means of two variables was investigated.
In [3] and [4] a problem was posed: Does inequality (1.4) hold for arbitrary
n, r ∈ N?
In 1997, Zh.-H. Zhang generalized (1.3) in [7] and also proved (1.4) by a
similar proof as in [5].
In [6], some inequalities of weighted symmetric mean were established.
P [r]
[r]
In this paper, we shall obtain an identity relating n (a) to En (a) and give
an elementary proof of an inequality which generalizes (1.4). Our main result
is as follows.
Zhi-Hua Zhang and
Zhen-Gang Xiao
Theorem 1.1. If r, s ∈ N and r > s, then
(1.5)
[s]
X
n
[r+1]
(a)
X
n
(a) ≥
A Generalization of an
Inequality Involving the
Generalized Elementary
Symmetric Mean
[r]
X
n
[s+1]
(a)
X
(a).
n
The equality in (1.5) holds if and only if a1 = a2 = · · · = an .
Letting s = r − 1 in inequality (1.5) leads to inequality (1.4).
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 13
J. Ineq. Pure and Appl. Math. 5(4) Art. 108, 2004
2.
Proof of Theorem 1.1
[r]
To prove inequality (1.5), the following properties for En are necessary.
Property 1. If n, r ∈ N, then
[r]
En[r] = En−1 + an En[r−1]
(2.1)
and
En[r]
(2.2)
=
r
X
A Generalization of an
Inequality Involving the
Generalized Elementary
Symmetric Mean
[r−j]
ajn En−1 .
j=0
Zhi-Hua Zhang and
Zhen-Gang Xiao
Proof. If n = 1 or r = 0, (2.1) holds trivially.
When n > 1 and r ≥ 1, we have
(2.3)
i1 +i2 +···+i
n
X n =r Y
i1 ,i2 ,··· ,in ≥0 k=1
aikk
=
i1 +i2 +···+i
n
X n =r Y
i1 ,i2 ,··· ,in−1 ≥0
in =0
aikk
+ an
i1 +i2 +···+i
n
Xn =r−1 Y
i1 ,i2 ,··· ,in ≥0
k=1
[r]
En
Combining the definition of
and (2.3), identity (2.1) follows.
Identity (2.2) can be deduced from the recurrence of (2.1).
Property 2. If r is an integer, then
(2.4)
(r + 1)En[r+1] =
k=1
aikk .
Title Page
Contents
JJ
J
II
I
Go Back
Close
n
r
X
X
k=0
i=1
!
ak+1
i
En[r−k] .
Quit
Page 5 of 13
J. Ineq. Pure and Appl. Math. 5(4) Art. 108, 2004
Proof. It will be verified by induction. It is clear that identity (2.4) holds trivially for n = 1. Suppose identity (2.4) is true for n − 1 and nonnegative integers
r.
By (2.2), for 0 ≤ k ≤ r, we have
En[r−k]
=
r−k
X
[r−k−j]
ajn En−1
,
j=0
and
r
X
[r−j]
[r−1]
[r]
[1]
[0]
r+1
r
2
(j + 1)aj+1
n En−1 = an En−1 + an En−1 + · · · + an En−1 + an En−1
j=0
+
[r−1]
a2n En−1
[1]
arn En−1
+ ··· +
······
[1]
+
[0]
ar+1
n En−1
[0]
+ arn En−1 + ar+1
n En−1
[0]
+ ar+1
n En−1
=
r X
r−k
X
[r−k−j]
anj+k+1 En−1 .
A Generalization of an
Inequality Involving the
Generalized Elementary
Symmetric Mean
Zhi-Hua Zhang and
Zhen-Gang Xiao
Title Page
Contents
JJ
J
II
I
k=0 j=0
Go Back
According to the inductive hypothesis, for nonnegative integers r and 0 ≤ j ≤
r, we have
!
r−j
n−1
X
X
[r+1−j]
[r−k−j]
=
(2.5)
(r − j + 1)En−1
ak+1
En−1 .
i
k=0
Close
Quit
Page 6 of 13
i=1
J. Ineq. Pure and Appl. Math. 5(4) Art. 108, 2004
From Property 1 and the above formula, we have
(2.6)
(r + 1)En[r+1]
= (r + 1)
r+1
X
[r+1−j]
ajn En−1
j=0
=
r
X
(r − j +
[r−j+1]
1)ajn En−1
+
j=0
=
r
X
ajn
X
n−1
X
k=0
i=1
r X
r−k
X
ajn
k=0 j=0
=
=
[r−j+1]
jajn En−1
j=1
r−j
j=0
=
r+1
X
r
X
n−1
X
k=0
i=1
r
n
X
X
k=0
n−1
X
!
[r−j−k]
En−1
ak+1
i
+
r
X
(j +
[r−j]
1)aj+1
n En−1
j=0
!
ak+1
i
[r−j−k]
En−1
i=1
+
r X
r−k
X
ak+1
+ ak+1
n
i
r−k
X
!
[r−k−j]
ajn En−1
j=0
!
En[r−k] .
i=1
This shows that (2.4) holds for n. The proof is complete.
Property 3. If r, s ∈ N and r > s, then
n+r
(2.7) (r + 1)(s + 1)
r+1
Zhi-Hua Zhang and
Zhen-Gang Xiao
k=0 j=0
!
ak+1
i
[r−k−j]
En−1
aj+k+1
n
A Generalization of an
Inequality Involving the
Generalized Elementary
Symmetric Mean


X
[s] [r+1]
[r] [s+1]
X
X
X
n+s 

−
s+1
n
n
n
n
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 7 of 13
J. Ineq. Pure and Appl. Math. 5(4) Art. 108, 2004
=
j
r X
X
j=0 k=0
"
X
En[s−k] En[r−j] − En[s−j] En[r−k]
1≤v<u≤n
j−k−1
×
X
!
aj−1−t
ak+t
v
u
#
2
(av − au )
.
t=0
Proof. When j > k, we have
(2.8)
n
X
i=1
aki
n
X
aj+1
i
i=1
−
n
X
aji
i=1
=
n
n
1 XX
2
n
X
A Generalization of an
Inequality Involving the
Generalized Elementary
Symmetric Mean
ak+1
i
i=1
k
j k+1
j
akv aj+1
+ aj+1
− ak+1
u
v au − av au
v au
v=1 u=1
n
n
1 X X k k j−k+1
j−k
=
av au av
+ auj−k+1 − aj−k
a
−
a
a
u
v
v
u
2 v=1 u=1
n
n
1 X X k k j−k
av au av − aj−k
(av − au )
u
2 v=1 u=1
X =
akv aku aj−k
− aj−k
(av − au )
v
u
=
1≤v<u≤n
Title Page
Contents
JJ
J
II
I
Go Back
and
(2.9)
Zhi-Hua Zhang and
Zhen-Gang Xiao
Close
j−k−1
X
j−k
aj−k
−
a
=
(a
−
a
)
aj−k−1−t
atu .
v
u
v
u
v
t=0
Quit
Page 8 of 13
J. Ineq. Pure and Appl. Math. 5(4) Art. 108, 2004
Therefore
(2.10)
n
X
aki
i=1
n
X
aj+1
i
−
i=1
n
X
aji
i=1
n
X
ak+1
i
i=1
k−j−1
"
X
X
1≤v<u≤n
t=0
=
!
#
ak+t
(av − au )2 .
aj−1−t
u
v
When k > j, we have
(2.11)
n
X
i=1
aki
n
X
aj+1
i
i=1
−
n
X
aji
i=1
n
X
A Generalization of an
Inequality Involving the
Generalized Elementary
Symmetric Mean
ak+1
i
i=1
k−j−1
"
X
X
1≤v<u≤n
t=0
=−
!
(2.12)
(r +
=
Title Page
Contents
r
n
X
X
j=0
Zhi-Hua Zhang and
Zhen-Gang Xiao
k−1−t
(av − au )2 .
aj+t
v au
From Property 2, it is deduced that
1)En[r+1]
#
!
aj+1
i
En[r−j]
JJ
J
II
I
i=1
Go Back
and
(2.13)
Close
(n + r)En[r] = nEn[r] + rEn[r] =
r
X
n
X
j=0
i=1
!
aji
En[r−j] .
Quit
Page 9 of 13
J. Ineq. Pure and Appl. Math. 5(4) Art. 108, 2004
[r−k]
Hence, using the above formulas and noting En
= 0 for k > r yields


X
[s] [r+1]
[r] [s+1]
X
X
X
n+r n+s 

(2.14) (r + 1)(s + 1)
−
r+1
s+1
n
n
n
n
= (n + s)(r + 1)En[s] En[r+1] − (n + r)(s + 1)En[r] En[s+1]
!
!
r
n
n
s
X
X
X
X
j+1
En[r−j]
aki En[s−k]
ai
=
k=0
−
=
=
j=0
i=1
r
X
n
X
j=0
i=1
aji
n
X
k=0 j=0
i=1
j
r X
X
−
−
i=1
aj+1
i
−
ak+1
i
En[s−k]
i=1
n
X
n
X
j
ai
i=1
En[s−k] En[r−j]
ak+1
i
En[s−k]
X
En[s−k] En[r−j]
X
Title Page
#
!
2
aj−1−t
auk+t
v
(av − au )
Contents
X
!
k−1−t
aj+t
v au
(av − au )
!
#
j−k−1
En[s−k] En[r−j]
"
1≤v<u≤n
2
X
aj−1−t
auk+t
v
JJ
J
II
I
Go Back
2
(av − au )
Close
t=0
1≤v<u≤n
X
#
t=0
1≤v<u≤n
"
j=0 k=0
·
En[r−j]
i=1
k−j−1
"
X
Zhi-Hua Zhang and
Zhen-Gang Xiao
!
t=0
1≤v<u≤n
j
r X
X
A Generalization of an
Inequality Involving the
Generalized Elementary
Symmetric Mean
!
j−k−1
X
k=0 j=0
j=0 k=0
aki
n
X
n
X
"
s X
k
X
j
r X
X
En[r−j]
i=1
s
X
k=0
s X
r
X
j=0 k=0
=
!
En[s−j] En[r−k]
j−k−1
!
X
aj−1−t
ak+t
(av − au )2
v
u
#
Quit
Page 10 of 13
t=0
J. Ineq. Pure and Appl. Math. 5(4) Art. 108, 2004
=
j
r X
X
"
j=0 k=0
X
En[s−k] En[r−j] − En[s−j] En[r−k]
1≤v<u≤n
j−k−1
×
X
!
#
aj−1−t
auk+t (av − au )2 ,
v
t=0
which implies the expression (2.7).
Property 4. If r, s ∈ N and r > s, then
A Generalization of an
Inequality Involving the
Generalized Elementary
Symmetric Mean
En[r−1] En[s] ≥ En[r] En[s−1] .
(2.15)
The equality in (2.15) holds if and only if at least n − 1 numbers equal zero
among {a1 , a2 , . . . , an }.
Zhi-Hua Zhang and
Zhen-Gang Xiao
Proof. From Property 1, we have
Title Page
(2.16) En[r−1] En[s] − En[r] En[s−1]
[s]
[r]
[r−1]
[s−1]
[r−1]
= En
En−1 + an En
− En−1 + an En
En[s−1]
Contents
[s]
[r]
= En[r−1] En−1 − En−1 En[s−1]
!
r−1
X
[r−1−j]
[s]
[r]
=
ajn En−1
En−1 − En−1
j=0
JJ
J
II
I
Go Back
s−1
X
j=0
!
[s−1−j]
ajn En−1
Close
Quit
Page 11 of 13
J. Ineq. Pure and Appl. Math. 5(4) Art. 108, 2004
=
s−1
X
[r−1−j] [s]
[r]
[s−1−j]
ajn En−1 En−1 − En−1 En−1
j=0
[s]
+ En−1
r−1
X
!
[r−1−j]
ajn En−1
.
j=s
Since (2.15) holds for n = 1, it follows by induction that (2.15) holds for n.
Property 5. If r, s, j, k ∈ N and r > s > j > k, then
En[s−k] En[r−j] ≥ En[s−j] En[r−k] .
(2.17)
The equality in (2.17) is valid if and only if at least n − 1 numbers equal zero
among {a1 , a2 , . . . , an }
Proof. From Property 4, if r − (k + 1) > s − (k + 1), r − (k + 2) > s − (k +
2), . . . , r − j > s − j, then
(2.18)
j
Y
m=k+1
En[r−m] En[s−m+1]
≥
j
Y
A Generalization of an
Inequality Involving the
Generalized Elementary
Symmetric Mean
Zhi-Hua Zhang and
Zhen-Gang Xiao
Title Page
En[r−m+1] En[s−m]
.
Contents
m=k+1
This implies (2.17).
It is easy to see that the equality in (2.17) is valid. The proof is completed.
JJ
J
II
I
Go Back
Proof of Theorem 1.1. Combination of Property 3 and Property 5 easily leads
to Theorem 1.1.
Remark 2.1. Finally, we pose an open problem: Give an explicit expression of
[r−1] [s]
[r] [s−1]
En En − En En
in terms of a1 , a2 , . . . , an .
Close
Quit
Page 12 of 13
J. Ineq. Pure and Appl. Math. 5(4) Art. 108, 2004
References
[1] D.W. DETEMPLE AND J.M. ROBERTSON, On generalized symmetric
means of two variables, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat.
Fiz., 634–672 (1979), 236–238
[2] K.V. MENON, Inequalities for symmetric functions, Duke Math. J., 35
(1968), 37–45.
[3] D.S. MITRINOVIĆ, Analytic Inequalities, Springer, 1970.
[4] J.-Ch. KUANG, Chángyòng Búděngshì (Applied Inequalities), 2nd Ed. Hunan Education Press, Changsha, China, 1993. (Chinese)
[5] G.H. HARDY, J.E. LITTLEWOOD AND G. POLYA, Inequalities, 2nd Ed.
Cambridge Univ. Press, 1952.
[6] Zh.-G. XIAO, Zh.-H. ZHANG AND X.-N. LU, A class of inequalities for
weighted symmetric mean, J. Hunan Educational Institute, 17(5) (1999),
130–134. (Chinese)
[7] Zh.-H. ZHANG, Three classes of new means in n + 1 variables and their
applications, J. Hunan Educational Institute, 15(5) (1997), 130–136. (Chinese)
A Generalization of an
Inequality Involving the
Generalized Elementary
Symmetric Mean
Zhi-Hua Zhang and
Zhen-Gang Xiao
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 13 of 13
J. Ineq. Pure and Appl. Math. 5(4) Art. 108, 2004