Journal of Inequalities in Pure and
Applied Mathematics
SOME INEQUALITIES FOR A CLASS OF GENERALIZED MEANS
KAI-ZHONG GUAN
Department of Mathematics and Physics
Nanhua University, Hengyang, Hunan 421001
The Peoples’ Republic of China
EMail: kaizhongguan@yahoo.com.cn
volume 5, issue 3, article 69,
2004.
Received 03 March, 2004;
accepted 30 March, 2004.
Communicated by: D. Stefanescu
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
054-04
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Abstract
In this paper, we define a symmetric function, show its properties, and establish
several analytic inequalities, some of which are "Ky Fan" type inequalities. The
harmonic-geometric mean inequality is refined.
2000 Mathematics Subject Classification: 05E05, 26D20.
Key words: Symmetric function ; Ky Fan inequality ; Harmonic-geometric mean inequality.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
Refinement of the Harmonic-Geometric Mean Inequality . . . 8
4
Schur-convexity of the Function Hnr (x) . . . . . . . . . . . . . . . . . . 11
5
Some “Ky Fan” Type Inequalities . . . . . . . . . . . . . . . . . . . . . . 16
References
Some Inequalities for a Class of
Generalized Means
Kai-Zhong Guan
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1.
Introduction
Let x = (x1 , x2 , . . . , xn ) be an n-tuple of positive numbers. The un-weighted
arithmetic, geometric and harmonic means of x, denoted by An (x) , Gn (x),
Hn (x), respectively, are defined as follows
n
1X
An (x) =
xi , Gn (x) =
n i=1
n
Y
! n1
xi
n
, Hn (x) = Pn
i=1
1
i=1 xi
.
Assume that 0 ≤ xi < 1, 1 ≤ i ≤ n and define 1 − x = (1 − x1 , 1 − x2 , . . . , 1 −
xn ). Throughout the sequel the symbols An (1 − x), Gn (1 − x), Hn (1 − x) will
stand for the un-weighted arithmetic, geometric, harmonic means of 1 − x.
A remarkable new counterpart of the inequality Gn ≤ An has been published
in [1].
Theorem 1.1. If 0 < xi ≤ 12 , for all i = 1, 2, . . . , n, then
(1.1)
Gn (x)
An (x)
≤
Gn (1 − x)
An (1 − x)
with equality if and only if all the xi are equal.
This result, commonly referred to as the Ky Fan inequality, has stimulated
the interest of many researchers. New proofs, improvements and generalizations of the inequality (1.1) have been found. For more details, interested readers can see [2], [3] and [4].
W.-L. Wang and P.-F. Wang [5] have established a counterpart of the classical
inequality Hn ≤ Gn ≤ An . Their result reads as follows.
Some Inequalities for a Class of
Generalized Means
Kai-Zhong Guan
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Theorem 1.2. If 0 < xi ≤ 12 , for all i = 1, 2, . . . , n, then
Hn (x)
Gn (x)
An (x)
≤
≤
.
Hn (1 − x)
Gn (1 − x)
An (1 − x)
All kinds of means about numbers and their inequalities have stimulated the
interest of many researchers. Here we define a new mean, that is:
(1.2)
Definition 1.1. Let x ∈ Rn+ = {x|x = (x1 , x2 , . . . , xn )|xi > 0, i = 1, 2, . . . , n},
we define the symmetric function as follows
!# n1
"
(r)
Y
r
r
r
Pr
Hn (x) = Hn (x1 , x2 , . . . , xn ) =
.
−1
i=1 xij
1≤i1 <···ir ≤n
n!
Clearly Hnn (x) = Hn (x) , Hn1 (x) = Gn (x), where nr = r!(n−r)!
.
The Schur-convex function was introduced by I. Schur in 1923 [7]. Its definition is as follows:
Definition 1.2. f : I n → R(n > 1) is called Schur-convex if x ≺ y, then
(1.3)
f (x) ≤ f (y)
for all x, y ∈ I n = I ×I ×· · ·×I (n copies). It is called strictly Schur-convex if
the inequality is strict; f is called Schur-concave (resp. strictly Schur-concave)
if the inequality (1.3) is reversed. For more details, interested readers can see
[6], [7] and [8].
The paper is organized as follows. A refinement of harmonic-geometric
mean inequality is obtained in Section 3. In Section 4, we investigate the Schurconvexity of the symmetric function. Several “Ky Fan” type inequalities are
obtained in Section 5.
Some Inequalities for a Class of
Generalized Means
Kai-Zhong Guan
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2.
Lemmas
In this section, we give the following lemmas for the proofs of our main results.
Lemma 2.1. ([5]) If 0 < xi ≤ 12 , for all i = 1, 2, . . . , n, then
" Qn 1 # n1
Pn
1
Hn (x)
Gn (x)
i=1
i=
i
i
Pn 1−x
or
≤
.
(2.1)
≤ Qn 1−x
1
1
Hn (1 − x)
Gn (1 − x)
i=1 xi
i= xi
P
Lemma 2.2. If 0 < xi ≤ 21 , for all i = 1, 2, . . . , n + 1, and Sn+1 = n+1
i=1
Pn+1 1
S n+1 = i=1 1−xi , then
1
n  Q
 n+1
 Pn+1 n+1
1
1
S
−
S
−
n+1
n+1
i=1
i=1
1−xi
1−xi

 ≤ Q

(2.2)
.
Pn+1 n+1
1
1
S
−
S
−
n+1
n+1
i=1
i=1
xi
xi
Proof. Inequality (2.2) is equivalent to the following

 Qn+1 1
S
−
n+1
i=1
1−xi
S n+1
1

n ln
≤
ln  Qn+1
1
Sn+1
n+1
(S
−
)
n+1
i=1
xi
Since 0 < xi ≤ 12 , and 1 − xi ≥ xi , it follows that
1
S n+1 − 1−x
j
1
Sn+1 − xj
=
≥
1
1−x1
+ · · · + 1−x1j−1 + 1−x1j+1 + · · · + 1−x1n+1
1
1
1
1
+ · · · + xj−1
+ xj+1
+ · · · + xn+1
x1
1
· · · 1−x1j−1 1−x1j+1 · · · 1−x1n+1
1−x1
.
1
1
1
1
·
·
·
·
·
·
x1
xj−1 xj+1
xn+1
1
,
xi
Some Inequalities for a Class of
Generalized Means
Kai-Zhong Guan
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By the above inequality and Lemma 2.1, we have
n+1
n+1
1
Y 1 1 n
Y S n+1 − 1−x
1
1
i
≥
ln
ln
n + 1 i=1
1 − xi
xi
n + 1 i=1 Sn+1 − x1i
n+1
Y 1 1 n+1
= n ln
1 − xi
xi
i=1
≥ n ln
or
1
1−x1
1
x1
+ ··· +
+ ··· +
1
1−xn+1
1
xn+1
,
Some Inequalities for a Class of
Generalized Means
Kai-Zhong Guan
 Qn+1 
1
i=1 S n+1 − 1−xi
S n+1
1
.
n ln
≤
ln  Q
n+1
Sn+1
n+1
S
− 1
i=1
n+1
xi
Lemma 2.3. [6, p. 259]. Let f (x) = f (x1 , x2 , . . . , xn ) be symmetric and
have continuous partial derivatives on I n , where I is an open interval. Then
f : I n → R is Schur-convex if and only if
∂f
∂f
−
≥0
(2.3)
(xi − xj )
∂xi ∂xj
on I n . It is strictly Schur-convex if (2.3) is a strict inequality for xi 6= xj ,
1 ≤ i, j ≤ n.
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Since f (x) is symmetric, Schur’s condition can be reduced as [7, p. 57]
∂f
∂f
(2.4)
(x1 − x2 )
−
≥ 0,
∂x1 ∂x2
and f is strictly Schur-convex if (2.4) is a strict inequality for x1 6= x2 . The
Schur condition that guarantees a symmetric function being Schur-concave is
the same as (2.3) or (2.4) except the direction of the inequality.
In Schur’s condition, the domain of f (x) does not have to be a Cartesian
product I n . Lemma 2.3 remains true if we replace I n by a set A ⊆ Rn with the
following properties ([7, p. 57]):
Some Inequalities for a Class of
Generalized Means
Kai-Zhong Guan
(i) A is convex and has a nonempty interior;
(ii) A is symmetric in the sense that x ∈ A implies P x ∈ A for any n × n
permutation matrix P .
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3.
Refinement of the Harmonic-Geometric Mean
Inequality
The goal of this section is to obtain the basic inequality of Hnr (x), and give a
refinement of the Harmonic-Geometric mean inequality.
Theorem 3.1. Let x ∈ Rn+ = {x|x = (x1 , x2 , . . . , xn )|xi > 0, i = 1, 2, . . . , n},
then
(3.1)
Hnr+1 (x) ≤ Hnr (x),
r = 1, 2, . . . , n − 1.
Proof. By the arithmetic-geometric mean inequality and the monotonicity of
the function y = ln x, we have
n
ln Hnr+1 (x)
r+1
X
r+1
=
ln Pr+1 −1
j=1 xij
1≤i1 <···<ir+1 ≤n
#
"
X
(r + 1)r
=
ln
P
Pr+1 −1
−1
(r + 1) r+1
k=1 xik −
j=1 xij
1≤i1 <···<ir+1 ≤n




X
r
i.
=
ln hP P
r+1
r+1 −1

x − x−1
(r + 1) 
1≤i1 <···<ir+1 ≤n
j=1
k=1
ik
ij
Some Inequalities for a Class of
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Kai-Zhong Guan
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
X
≤
1≤i1 <···<ir+1 ≤n
X
=


ln  r
Qr+1 Pr+1
j=1
ln
1≤i1 <···<ir+1 ≤n
"r+1
Y
r
Pr+1
−1
k=1 xik
j=1
" r+1
X
X
1
=
r + 1 1≤i <···<i
r+1 ≤n
1
k=1
−1
x−1
ik − xij
1
# r+1
− x−1
ij
ln Pr+1
k=1
j=1

1 
r+1
#
r
Some Inequalities for a Class of
Generalized Means
−1
x−1
ik − xij
i1 ,...,ir 6=j
n
X
r
1 X
ln Pr
=
−1 .
r + 1 j=1 1≤i <···<i ≤n
k=1 xik
Kai-Zhong Guan
r
1
Title Page
Let
i1 ,...,ir 6=j
X
Sj =
ln Pr
1≤i1 <···<ir ≤n
Contents
r
−1
k=1 xik
,
j = 1, 2, . . . , n.
We can easily get
n
X
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Sj = (n − r)
X
1≤i1 <···<ir ≤n
j=1
ln Pr
r
−1
k=1 xik
= (n − r)
n
r
ln Hnr (x).
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Thus
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n
r+1
ln Hnr+1 (x) ≤
n − r n
ln Hnr (x) =
r+1 r
n
r+1
ln Hnr (x),
J. Ineq. Pure and Appl. Math. 5(3) Art. 69, 2004
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or
Hnr+1 (x) ≤ Hnr (x),
r = 1, 2, . . . , n − 1.
Corollary 3.2. Let x ∈ Rn+ = {x|x = (x1 , x2 , . . . , xn )|xi > 0, i = 1, 2, . . . , n},
then
(3.2)
Hn (x) ≤ Hnn−1 (x) ≤ · · · ≤ Hn2 (x) ≤ Hn1 (x) = Gn (x).
Remark 3.1. The corollary refines the harmonic-geometric mean inequality.
Some Inequalities for a Class of
Generalized Means
Kai-Zhong Guan
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4.
Schur-convexity of the Function Hnr (x)
In this section, we investigate the Schur-convexity of the function Hnr (x), and
establish several analytic inequalities by use of the theory of majorization.
Theorem 4.1. Let Rn+ = {x|x = (x1 , x2 , . . . , xn )|xi > 0, i = 1, 2, . . . , n}, then
the function Hnr (x) is Schur-concave in Rn+ .
Proof. It is clear that Hnr (x) is symmetric and has continuous partial derivatives
on Rn+ . By Lemma 2.3, we only need to prove
∂Hnr (x) ∂Hnr (x)
(x1 − x2 )
−
≤ 0.
∂x1
∂x2
As matter of fact, we can easily derive
X
X
1
r
r
ln Hnr (x) = n ln Pr
+
ln −1 Pr−1 −1 .
−1
x1 + j=1 xij
j=1 xij
r 2≤i1 <···<ir ≤n
2≤i1 <···<ir−1 ≤n
ln Hnr (x)
with respect to x1 , we have


r
r
X
∂Hn (x)
1
H (x)
1
= nn 
Pr−1 −1  · 2
−1
∂x1
x1
x + j=1 xij
r
2≤i1 <···<ir−1 ≤n 1


r
X
H (x) 1
1

= nn · 2 
P
r−1 −1
−1
x
x
+
x
1
1
r
j=1 ij
3≤i1 <···<ir−1 ≤n
Differentiating
+
1
3≤i1 <···<ir−2 ≤n
x−1
1
+
x−1
2
+
Pr−2
−1
j=1 xij
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
X
Some Inequalities for a Class of
Generalized Means
 .
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Similar to the above, we can also obtain


r
r
X
∂Hn (x)
H (x)
1
1
= nn 
Pr−1 −1  · 2
−1
∂x2
x2
x + j=1 xij
r
2≤i1 <···<ir−1 ≤n 2


r
X
1
H (x) 1
= nn · 2 
Pr−1 −1 
−1
x2
x + j=1 xij
r
3≤i <···<i
≤n 2
1
r−1

1
X
+
3≤i1 <···<ir−2 ≤n
−1
x−1
1 + x2 +
Pr−2
−1
j=1 xij
 .
Thus
Kai-Zhong Guan
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(x1 − x2 )
Some Inequalities for a Class of
Generalized Means
∂Hnr (x)
∂Hnr (x)
Contents
−
∂x2

X
H r (x)
= (x1 − x2 ) n 
∂x1
n
r

−
2≤i1
+
JJ
J

1
Pr−1
· 1
−1
−1
x21
x + j=1 xij
2≤i1 <···<ir−1 ≤n 1

X
1
1
Pr−1 −1  · 2
−1
x2
x + j=1 xij
<···<i
≤n 2
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r−1
3≤i1 <···<ir−2 ≤n
1
X
−1
x−1
1 + x2 +
Pr−2
−1
j=1 xij
II
I

1
1 
−
x21 x22
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
= −(x1 − x2 )2 
X
(x1 + x2 )
x21 x22 3≤i <···<i
x−1
1
r−2 ≤n
1
+
x−1
2
1
P
−1
+ r−2
j=1 xij
Pr−1
+
−1
j=1 xij
X
2≤i1 <···<ir−1 ≤n
x21 x22

1 + (x1 + x2 )
Pr−1 −1 −1 Pr−1 −1 
−1
x1 + j=1 xij
x2 + j=1 xij
≤ 0.
Some Inequalities for a Class of
Generalized Means
P
Corollary 4.2. Let xi > 0, i = 1, 2, . . . , n, n ≥ 2 ,and ni=1 xi = s, c > 0,
then
n1
Hnr (c + x) nc
( )
(4.1)
≥
+1 r ,
r = 1, 2, . . . , n,
r
Hn (x)
s
where c + x = (c + x1 , c + x2 , . . . , c + xn ).
Proof. By [9], we have
c+x
c + x1
c + xn
x1
xn x
=
,...,
≺
,...,
= .
nc + s
nc + s
nc + s
s
s
s
Using Theorem 4.1, we obtain
x
c+x
r
Hn
≥ Hnr
.
nc + s
s
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Or
n1
Hnr (c + x) nc
( )
≥
+1 r .
r
Hn (x)
s
P
Corollary 4.3. Let xi > 0, i = 1, 2, . . . , n, n ≥ 2, and ni=1 xi = s, c ≥ s,
then
n1
Hnr (c − x) nc
(r)
(4.2)
≥
−
1
, r = 1, 2, . . . , n,
Hnr (x)
s
where c − x = (c − x1 , c − x2 , . . . , c − xn ).
Proof. By [9], we have
c−x
c − x1
c − xn
x1
xn x
=
,...,
≺
,...,
= .
nc − s
nc − s
nc − s
s
s
s
Using Theorem 4.1, we obtain
x
c−x
r
≥ Hnr
,
Hn
nc − s
s
or
n1
Hnr (c − x) nc
(r)
≥
−
1
.
Hnr (x)
s
Some Inequalities for a Class of
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Remark 4.1. Let c = s = 1, we can obtain
1
Hnr (1 − x)
(n
r),
≥
(n
−
1)
Hnr (x)
r = 1, 2, . . . , n.
In particular,
Hn (1 − x)
≥ (n − 1),
Hn (x)
√
Gn (1 − x)
≥ n n − 1.
Gn (x)
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5.
Some “Ky Fan” Type Inequalities
In this section, some “Ky Fan” type inequalities are established, the Ky Fan
inequality is generalized.
Theorem 5.1. Assume that 0 < xi ≤ 21 , i = 1, 2, . . . , n, then
(5.1)
r1
Hnr (x)
Hnr+1 (x)
≤
,
Hnr+1 (1 − x)
Hnr (1 − x)
r = 1, 2, . . . , n − 1.
Some Inequalities for a Class of
Generalized Means
Proof. Set
ϕr =
Y
Hnr (x)
=
Hnr (1 − x) 1≤i <···<i
1
r ≤n
 Pr



1
j=1 1−xij
Pr
1
j=1 xij
Kai-Zhong Guan
1
(n
r)
.
By Lemma 2.2 and the monotonicity of the function y = ln x , we have
Pr+1 1
X
j=1 1−xi
n
ln φr+1 =
ln Pr+1 1 j
r+1
j=1 xij
1≤i1 <···<ir+1 ≤n
Pr+1 Pr+1 1
1
−
X
j=1
k=1 1−xik
1−xij
=
ln P P
r+1
r+1 1
1
1≤i1 <···<ir+1 ≤n
j=1
k=1 xik − xij
1

 r(r+1)
Pr+1 1
1
r+1
X
Y
k=1 1−xik − 1−xij

≤
ln 
Pr+1 1
1
−
k=1 xi
xi
j=1
1≤i1 <···<ir+1 ≤n
k
j
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Pr
1
i1 ,...,ir 6=j
n
X
X
k=1 1−xik
1
ln Pr
=
1 .
r(r + 1) j=1 1≤i <···<i ≤n
k=1 xi
r
1
k
Similar to Theorem 3.1, we can derive
n
n
1
1
n
ln φr+1 ≤
(n − r)
ln φr =
ln φr .
r+1
r(r + 1)
r
r r+1
Thus
Some Inequalities for a Class of
Generalized Means
1
(φr ) r ≥ φr+1 ,
or
Hnr+1 (x)
Hnr+1 (1 − x)
≤
Hnr (x)
Hnr (1 − x)
Kai-Zhong Guan
r1
,
r = 1, 2, . . . , n − 1.
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Remark 5.1. By Theorem 5.1, we can obtain
(5.2)
Hn2 (x)
Hn1 (x)
Gn (x)
An (x)
≤
=
≤
.
Hn2 (1 − x)
Hn1 (1 − x)
Gn (1 − x)
An (1 − x)
This is a generalization of the “Ky Fan” inequality.
By Lemma 2.1 and the proof of Theorem 3.1, we have the following
1
,
2
Theorem 5.2. If 0 < xi ≤ i = 1, 2, . . . , n, then
Qn
(x )
Hn (x)
Gn (x)
An (x)
Qn i=1 i
(5.3)
≤
≤
≤
.
Hn (1 − x)
Gn (1 − x)
An (1 − x)
i=1 (1 − xi )
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The inequality (5.3) generalizes the inequality (1.2).
Theorem 5.3. If 0 < xi ≤ 12 , i = 1, 2, . . . , n, then
(5.4)
Hnr (x)
Hn1 (x)
≤
Hnr (1 − x)
Hn1 (1 − x)
An (x)
Gn (x)
=
≤
,
Gn (1 − x)
An (1 − x)
r = 2, . . . , n.
Some Inequalities for a Class of
Generalized Means
Proof. Set
ϕr =
Hnr (x)
Hnr (1 − x)
 Pr



Y
=
1≤i1 <···<ir ≤n
1
j=1 1−xij
Pr
1
j=1 xij
1
Kai-Zhong Guan
(n
r)
.
By Lemma 2.1 and the monotonicity of the function y = ln x , we have
Pr
1
n
X
j=1 1−xij
ln φr =
ln Pr
1
r
j=1 xij
1≤i1 <···<ir+1 ≤n
 r1
 Qr
1
X
j=1 1−xij

≤
ln  Qr
1
j=1 xij
1≤i1 <···<ir+1 ≤n
X
1
=
r 1≤i <···<i
1
r
X
r ≤n
j=1
ln
1
1−xij
1
xij
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By knowledge of combinations, we can easily find
" n 1 #( n−1
r−1 )
Y 1−x
1
i
ln φr ≤ ln
1
r
r
i=1 xi
#
"Y
n
1
1 n−1
1−xi
ln
=
1
r r−1
i=1 xi
n
1 n−1
ln φ1 =
ln φ1 .
=
r r−1
r
n
Some Inequalities for a Class of
Generalized Means
Kai-Zhong Guan
Thus
φr ≤ φ1 ,
r = 2, . . . , n,
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or
(5.5)
Hnr (x)
Gn (x)
≤
, r = 2, . . . , n.
r
Hn (1 − x)
Gn (1 − x)
The inequality (5.5) generalizes the “Ky Fan” inequality.
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References
[1] E.F. BECKENBACH
New York, 1965.
AND
R. BELLMAN, Inequalities, Springer-Verlag,
[2] H. ALZER, The inequality of Ky Fan and related results, Acta. Appl. Math.,
38 (1995), 305–354.
[3] J. SÁNDOR, On an inequaliy of Ky Fan, Babeş-Bolyai Univ., Fac. Math.
Phys., Res. Semin., 7 (1990), 29–34.
[4] J. SÁNDOR AND T. TRIF, A new refinement of the Ky Fan inequality,
Math. Inequal. Appl., 2 (1999), 529–533.
[5] W.-L. WANG AND P.-F. WANG, A class of inequalities for symmetric functions (in chinese), Acta. Math .Sinica, 27 (1984), 485–497.
[6] A.W. ROBERTS AND D.E. VARBERG, Convex Function, Academic Press,
New York, San Francisco, London, 1973.
[7] A.W. MARSHALL AND I. OLKIN, Inequalities: Theory of Majorization
and Its Applications, Academic Press, 1979.
Some Inequalities for a Class of
Generalized Means
Kai-Zhong Guan
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[8] D.S. MITRINOVIĆ, Analytic Inequalities, Springer-Verlag, New York,
1970.
[9] HUANNAN SHI, Refinement and generalization of a class of inequalities
for symmetric functions (in Chinese), Mathematics in Practices and Theory, 4 (1999), 81–84.
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