ON SOME NEW MEAN VALUE INEQUALITIES
LIANG-CHENG WANG
CAI-LIANG LI
School of Mathematics Science
Chongqing Institute of Technology
No. 4 of Xingsheng Lu, Yangjia Ping 400050
Chongqing, The People’s Republic of China.
EMail: wangliangcheng@163.com
Chengdu Electromechanical College
Chengdu 610031,
Sichuan Province
The People’s Republic of China.
EMail: dzlcl@163.com
Some New Mean Value
Inequalities
Liang-Cheng Wang and Cai-Liang Li
vol. 8, iss. 3, art. 87, 2007
Title Page
Contents
Received:
04 March, 2007
Accepted:
27 April, 2007
Communicated by:
P.S. Bullen
2000 AMS Sub. Class.:
26D15.
Key words:
Mean value inequality, Hölder’s inequality, Continuous positive function, Extension.
Abstract:
In this paper, using the arithmetic-geometric mean inequality, we obtain some
new mean value inequalities. Finally, some applications are given, they are extension of Hölder’s inequalities.
Acknowledgements:
The first author is partially supported by the Key Research Foundation of the
Chongqing Institute of Technology under Grant 2004DZ94.
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Contents
1
Introduction and Main Results
3
2
Proof of Theorem and Corollary
7
3
Applications
11
Some New Mean Value
Inequalities
Liang-Cheng Wang and Cai-Liang Li
vol. 8, iss. 3, art. 87, 2007
Title Page
Contents
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1.
Introduction and Main Results
Let a > 0, b > 0 and t ∈ (0, 1). It is well-known that the following arithmeticgeometric mean inequality holds
at b1−t ≤ ta + (1 − t)b.
(1.1)
The arithmetic-geometric mean inequality is a classical inequality with many applications. Also, there exist extensive works devoted to generalizing or improving the
arithmetic-geometric mean inequality. In this respect, we refer the reader to [1] – [7]
and the references cited therein for updated results.
In this paper, by (1.1), we obtain some new mean value inequalities. Finally,
some applications are given.
In this paper, we agree
q
X
(bi ∈ R,
bi = 0,
q ∈ N).
i=q+1
Theorem 1.1. Let xi > 0 (i = 1, 2, . . . , n; n ≥ 2) and t ∈ (0, 1).
1. For the following
" k
X
1
B(k) 4 2 k
xi +
n
i=1
+
Some New Mean Value
Inequalities
Liang-Cheng Wang and Cai-Liang Li
vol. 8, iss. 3, art. 87, 2007
Title Page
Contents
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n
X
!
xti
i=1
n
X
i=k+1
n
X
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!
x1−t
i
Close
i=k+1
!
xti
k
X
i=1
!#
x1−t
i
,
(k = 1, 2, . . . , n)
and
"
n
X
1
C(j) 4 2 (n − j + 1)
xi +
n
i=j
j−1
+
X
n
X
!
xti
i=1
!
xti
!
x1−t
i
i=1
n
X
i=1
j−1
X
!#
x1−t
i
,
(j = 1, 2, . . . , n),
i=j
Some New Mean Value
Inequalities
Liang-Cheng Wang and Cai-Liang Li
we have
vol. 8, iss. 3, art. 87, 2007
(1.2)
n
1X t
x
n i=1 i
!
n
1 X 1−t
x
n i=1 i
!
Title Page
= B(1) ≤ B(2) ≤ · · · ≤ B(k) ≤ B(k + 1) ≤ · · · ≤ B(n) =
1
n
n
X
i=1
and
n
(1.3)
1X t
x
n i=1 i
!
n
1 X 1−t
x
n i=1 i
!
i=j
xi + (l − k + 1)
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1
n
n
X
i=1
2. For 1 ≤ j < k < l ≤ n (n ≥ 3), we have
(1.4) (k − j + 1)
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= C(n) ≤ C(n − 1) ≤ · · · ≤ C(j) ≤ C(j − 1) ≤ · · · ≤ C(1) =
k
X
Contents
xi
l
X
i=k
xi +
l
X
i=j
!
xti
l
X
i=j
!
x1−t
i
xi .
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Close
≤ (l − j + 1)
l
X
k
X
xi +
i=j
!
k
X
xti
i=j
!
x1−t
i
l
X
+
i=j
!
xti
l
X
!
.
x1−t
i
i=k
i=k
Corollary 1.2. Let xi > 0 (i = 1, 2, . . . , n, n ≥ 2) and p, q be any two positive
numbers.
1. For
"
k
n
X
X p+q
1
D(k) 4 2 k
xi +
n
i=1
!
n
X
xpi
i=1
Liang-Cheng Wang and Cai-Liang Li
xqi
vol. 8, iss. 3, art. 87, 2007
i=k+1
!
n
X
+
Some New Mean Value
Inequalities
!
xpi
k
X
!#
xqi
,
(k = 1, 2, . . . , n)
Title Page
i=1
i=k+1
Contents
and
"
n
X
1
E(j) 4 2 (n − j + 1)
xp+q
+
i
n
i=j
+
j−1
X
i=1
n
X
xpi
i=1
!
xpi
!
n
X
j−1
X
!
xqi
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i=1
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!#
xqi
,
(j = 1, 2, . . . , n),
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i=j
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we have
n
(1.5)
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1X p
x
n i=1 i
!
n
1X q
x
n i=1 i
Close
!
n
1 X p+q
x
= D(1) ≤ D(2) ≤ · · · ≤ D(k) ≤ D(k + 1) ≤ · · · ≤ D(n) =
n i=1 i
and
n
(1.6)
1X p
x
n i=1 i
!
n
1X q
x
n i=1 i
!
n
1 X p+q
= E(n) ≤ E(n−1) ≤ · · · ≤ E(j) ≤ E(j−1) ≤ · · · ≤ E(1) =
x .
n i=1 i
Some New Mean Value
Inequalities
Liang-Cheng Wang and Cai-Liang Li
2. For 1 ≤ j < k < l ≤ n (n ≥ 3), we have
vol. 8, iss. 3, art. 87, 2007
(1.7) (k − j + 1)
k
X
xp+q
+ (l − k + 1)
i
i=j
≤ (l − j + 1)
l
X
i=j
l
X
xp+q
+
i
xp+q
i
+
i=j
!
xpi
k
X
i=j
xpi
i=j
i=k
k
X
l
X
!
!
xqi
+
l
X
i=k
l
X
!
xqi
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i=j
!
xpi
l
X
i=k
Contents
!
xqi
.
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2.
Proof of Theorem and Corollary
Proof of Theorem 1.1. (1) Two equalities are clear in (1.2). To complete the proof of
(1.2), we only need to prove that B(k) ≤ B(k + 1) (1 ≤ k ≤ n − 1). Indeed, from
(1.1) we have
(2.1)
xtk+1
k
X
x1−t
i
=
i=1
k
X
xtk+1 x1−t
i
≤
i=1
k
X
Some New Mean Value
Inequalities
(txk+1 + (1 − t)xi ) ,
i=1
Liang-Cheng Wang and Cai-Liang Li
and
(2.2)
vol. 8, iss. 3, art. 87, 2007
x1−t
k+1
k
X
xti =
i=1
k
X
t
x1−t
k+1 xi ≤
k
X
((1 − t)xk+1 + txi ) .
Title Page
i=1
i=1
Contents
Using (2.1) and (2.2), after a simple manipulation we get
(2.3)
xtk+1
k
X
x1−t
i
+
i=1
x1−t
k+1
k
X
xti
≤ kxk+1 +
i=1
k
X
xi .
i=k+2
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i=1
For k = 1, 2, . . . , n − 1, by (2.3) we get
" k
!
!
! k
!#
n
n
n
X
X
X
X
X
1
1−t
1−t
t
t
B(k) = 2 k
xi +
xi
xi
+
xi
xi
n
i=1
i=1
i=1
i=k+1
i=k+1
" k
!
!
n
n
X
X
X
1
= 2 k
xi + xk+1 +
xti
x1−t
i
n
i=1
i=1
i=k+2
!
!
!
#
n
k
k
k
n
X
X
X
X
X
+
xti +
xti x1−t
xti
xi1−t + xtk+1
x1−t
i
k+1 +
i=1
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i=k+2
i=1
i=1
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"
k
k
k
X
X
X
1
1−t
1−t
t
= 2 k
xi + xk+1 + xk+1
xi + xk+1
xti
n
i=1
i=1
i=1
!
! k+1
!#
!
n
n
n
X
X
X
X
+
xti
x1−t
+
xti
x1−t
i
i
i=1
"
≤
1
k
n2
k
X
i=k+2
xi + xk+1 + kxk+1 +
i=1
+
Some New Mean Value
Inequalities
xi
Liang-Cheng Wang and Cai-Liang Li
i=1
n
X
!
xti
i=1
"
i=1
i=k+2
k
X
1
= 2 (k + 1)
n
k+1
X
!
n
X
x1−t
i
n
X
+
i=k+2
xi +
i=1
n
X
!
xti
xti
i=1
n
X
!
x1−t
i
i=k+2
vol. 8, iss. 3, art. 87, 2007
x1−t
i
i=1
i=k+2
!
k+1
X
!#
+
!
n
X
xti
i=k+2
k+1
X
!#
xi1−t
i=1
= B(k + 1).
By same arguments of proof for (1.2), we can also get inequalities in (1.3).
≤
l
k−1 X
X
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(txi + (1 − t)xs )
i=j s=k+1
= (l − k)
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i=j s=k+1
i=k+1
Contents
Page 8 of 17
(2) For 1 ≤ j < k < l ≤ n, from (1.1) we have
!
! k−1 l
k−1
l
X
X
X X
(2.4)
xti
x1−t
=
xti x1−t
s
i
i=j
Title Page
k−1
X
i=j
txi + (k − j)
l
X
i=k+1
(1 − t)xi
and
!
l
X
(2.5)
k−1
X
xti
!
x1−t
i
≤ (l − k)
k−1
X
i=j
i=k+1
(1 − t)xi + (k − j)
i=j
l
X
txi .
i=k+1
Using (2.4) and (2.5), after a simple manipulation we have
!
!
! k−1
!
k−1
l
l
X
X
X
X
(2.6)
xti
x1−t
+
xti
x1−t
i
i
i=j
i=k+1
Some New Mean Value
Inequalities
Liang-Cheng Wang and Cai-Liang Li
i=j
i=k+1
≤ (l − k)
k−1
X
xi + (k − j)
i=j
l
X
i=k+1
vol. 8, iss. 3, art. 87, 2007
xi .
Title Page
From (2.6) we obtain
(k − j + 1)
k
X
xi + (l − k + 1)
l
X
i=j
= (k − j + 1)
k
X
xi + (l − k + 1)
k
X
!
xti
k
X
!
x1−t
i
i=j
Contents
!
xi1−t
i=j
xi + (l − k)xk
l
X
+
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xti
l
X
l
X
!
xi1−t
l
X
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xti + xk
i=k+1
!
x1−t
i
Go Back
i=k+1
i=k+1
!
xti
!
i=k+1
x1−t
+ x1−t
i
k
i=k+1
+
l
X
i=j
l
X
k−1
X
xti
l
X
i=k+1
i=j
+ xtk
!
i=j
i=k
i=j
+
xi +
l
X
+
l
X
i=k+1
!
xti
k−1
X
i=j
!
xi1−t
≤ (k − j + 1)
k
X
xi + (l − k + 1)
i=j
+
k
X
!
xti
k
X
!
x1−t
i
k−1
X
xi +
l
X
!
xti
i=k
xi + (k − j)
i=j
i=j
+
i=j
+ (l − k)
= (l − j + 1)
xi + (l − k)xk
i=k+1
i=j
l
X
l
X
l
X
l
X
!
xi1−t
i=k
Some New Mean Value
Inequalities
xi
Liang-Cheng Wang and Cai-Liang Li
i=k+1
k
X
i=j
!
xti
k
X
i=j
!
x1−t
i
+
l
X
!
xti
i=k
l
X
vol. 8, iss. 3, art. 87, 2007
!
xi1−t
,
i=k
Title Page
which implies (1.4).
This completes the proof of Theorem 1.1.
Proof of Corollary 1.2. Replace t, 1 − t and xi in Theorem 1.1 by
respectively. We obtain Corollary 1.2.
Contents
p
, q
p+q p+q
,
and xp+q
i
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3.
Applications
Proposition 3.1. Let xir > 0 (i = 1, 2, . . . , n, n ≥ 2; r = 1, 2, . . . , m, m ≥ 2) and
t ∈ (0, 1). For

!
!t !  n
!1−t 
k
m
n
m
m
X
X
X
X
X
X
1


F (k) 4 2 k
xir +
xir
xir
n
r=1
r=1
r=1
i=1
i=1
i=k+1


!
!
!
t
1−t
m
k
n
m
X
X
X
X

 , (k = 1, 2, . . . , n)
+
xir
xir
i=k+1
r=1
i=1
Some New Mean Value
Inequalities
Liang-Cheng Wang and Cai-Liang Li
vol. 8, iss. 3, art. 87, 2007
r=1
Title Page
and
Contents



!t
!1−t
n X
n
h
h
m
X
X
X
X
1

 ,
xir
xjr
+
xtir x1−t
G(h) 4 2 
jr
n i=1 j=1
r=1
r=1
r=h+1
(h = 1, 2, . . . , m),
we have
(3.1)
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1
n2
n X
n
X
m
X
i=1 j=1
r=1
!
xtir x1−t
jr
= G(1) ≤ G(2) ≤ · · · ≤ G(h) ≤ G(h + 1) ≤ · · · ≤ G(m)
!t !  n
!1−t 
n
m
m
X
X
X
X
1
1

=
xir
xir
n i=1 r=1
n i=1 r=1
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Close
= F (1) ≤ F (2) ≤ · · · ≤ F (k) ≤ F (k + 1) ≤ · · · ≤ F (n)
n
m
1 XX
xir .
=
n i=1 r=1
Proof. For xir > 0, xjr > 0 (1 ≤ i, j ≤ n, r = 1, 2, . . . , m) and t ∈ (0, 1). We
write
!t
!1−t
h
h
m
X
X
X
+
xtir x1−t
(h = 1, 2, . . . , m).
P (i, j; h) 4
xir
xjr
jr
r=1
r=1
r=h+1
The first named author of this paper showed in [8] that the following chain of
Hölder’s inequalities holds
(3.2)
m
X
Some New Mean Value
Inequalities
Liang-Cheng Wang and Cai-Liang Li
vol. 8, iss. 3, art. 87, 2007
Title Page
Contents
xtir x1−t
jr
r=1
= P (i, j; 1)
≤ P (i, j; 2) ≤ · · · ≤ P (i, j; h) ≤ P (i, j; h + 1) ≤ · · · ≤ P (i, j; m)
!t m
!1−t
m
X
X
=
xir
xjr
.
r=1
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r=1
From the properties of inequality and (3.2), we have
!
n
n
m
1 X X X t 1−t
(3.3)
x x
n2 i=1 j=1 r=1 ir jr
=
n
n
1 XX
P (i, j; 1)
n2 i=1 j=1
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Close
n
n
1 XX
≤ 2
P (i, j; 2)
n i=1 j=1
n
n
1 XX
≤ ··· ≤ 2
P (i, j; h)
n i=1 j=1
n
n
1 XX
≤ 2
P (i, j; h + 1) ≤ · · ·
n i=1 j=1
n
n
1 XX
P (i, j; m)
≤ 2
n i=1 j=1
!t m
!1−t
n
n
m
X
1 XX X
= 2
xir
xjr
n i=1 j=1 r=1
r=1
!t !  n
!1−t 
n
m
m
X
X
X
X
1
1
.
xir
xir
=
n i=1 r=1
n i=1 r=1
Some New Mean Value
Inequalities
Liang-Cheng Wang and Cai-Liang Li
vol. 8, iss. 3, art. 87, 2007
Title Page
Contents
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It is easy to see that
(3.4)
n
n
1 XX
P (i, j; h),
G(h) = 2
n i=1 j=1
Go Back
h = 1, 2, . . . , m.
(3.3) and (3.4) imply inequalities between the first equality and the second equality
in (3.1).
P
Replacing xi in (1.2) by m
r=1 xir , we obtain inequalities between the third equality and the fourth equality in (3.1).
This completes the proof of Proposition 3.1.
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Close
Proposition 3.2. Let fi : [a, b] 7→ (0, +∞) (a < b) be continuous functions (i =
1, 2, . . . , n, n ≥ 2) and t ∈ (0, 1). For
" k Z
b
X
1
H(k) = 2 k
fi (x)dx
n
a
i=1
t !
n Z b
X
fi (x)dx
+
i=1
+
n
X
i=k+1
Z
n Z
X
a
b
fi (x)dx
a
k Z b
X
i=1
fi (x)dx
a
i=k+1
t !
1−t !
b
Some New Mean Value
Inequalities
Liang-Cheng Wang and Cai-Liang Li
1−t !#
fi (x)dx
vol. 8, iss. 3, art. 87, 2007
,
(k = 1, 2, . . . , n)
a
Title Page
and any y ∈ [a, b], we have
n
n Z b
1 XX
t
1−t
(3.5)
(fi (x)) (fj (x)) dx
n2 i=1 j=1
a
" n n Z
t Z y
1−t
y
1 XX
fi (x)dx
fj (x)dx
≤ 2
n i=1 j=1
a
a
#
Z b
+
(fi (x))t (fj (x))1−t dx
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y
n
≤
1X
n i=1
Z
t !
b
fi (x)dx
a
n
1X
n i=1
Z
1−t !
b
fi (x)dx
a
Close
= H(1) ≤ H(2) ≤ · · · ≤ H(k) ≤ H(k + 1) ≤ · · · ≤ H(n)
n Z
1X b
fi (x)dx.
=
n i=1 a
Proof. For 1 ≤ i, j ≤ n, t ∈ (0, 1), y ∈ [a, b] and continuous functions fi : [a, b] 7→
(0, +∞) (i = 1, 2, . . . , n; n ≥ 2), in [8], Wang also obtained the following refinement for the integral form of Hölder’s inequalities:
Z b
(3.6)
(fi (x))t (fj (x))1−t dx
a
Z y
t Z y
1−t Z b
≤
fi (x)dx
fj (x)dx
+
(fi (x))t (fj (x))1−t dx
a
Z
≤
a
t Z
b
fi (x)dx
a
y
Some New Mean Value
Inequalities
Liang-Cheng Wang and Cai-Liang Li
vol. 8, iss. 3, art. 87, 2007
Title Page
1−t
b
fj (x)dx
Contents
.
a
Using the properties of inequality and (3.6), we have
n
n Z b
1 XX
t
1−t
(fi (x)) (fj (x)) dx
n2 i=1 j=1
a
!
Z y
t Z y
1−t Z b
n
n
1 XX
≤ 2
fi (x)dx
fj (x)dx
+
(fi (x))t (fj (x))1−t dx
n i=1 j=1
a
a
y
Z
Z
t
1−t
n
n
b
b
1 XX
fi (x)dx
fj (x)dx
≤ 2
n i=1 j=1
a
a
t !
1−t !
n Z b
n Z b
X
X
1
1
=
fi (x)dx
fi (x)dx
,
n i=1
n i=1
a
a
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which is two inequalities of left
R bhand in (3.2).
Replacing xi in (1.2) by a fi (x)dx, we obtain inequalities between the two
equalities in (3.2).
This completes the proof of Proposition 3.2.
Remark 1. (3.1) and (3.2) are extensions of Hölder’s inequalities.
Some New Mean Value
Inequalities
Liang-Cheng Wang and Cai-Liang Li
vol. 8, iss. 3, art. 87, 2007
Title Page
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References
[1] D.S. MITRINOVIĆ
Beograd, 40 (1969).
AND
P.M. VASIĆ, Sredine, Matematička Biblioteka,
[2] D.S. MITRINOVIĆ, Analytic Inequalities, Springer-Verlag, Berlin/New York,
1970.
[3] J.-C. KUANG, Applied Inequalities, Shandong Science and Technology Press,
2004. (Chinese).
Some New Mean Value
Inequalities
Liang-Cheng Wang and Cai-Liang Li
vol. 8, iss. 3, art. 87, 2007
[4] L.-C. WANG, Convex Functions and Their Inequalities, Sichuan University
Press, Chengdu, China, 2001. (Chinese).
Title Page
[5] C.L. WANG, Inequalities of the Rado-Popoviciu type for functions and their
applications, J. Math. Anal. Appl., 100 (1984), 436–446.
[6] G.H. HARDY, J.E. LITTLEWOOD AND G PÓLYA, Inequalities, 2nd ed., Cambridge, 1952.
[7] FENG QI, Generalized weighted mean values with two parameters , Proc. R.
Soc. Lond. A., 454 (1998), 2723–2732.
[8] L.C. WANG, Two mappings related to Hölder’s inequalities, Univ. Beograd.
Publ. Elektrotehn. Fak. Ser. Mat., 15 (2004), 92–97.
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