Journal of Inequalities in Pure and
Applied Mathematics
SOME NEW INEQUALITIES SIMILAR TO HILBERT-PACHPATTE
TYPE INEQUALITIES
volume 4, issue 2, article 33,
2003.
ZHONGXUE LÜ
Department of Basic Science of Technology College,
Xuzhou Normal University, 221011,
People’s Republic of China.
E-Mail: lvzx1@163.net
Received 25 May, 2002;
accepted 7 April, 2003.
Communicated by: S.S. Dragomir
Abstract
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2000
Victoria University
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Abstract
In this paper, some new inequalities similar to Hilbert-Pachpatte type inequalities are given.
2000 Mathematics Subject Classification: 26D15.
Key words: Inequalities, Hilbert-Pachpatte inequalities, Hölder inequality.
Contents
1
2
3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Discrete Analogues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Some New Inequalities Similar
to Hilbert-Pachpatte Type
Inequalities
Zhongxue Lü
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1.
Introduction
In [1, Chap. 9], the well-known Hardy-Hilbert inequality is given as follows.
P
p
Theorem 1.1. Let p > 1, q > 1, p1 + 1q = 1, am , bn ≥ 0, 0 < ∞
n=1 an < ∞,
P∞ q
0 < n=1 bn < ∞. Then
! p1 ∞ ! 1q
∞ X
∞
∞
X
X
X
π
am b n
p
(1.1)
≤
a
bqn
m
λ
(m
+
n)
sin(π/p)
m=1 n=1
m=1
n=1
where
π
sin(π/p)
Some New Inequalities Similar
to Hilbert-Pachpatte Type
Inequalities
is best possible.
The integral analogue of the Hardy-Hilbert inequality can be stated as follows
R∞
Theorem 1.2. Let p > 1, q > 1, p1 + 1q = 1, f (x), g(y) ≥ 0, 0 < 0 f p (x)dx <
R∞
∞, 0 < 0 g q (y)dy < ∞. Then
1q
Z ∞
p1 Z ∞
Z ∞Z ∞
π
f (x)g(y)
p
q
dxdy ≤
f (x)dx
g (y)dy ,
(1.2)
x+y
sin(π/p) 0
0
0
0
where
π
sin(π/p)
is best possible.
In [1, Chap. 9] the following extension of Hardy-Hilbert’s double-series
theorem is given.
Theorem 1.3. Let p > 1, q > 1,
Then
∞ X
∞
X
am b n
m=1 n=1
(m + n)λ
1
p
+
≤K
1
q
1
p
≥ 1, 0 < λ = 2 − −
∞
X
m=1
! p1
apm
∞
X
n=1
1
q
+
1
q
≤ 1.
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! 1q
bqn
=
1
p
Zhongxue Lü
,
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where K = K(p, q) depends on p and q only.
The following integral analogue of Theorem 1.3 is also given in [1, Chap.
9].
Theorem 1.4. Under the same conditions as in Theorem 1.1 we have
Z ∞
p1 Z ∞
1q
Z ∞Z ∞
f (x)g(y)
p
q
dxdy ≤ K
f dx
g dy ,
(x + y)λ
0
0
0
0
where K = K(p, q) depends on p and q only.
The inequalities in Theorems 1.1 and 1.2 were studied by Yang and Kuang
(see [2, 3]). In [4, 5], some new inequalities similar to the inequalities given in
Theorems 1.1, 1.2, 1.3 and 1.4 were established.
In this paper, we establish some new inequalities similar to the HilbertPachpatte inequality.
Some New Inequalities Similar
to Hilbert-Pachpatte Type
Inequalities
Zhongxue Lü
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2.
Main Results
In what follows we denote by R the set of real numbers. Let N = {1, 2, . . . },
N0 = {0, 1, 2, . . . }. We define the operator ∇ by ∇u(t) = u(t) − u(t − 1) for
any function u defined on N . For any function u(t) : [0, ∞) → R, we denote
by u0 the derivatives of u.
First we introduce some Lemmas.
Lemma 2.1. (see [2]). Let p > 1, q > 1, p1 + 1q = 1, λ > 2 − min{p, q}, define
the weight function ω1 (q, x) as
2−λ
Z ∞
x q
1
dy, x ∈ [0, ∞).
ω1 (q, x) :=
(x + y)λ y
0
Some New Inequalities Similar
to Hilbert-Pachpatte Type
Inequalities
Zhongxue Lü
Then
(2.1)
ω1 (q, x) = B
q+λ−2 p+λ−2
,
q
p
Title Page
x1−λ ,
Contents
where B(p, q) is β-function.
Lemma 2.2. (see [3]). Let p > 1, q > 1, p1 + 1q = 1, λ > 2 − min{p, q}, define
the weight function ω2 (q, x) as
2−λ
Z ∞
x q
1
ω2 (q, x) :=
dy, x ∈ [0, ∞).
xλ + y λ y
0
Then
(2.2)
1
ω2 (q, x) = B
λ
q+λ−2 p+λ−2
,
qλ
pλ
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x
1−λ
.
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Lemma 2.3. Let p > 1, q > 1,
weight function ω3 (q, m) as
ω3 (q, m) :=
∞
X
n=1
1
p
+
1
q
= 1, λ > 2 − min{p, q}, define the
m 2−λ
1
q
, m ∈ {1, 2, . . . }.
λ
(m + n)
n
Then
(2.3)
ω3 (q, m) < B
q+λ−2 p+λ−2
,
q
p
m1−λ ,
where B(p, q) is β-function.
Some New Inequalities Similar
to Hilbert-Pachpatte Type
Inequalities
Zhongxue Lü
Proof. By Lemma 2.1,we have
2−λ
m q
1
ω3 (q, m) <
dy
(m + y)λ y
0
q+λ−2 p+λ−2
=B
,
m1−λ .
q
p
Z
∞
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The proof is completed.
Lemma 2.4. Let p > 1, q > 1,
weight function ω4 (q, m) as
ω4 (q, m) :=
Title Page
∞
X
n=1
1
p
+
1
q
= 1, λ > 2 − min{p, q}, define the
m 2−λ
1
q
, m ∈ {1, 2, . . . }.
λ
λ
m +n
n
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Then
1
ω4 (q, m) < B
λ
(2.4)
q+λ−2 p+λ−2
,
qλ
pλ
m1−λ .
Proof. By Lemma 2.2, we have
2−λ
m q
1
ω4 (q, m) <
dy
mλ + y λ y
0
1
q+λ−2 p+λ−2
= B
,
m1−λ .
λ
qλ
pλ
Z
∞
Zhongxue Lü
The proof is completed.
Our main result is given in the following theorem.
1
p
1
q
Theorem 2.5. Let p > 1, + = 1, and f (x), g(y) be real-valued continuous
functions defined on [0, ∞), respectively, and let f (0) = g(0) = 0, and
Z ∞Z x
Z ∞Z y
p
q
0
0<
|f (τ )| dτ dx < ∞, 0 <
|g 0 (δ)| dδdy < ∞.
0
0
Some New Inequalities Similar
to Hilbert-Pachpatte Type
Inequalities
0
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Then
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∞
∞
|f (x)| |g(y)|
dxdy
(2.5)
p−1
(qx
+ py q−1 )(x+y)
0
0
Z ∞ Z x
p1 Z ∞ Z y
1q
π
p
q
0
0
≤
|f (τ )| dτ dx
|g (δ)| dδdy .
sin(π/p)pq
0
0
0
0
Z
Z
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In particular, when p = q = 2, we have
Z
∞
Z
(2.6)
0
0
∞
|f (x)| |g(y)|
dxdy
(x + y)2
Z ∞ Z x
12 Z ∞ Z y
12
π
2
2
0
0
≤
|f (τ )| dτ dx
|g (δ)| dδdy .
2
0
0
0
0
Proof. From the hypotheses, we have the following identities
Z x
(2.7)
f (x) =
f 0 (τ )dτ,
Some New Inequalities Similar
to Hilbert-Pachpatte Type
Inequalities
0
Zhongxue Lü
and
y
Z
(2.8)
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g 0 (δ)dδ
g(y) =
0
Contents
for x, y ∈ (0, ∞). From (2.7) and (2.8) and using Hölder’s integral inequality,
respectively, we have
(2.9)
|f (x)| ≤ x
1
q
x
Z
0
p
p1
|f (τ )| dτ
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0
and
(2.10)
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|g(y)| ≤ y
1
p
Z
y
0
q
1q
Page 8 of 19
|g (δ)| dδ
0
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for x, y ∈ (0, ∞). From (2.9) and (2.10) and using the elementary inequality
z1 z2 ≤
(2.11)
z1p z2q
1 1
+ , z1 ≥ 0, z2 ≥ 0, + = 1, p > 1,
p
q
p q
we observe that
1
q
|f (x)| |g(y)| ≤ x y
1
p
Z
x
p
|f 0 (τ )| dτ
p1 Z
0
≤
(2.12)
x
p−1
p
+
y
q
|g 0 (δ)| dδ
1q
0
y
q−1
Z
q
x
p
|f 0 (τ )| dτ
p1 Z
y
q
|g 0 (δ)| dδ
1q
Some New Inequalities Similar
to Hilbert-Pachpatte Type
Inequalities
0
0
Zhongxue Lü
for x, y ∈ (0, ∞). From (2.12) we observe that
1
|f (x)| |g(y)|
≤
p−1
q−1
qx
+ py
pq
(2.13)
Z
x
0
p
p1 Z
|f (τ )| dτ
0
y
0
q
|g (δ)| dδ
Title Page
1q
.
0
Hence
Z
∞
Z
(2.14)
0
0
∞
|f (x)| |g(y)|
dxdy
+ py q−1 )(x + y)
1 R y
1
Z ∞Z ∞ Rx 0
|f (τ )|p dτ p 0 |g 0 (δ)|q dδ q
1
0
≤
dxdy.
pq 0
x+y
0
(qxp−1
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By Hölder’s integral inequality and (2.1), we have
Z
∞
∞
Z
0
0
(2.15)
p1 R y 0
1q
q
p
0
|f
(τ
)|
dτ
|g
(δ)|
dδ
0
0
dxdy
x+y
p1 1 R y 0
1q 1
Z ∞Z ∞ Rx 0
p
q
pq
(τ
)|
dτ
|f
|g
(δ)|
dδ
x
y pq
0
0
=
dxdy
1
1
y
x
0
0
(x + y) p
(x + y) q
! p1
1q
Z ∞Z ∞ Rx 0
p
|f
(τ
)|
dτ
x
0
dxdy
≤
x+y
y
0
0
1q
Z ∞ Z ∞ R y 0
|g (δ)|q dδ y p1
0
dxdy
×
x+y
x
0
0
Z ∞ Z x
p1 Z ∞ Z y
1q
π
p
q
0
0
≤
|f (τ )| dτ dx
|g (δ)| dδdy
sin(π/p)
0
0
0
0
Rx
by (2.14) and (2.15), we get (2.5). The proof of Theorem 2.5 is complete.
In a similar way to the proof of Theorem 2.5, we can prove the following
theorems.
1
p
1
q
Theorem 2.6. Let p > 1, + = 1, λ > 2 − min{p, q}, and f (x), g(y) be
real-valued continuous functions defined on [0, ∞),respectively, and let f (0) =
g(0) = 0, and
Z ∞Z x
Z ∞Z y
p
q
1−λ
0
0<
x
|f (τ )| dτ dx < ∞, 0 <
y 1−λ |g 0 (δ)| dδdy < ∞,
0
0
0
0
Some New Inequalities Similar
to Hilbert-Pachpatte Type
Inequalities
Zhongxue Lü
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then
∞
Z
∞
|f (x)| |g(y)|
dxdy
(qxp−1 + py q−1 )(x + y)λ
0
q+λ−2 p+λ−2 Z
p1
∞Z x
B
, p
q
p
1−λ
0
≤
x
|f (τ )| dτ dx
pq
0
0
Z ∞ Z y
1q
q
×
y 1−λ |g 0 (δ)| dδdy .
(2.16)
0
Z
0
0
In particular, when p = q = 2,
Some New Inequalities Similar
to Hilbert-Pachpatte Type
Inequalities
Zhongxue Lü
∞
Z
Z
(2.17)
0
0
∞
|f (x)| |g(y)|
dxdy
(x + y)1+λ
Z ∞ Z x
12
B λ2 , λ2
2
1−λ
0
≤
x
|f (τ )| dτ dx
2
0
0
Z ∞ Z y
12
2
1−λ 0
×
y
|g (δ)| dδdy .
0
0
0
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Theorem 2.7. Let p > 1, p1 + 1q = 1, λ > 2 − min{p, q}, and f (x), g(y) be
real-valued continuous functions defined on [0, ∞), respectively, and let f (0) =
g(0) = 0, and
Z ∞Z x
Z ∞Z y
p
q
1−λ
0
0<
x
|f (τ )| dτ dx < ∞, 0 <
y 1−λ |g 0 (δ)| dδdy < ∞.
0
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Then
Z
(2.18)
0
∞
∞
|f (x)| |g(y)|
dxdy
(qxp−1 + py q−1 )(xλ + y λ )
0
q+λ−2 p+λ−2 Z
p1
∞Z x
B
, pλ
qλ
p
1−λ
0
≤
x
|f (τ )| dτ dx
λpq
0
0
Z ∞ Z y
1q
q
×
y 1−λ |g 0 (δ)| dδdy .
Z
0
0
In particular, when p = q = 2,
Some New Inequalities Similar
to Hilbert-Pachpatte Type
Inequalities
Zhongxue Lü
Z
∞
Z
(2.19)
0
0
∞
|f (x)| |g(y)|
dxdy
+ y λ )(x + y)
Z ∞ Z x
12
π
2
1−λ
0
≤
x
|f (τ )| dτ dx
2λ
0
0
Z ∞ Z y
12
2
1−λ 0
×
y
|g (δ)| dδdy .
(xλ
0
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3.
Discrete Analogues
Theorem 3.1. Let p > 1, p1 + 1q = 1, and {a(m)} and {b(n)} be two sequences
where m,
N0 , and a(0) = b(0) = 0, and 0 <
P∞ Pofm real numbers
Pn∞ ∈ P
p
n
q
|∇a(τ
)|
<
∞,
0
<
m=1
τ =1
n=1
δ=1 |∇b(δ)| < ∞, then
(3.1)
∞ X
∞
X
m=1 n=1
(qmp−1
|am | |bn |
+ pnq−1 )(m + n)
π
≤
sin(π/p)pq
∞ X
m
X
! p1
apk
m=1 k=1
∞ X
n
X
! 1q
bqr
.
n=1 r=1
Some New Inequalities Similar
to Hilbert-Pachpatte Type
Inequalities
Zhongxue Lü
In particular, when p = q = 2, we have
(3.2)
∞ X
∞
X
|am | |bn |
π
≤
2
(m + n)
2
m=1 n=1
∞ X
m
X
! 12
a2k
m=1 k=1
∞ X
n
X
! 12
b2r
Title Page
.
n=1 r=1
Proof. From the hypotheses, it is easy to observe that the following identities
hold
(3.3)
am =
m
X
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and
(3.4)
bn =
n
X
δ=1
Page 13 of 19
∇b(δ)
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for m, n ∈ N. From (3.3) and (3.4) and using Hölder’s inequality, we have
1
q
|am | ≤ m
(3.5)
m
X
! p1
|∇a(τ )|p
,
τ =1
and
|bn | ≤ n
(3.6)
1
p
n
X
! 1q
|∇b(δ)|q
Some New Inequalities Similar
to Hilbert-Pachpatte Type
Inequalities
δ=1
for m, n ∈ N. From (3.5) and (3.6) and using the elementary inequality (2.11),
we observe that
! p1
! 1q
m
n
X
X
1
1
p
q
|∇b(δ)|
|am | |bn | ≤ m q n p
|∇a(τ )|
τ =1
(3.7)
≤
δ=1
mp−1 nq−1
+
p
q
X
m
! p1
p
|∇a(τ )|
τ =1
n
X
! 1q
q
|∇b(δ)|
δ=1
(3.8)
|am | |bn |
p−1
qm
+ pnq−1
≤
1
pq
τ =1
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! p1
|∇a(τ )|p
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for m, n ∈ N. From (3.7), we observe that
m
X
Zhongxue Lü
n
X
δ=1
! 1q
|∇b(δ)|q
.
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Hence
(3.9)
∞ X
∞
X
|am | |bn |
+ pnq−1 )(m + n)
m=1 n=1
P
∞ ∞
p p1 Pn
q 1q
1 XX( m
τ =1 |∇a(τ )| ) (
δ=1 |∇b(δ)| )
≤
.
pq m=1 n=1
m+n
(qmp−1
By the Hölder inequality and (2.3)
P
∞ X
∞
p p1 Pn
q 1q
X
( m
δ=1 |∇b(δ)| )
τ =1 |∇a(τ )| ) (
m+n
m=1 n=1
1
P
P
∞ X
∞
p p1 X
( m
m pq1 ( nδ=1 |∇b(δ)|q ) q n pq1
τ =1 |∇a(τ )| )
=
1
1
m
n
(m + n) q
(m + n) p
m=1 n=1
!1
∞ X
∞ Pm
p 1q p
X
|∇a(τ
)|
m
τ =1
≤
m+n
n
m=1 n=1
!1
∞ X
∞ Pn
q p1 q
X
|∇b(δ)|
n
δ=1
×
m+n
m
m=1 n=1
! p1 ∞ n
! 1q
∞ X
m
X
X
X
π
(3.10)
<
|∇a(τ )|p
|∇b(δ)|q
sin(π/p) m=1 τ =1
n=1 δ=1
by (3.9) and (3.10), we get (3.1). The proof of Theorem 3.1 is complete.
Some New Inequalities Similar
to Hilbert-Pachpatte Type
Inequalities
Zhongxue Lü
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In a similar manner to the proof of Theorem 3.1, we can prove the following
theorems.
Theorem 3.2. Let p > 1, p1 + 1q = 1, λ > 2−min{p, q}, and {a(m)} and {b(n)}
be two sequences of real numbers where m, n ∈ N0 ,and a(0) = b(0) = 0, and
0<
∞ X
m
X
m1−λ |∇a(τ )|p < ∞,
m=1 τ =1
0<
n
∞ X
X
Some New Inequalities Similar
to Hilbert-Pachpatte Type
Inequalities
n1−λ |∇b(δ)|q < ∞,
n=1 δ=1
Zhongxue Lü
then
(3.11)
∞ X
∞
X
m=1 n=1
≤
|am | |bn |
(qmp−1 + pnq−1 )(m + n)λ
q+λ−2 p+λ−2
∞ X
m
B
, p
X
q
pq
Title Page
Contents
! p1
m=1 τ =1
×
∞ X
n
X
n=1 δ=1
In particular, when p = q = 2, we have
(3.12)
∞ X
∞
X
|am | |bn |
(m + n)1+λ
m=1 n=1
JJ
J
m1−λ |∇a(τ )|p
! 1q
n1−λ |∇b(δ)|q
.
II
I
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≤
B
λ λ
,
2 2
2
∞ X
m
X
! 12
∞ X
n
X
2
m1−λ |∇a(τ )|
m=1 τ =1
! 12
2
n1−λ |∇b(δ)|
.
n=1 δ=1
Theorem 3.3. Let p > 1, p1 + 1q = 1, λ > 2−min{p, q}, and {a(m)} and {b(n)}
be two sequences of real numbers where m, n ∈ N0 , and a(0) = b(0) = 0, and
0<
m
∞ X
X
m1−λ |∇a(τ )|p < ∞,
m=1 τ =1
0<
∞ X
n
X
Some New Inequalities Similar
to Hilbert-Pachpatte Type
Inequalities
q
n1−λ |∇b(δ)| < ∞,
n=1 δ=1
Zhongxue Lü
then
(3.13)
∞ X
∞
X
m=1 n=1
≤
|am | |bn |
+ pnq−1 )(mλ + nλ )
p+λ−2
∞ X
m
B q+λ−2
,
X
qλ
pλ
Title Page
(qmp−1
λpq
Contents
! p1
JJ
J
m1−λ |∇a(τ )|p
m=1 τ =1
×
∞ X
n
X
! 1q
n1−λ |∇b(δ)|q
II
I
Go Back
.
Close
n=1 δ=1
In particular, when p = q = 2, we have
(3.14)
∞ X
∞
X
m=1 n=1
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|am | |bn |
(m + n)(mλ + nλ )
J. Ineq. Pure and Appl. Math. 4(2) Art. 33, 2003
http://jipam.vu.edu.au
π
≤
2λ
∞ X
m
X
m=1 τ =1
! 12
2
m1−λ |∇a(τ )|
∞ X
n
X
! 12
2
n1−λ |∇b(δ)|
.
n=1 δ=1
Some New Inequalities Similar
to Hilbert-Pachpatte Type
Inequalities
Zhongxue Lü
Title Page
Contents
JJ
J
II
I
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References
[1] G.H. HARDY, J.E. LITTLEWOOD
bridge Univ. Press, London, 1952.
AND
G. PÓLYA, Inequalities, Cam-
[2] BICHENG YANG, A generalized Hilbert’s integral inequality with the best
constant, Ann. of Math.(Chinese), 21A:4 (2000), 401–408.
[3] JICHANG KUANG,On new extensions of Hilbert’s integral inequality, J.
Math. Anal. Appl., 235 (1999), 608–614.
[4] B.G. PACHPATTE, Inequalities similar to certain extensions of Hilbert’s
inequality, J.Math. Anal. Appl., 243 (2000), 217–227.
[5] B.G. PACHPATTE, On some new inequalities similar to Hilbert’s inequality, J. Math. Anal. Appl., 226 (1998), 166–179.
[6] D.S. MITRINOVIĆ, Analytic Inequalities, Springer-Verlag, Berlin/New
York,1970.
[7] E.F. BECKENBACH
Berlin, 1983.
AND
R. BELLMAN, Inequalities, Springer-Verlag,
Some New Inequalities Similar
to Hilbert-Pachpatte Type
Inequalities
Zhongxue Lü
Title Page
Contents
JJ
J
II
I
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