Volume 8 (2007), Issue 3, Article 87, 8 pp.
ON SOME NEW MEAN VALUE INEQUALITIES
LIANG-CHENG WANG AND CAI-LIANG LI
S CHOOL OF M ATHEMATICS S CIENCE
C HONGQING I NSTITUTE OF T ECHNOLOGY
N O . 4 OF X INGSHENG L U
YANGJIA P ING 400050
C HONGQING , T HE P EOPLE ’ S R EPUBLIC OF C HINA .
wangliangcheng@163.com
C HENGDU E LECTROMECHANICAL C OLLEGE
C HENGDU 610031, S ICHUAN P ROVINCE
T HE P EOPLE ’ S R EPUBLIC OF C HINA .
dzlcl@163.com
Received 04 March, 2007; accepted 27 April, 2007
Communicated by P.S. Bullen
A BSTRACT. 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.
Key words and phrases: Mean value inequality, Hölder’s inequality, Continuous positive function, Extension.
2000 Mathematics Subject Classification. 26D15.
1. I NTRODUCTION AND M AIN R ESULTS
Let a > 0, b > 0 and t ∈ (0, 1). It is well-known that the following arithmetic-geometric
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 = 0,
(bi ∈ R, q ∈ N).
i=q+1
The first author is partially supported by the Key Research Foundation of the Chongqing Institute of Technology under Grant 2004DZ94.
073-07
2
L IANG -C HENG WANG AND C AI -L IANG L I
Theorem 1.1. Let xi > 0 (i = 1, 2, . . . , n; n ≥ 2) and t ∈ (0, 1).
(1) For the following
" k
!
!
n
n
X
X
X
1
x1−t
xi +
xti
B(k) 4 2 k
i
n
i=1
i=1
i=k+1
! k
!#
n
X
X
xti
x1−t
,
+
i
(k = 1, 2, . . . , n)
i=1
i=k+1
and
"
C(j) 4
1
(n − j + 1)
n2
n
X
n
X
xi +
i=j
j−1
X
!
xti
i=1
!
x1−t
i
i=1
j−1
+
X
!
n
X
xti
i=1
!#
x1−t
i
,
(j = 1, 2, . . . , n),
i=j
we have
n
(1.2)
1X t
x
n i=1 i
!
n
1 X 1−t
x
n i=1 i
!
n
1X
xi
n i=1
= B(1) ≤ B(2) ≤ · · · ≤ B(k) ≤ B(k + 1) ≤ · · · ≤ B(n) =
and
(1.3)
n
1X t
x
n i=1 i
!
n
1 X 1−t
x
n i=1 i
!
n
= C(n) ≤ C(n − 1) ≤ · · · ≤ C(j) ≤ C(j − 1) ≤ · · · ≤ C(1) =
1X
xi .
n i=1
(2) For 1 ≤ j < k < l ≤ n (n ≥ 3), we have
(1.4) (k − j + 1)
k
X
xi + (l − k + 1)
i=j
≤ (l − j + 1)
l
X
l
X
xi +
xi +
i=j
xti
i=j
i=k
l
X
!
k
X
!
xti
i=j
l
X
!
x1−t
i
i=j
k
X
!
xi1−t
i=j
+
l
X
i=k
!
xti
l
X
!
x1−t
.
i
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
n
X p+q
X
X
1
xi +
xpi
xqi
D(k) 4 2 k
n
i=1
i=1
i=k+1
! k
!#
n
X q
X
p
+
xi
xi
, (k = 1, 2, . . . , n)
i=k+1
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 87, 8 pp.
i=1
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O N S OME N EW M EAN VALUE I NEQUALITIES
3
and
"
n
X
1
xp+q
+
E(j) 4 2 (n − j + 1)
i
n
i=j
n
X
j−1
X
!
xpi
i=1
!
xqi
i=1
j−1
X
+
!
n
X
xpi
i=1
!#
xqi
,
(j = 1, 2, . . . , n),
i=j
we have
n
(1.5)
1X p
x
n i=1 i
!
n
1X q
x
n i=1 i
!
n
1 X p+q
= D(1) ≤ D(2) ≤ · · · ≤ D(k) ≤ D(k + 1) ≤ · · · ≤ D(n) =
x
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
(2) For 1 ≤ j < k < l ≤ n (n ≥ 3), we have
(1.7) (k − j + 1)
k
X
xp+q
i
+ (l − k + 1)
i=j
l
X
xp+q
i
l
X
+
≤ (l − j + 1)
l
X
i=j
i=k
l
X
!
xpi
k
X
xp+q
+
i
i=j
!
xpi
i=j
2. P ROOF OF T HEOREM
!
xqi
i=j
k
X
!
xqi
+
i=j
AND
l
X
i=k
!
xpi
l
X
!
xqi
.
i=k
C OROLLARY
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
(txk+1 + (1 − t)xi ) ,
i=1
and
(2.2)
x1−t
k+1
k
X
xti
=
i=1
k
X
t
x1−t
k+1 xi
≤
i=1
k
X
((1 − t)xk+1 + txi ) .
i=1
Using (2.1) and (2.2), after a simple manipulation we get
(2.3)
xtk+1
k
X
x1−t
i
+
i=1
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 87, 8 pp.
x1−t
k+1
k
X
i=1
xti
≤ kxk+1 +
k
X
xi .
i=1
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4
L IANG -C HENG WANG AND C AI -L IANG L I
For k = 1, 2, . . . , n − 1, by (2.3) we get
"
k
X
n
X
!
!
n
X
n
X
!
!#
k
X
1
k
xi +
xti
x1−t
+
xti
xi1−t
i
2
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
1−t
1−t
1−t
t
t
t
t
xi
xi
+ xk+1
xi
+
xi +
xi xk+1 +
B(k) =
i=1
i=k+2
i=1
i=k+2
i=1
"
=
k
k
k
X
X
X
1
1−t
1−t
t
k
x
+
x
+
x
x
+
x
xti
i
k+1
k+1
i
k+1
n2
i=1
i=1
i=1
!
!
! k+1
!#
n
n
n
X
X
X
X
+
xti
x1−t
+
xti
xi1−t
i
i=1
"
≤
1
k
n2
k
X
i=k+2
i=1
i=k+2
xi + xk+1 + kxk+1 +
i=1
k
X
xi
i=1
n
X
+
!
n
X
xti
i=1
"
1
= 2 (k + 1)
n
!
x1−t
i
+
i=k+2
k+1
X
xi +
i=1
n
X
!
n
X
xti
xti
i=1
n
X
!
x1−t
i
i=k+2
!#
xi1−t
i=1
i=k+2
!
k+1
X
+
!
n
X
xti
k+1
X
!#
x1−t
i
i=1
i=k+2
= B(k + 1).
By same arguments of proof for (1.2), we can also get inequalities in (1.3).
(2) For 1 ≤ j < k < l ≤ n, from (1.1) we have
(2.4)
k−1
X
!
xti
i=j
l
X
!
x1−t
i
=
k−1 X
l
X
xti x1−t
s
i=j s=k+1
i=k+1
≤
k−1 X
l
X
(txi + (1 − t)xs )
i=j s=k+1
= (l − k)
k−1
X
i=j
txi + (k − j)
l
X
(1 − t)xi
i=k+1
and
(2.5)
l
X
i=k+1
!
xti
k−1
X
!
x1−t
i
≤ (l − k)
i=j
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 87, 8 pp.
k−1
X
i=j
(1 − t)xi + (k − j)
l
X
txi .
i=k+1
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O N S OME N EW M EAN VALUE I NEQUALITIES
5
Using (2.4) and (2.5), after a simple manipulation we have
(2.6)
k−1
X
i=j
!
xti
l
X
!
x1−t
i
+
!
l
X
k−1
X
xti
x1−t
i
i=j
i=k+1
i=k+1
!
≤ (l − k)
k−1
X
l
X
xi + (k − j)
i=j
xi .
i=k+1
From (2.6) we obtain
(k − j + 1)
k
X
xi + (l − k + 1)
l
X
i=j
xi + (l − k + 1)
i=j
k
X
+ xtk
!
xti
k
X
x1−t
i
xti
!
l
X
x1−t
i
xti
x1−t
i
i=k+1
!
l
X
+
xti
k−1
X
l
X
!
x1−t
i
i=j
i=k+1
xi + (l − k + 1)
xi + (l − k)xk
i=k+1
!
xti
k
X
!
x1−t
i
+
i=j
+ (l − k)
k−1
X
xi +
l
X
!
xti
i=k
xi + (k − j)
i=j
i=j
!
l
X
xti + xk
i=k+1
i=j
= (l − j + 1)
!
i=k+1
!
k
X
l
X
xi1−t
i=j
i=k+1
l
X
i=j
+
l
X
+
x1−t
+ x1−t
i
k
i=j
k
X
!
xi + (l − k)xk
!
i=k+1
≤ (k − j + 1)
xti
l
X
i=j
l
X
i=j
l
X
k−1
X
!
i=k+1
i=j
+
xi +
i=k
k
X
= (k − j + 1)
+
l
X
l
X
l
X
!
xi1−t
i=k
xi
i=k+1
k
X
!
xti
i=j
k
X
i=j
!
x1−t
i
+
l
X
!
xti
i=k
l
X
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 87, 8 pp.
x1−t
,
i
i=k
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
tively. We obtain Corollary 1.2.
!
p
, q
p+q p+q
and xp+q
, respeci
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6
L IANG -C HENG WANG AND C AI -L IANG L I
3. A PPLICATIONS
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


xir +
xir
xir
F (k) 4 2 k
n
r=1
r=1
r=1
i=1
i=1
i=k+1


!
!
!
t
1−t
n
m
k
m
X
X
X
X

 , (k = 1, 2, . . . , n)
+
xir
xir
i=k+1
r=1
i=1
r=1
and



!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
n
n
m
1 X X X t 1−t
x x
n2 i=1 j=1 r=1 ir jr
(3.1)
!
= 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
= 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
xir
xjr
+
xtir x1−t
(h = 1, 2, . . . , m).
P (i, j; h) 4
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
m
X
(3.2)
xtir x1−t
jr = P (i, j; 1)
r=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
r=1
From the properties of inequality and (3.2), we have
!
n
n
m
n
n
1 XX
1 X X X t 1−t
(3.3)
x
x
=
P (i, j; 1)
n2 i=1 j=1 r=1 ir jr
n2 i=1 j=1
n
n
1 XX
≤ 2
P (i, j; 2)
n i=1 j=1
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 87, 8 pp.
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O N S OME N EW M EAN VALUE I NEQUALITIES
7
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
≤ 2
P (i, j; m)
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
It is easy to see that
(3.4)
n
n
1 XX
G(h) = 2
P (i, j; h),
n i=1 j=1
h = 1, 2, . . . , m.
(3.3) and (3.4) imply inequalities
Pm between the first equality and the second equality in (3.1).
Replacing xi in (1.2) by 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.
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
t ! X
1−t !
n Z b
n Z b
b
X
X
1
fi (x)dx +
fi (x)dx
fi (x)dx
H(k) = 2 k
n
a
a
a
i=1
i=1
i=k+1
t ! X
1−t !#
n Z b
k Z b
X
+
fi (x)dx
fi (x)dx
, (k = 1, 2, . . . , n)
i=k+1
a
i=1
a
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 Z b
y
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 b
Z b
t
1−t
n
n
1X
1X
≤
fi (x)dx
fi (x)dx
n i=1
n i=1
a
a
= H(1) ≤ H(2) ≤ · · · ≤ H(k) ≤ H(k + 1) ≤ · · · ≤ H(n)
n Z
1X b
=
fi (x)dx.
n i=1 a
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 87, 8 pp.
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8
L IANG -C HENG WANG AND C AI -L IANG L I
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
1−t
b
fj (x)dx
.
a
Using the properties of inequality and (3.6), we have
n Z b
n
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
t
1−t
≤ 2
fi (x)dx
fj (x)dx
+
(fi (x)) (fj (x)) dx
n i=1 j=1
a
a
y
t Z b
1−t
n
n Z b
1 XX
≤ 2
fi (x)dx
fj (x)dx
n i=1 j=1
a
a
t !
1−t !
n Z b
n Z b
1X
1X
=
fi (x)dx
fi (x)dx
,
n i=1
n i=1
a
a
which is two inequalities of left
R b hand 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 3.3. (3.1) and (3.2) are extensions of Hölder’s inequalities.
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[4] L.-C. WANG, Convex Functions and Their Inequalities, Sichuan University Press, Chengdu, China,
2001. (Chinese).
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Anal. Appl., 100 (1984), 436–446.
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[7] FENG QI, Generalized weighted mean values with two parameters , Proc. R. Soc. Lond. A., 454
(1998), 2723–2732.
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J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 87, 8 pp.
http://jipam.vu.edu.au/