Journal of Inequalities in Pure and
Applied Mathematics
FOUR INEQUALITIES SIMILAR TO HARDY-HILBERT’S
INTEGRAL INEQUALITY
volume 7, issue 2, article 76,
2006.
W.T. SULAIMAN
College of Computer Sciences and Mathematics
University of Mosul, Iraq.
Received 22 September, 2005;
accepted 05 January, 2006.
Communicated by: B. Yang
EMail: waadsulaiman@hotmail.com
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
Four new different types of inequalities similar to Hardy-Hilbert’s inequality are
given.
2000 Mathematics Subject Classification: 26D15.
Key words: Hardy-Hilbert integral inequality.
Four Inequalities Similar To
Hardy-Hilbert’s Integral
Inequality
The author is grateful to the referee who read through the paper very carefully and
correct many typing mistakes.
W.T. Sulaiman
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
New Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
6
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1.
Introduction
Suppose
f and g are real functions, such that 0 <
R ∞ that
2
0 < 0 g (t)dt < ∞, then
Z
∞
Z
(1.1)
0
0
∞
f (x)g(y)
<π
x+y
Z
∞
2
Z
∞
f (t)dt
0
R∞
0
f 2 (t)dt < ∞ and
12
g (t)dt ,
2
0
where π P
is best possible. If (an )P
and (bn ) are sequences of real numbers such
∞
2
2
that 0 < ∞
a
<
∞
and
0
<
n=1 n
n=1 bn < ∞, then
(1.2)
∞ X
∞
∞
∞
X
X
X
am b n
2
<π
an
b2n
m+n
n=1 m=1
n=1
n=1
! 12
Four Inequalities Similar To
Hardy-Hilbert’s Integral
Inequality
W.T. Sulaiman
.
Title Page
The inequalities (1.1) and (1.2) are called Hilbert’s inequalities. These inequalities play an important role in analysis (cf. [1, Chap. 9]). In their recent papers
Hu [5] and Gao [3] gave two distinct improvements of (1.1) and Gao [4] gave a
strengthened version of (1.2).
The following definitions are given:
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r+λ−2
ϕλ (r) =
(r = p, q),
r
kλ (p) = B (ϕλ (p), ϕλ (q)) ,
and B is the beta function.
Recently, by introducing some parameters, Yang and Debnath [2] gave the
following extensions:
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Theorem A. If f, g ≥ 0, p > 1,
1
p
+
1
q
= 1, λ > 2 − min {p, q} , such that
∞
Z
0<
t
∞
Z
1−λ p
f (t)dt < ∞
and
t1−λ g q (t)dt < ∞,
0<
0
0
then
Z
∞
∞
Z
(1.3)
0
0
<
f (x)g(y)
(Ax + By)λ
kλ (p)
ϕ
λ
A (p) B ϕλ (q)
dxdy
Z
∞
x1−λ f p (x)dx
0
p1 Z
∞
y 1−λ g q (y)dy
1q
,
0
where the constant factor [kλ (p)/A
ϕλ (p)
B
ϕλ (q)
∞
y (λ−1)(p−1)
Z
∞
f (x)
!p
dx dy
(Ax + By)λ
p Z ∞
kλ (p)
<
x1−λ f p (x)dx,
ϕ
(p)
ϕ
(q)
λ
λ
A
B
0
p
where the constant factor kλ (p)/Aϕλ (p) B ϕλ (q) is the best possible. The inequalities (1.3) and (1.4) are equivalent.
(1.4)
0
0
W.T. Sulaiman
] is the best possible.
Theorem B. If f ≥ 0, p > 1, p1 + 1q = 1, λ > 2 − min {p, q} , A, B > 0 such
R∞
that 0 < 0 t1−λ f p (t)dt < ∞, then
Z
Four Inequalities Similar To
Hardy-Hilbert’s Integral
Inequality
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Theorem C. If an , bn > 0 (n ∈ N), p > 1, p1 + 1q = 1, 2 − min {p, q} < λ < 2,
A, B > 0 such that
0<
∞
X
n1−λ apn
<∞
n=1
and
0<
∞
X
n1−λ bqn < ∞,
n=1
then
(1.5)
∞ X
∞
X
am b n
n=1 m=1
(Am + Bn)λ
! p1 ∞
! 1q
∞
X
X
kλ (p)
n1−λ bqn
,
< ϕ (p) ϕ (q)
n1−λ apn
A λ B λ
n=1
n=1
where the constant factor kλ (p)/Aϕλ (p) B ϕλ (q) is the best possible.
Theorem D. If an ≥ 0 (n ∈ N), p > 1, p1 + 1q = 1, 2 − min {p, q} < λ ≤ 2,
P
1−λ p
A, B > 0 such that 0 < ∞
an < ∞, then
n=1 n
!p
∞
∞
X
X
a
m
(1.6)
n(λ−1)(p−1)
λ
n=1
m=1 (Am + Bn)
p X
∞
kλ (p)
<
n1−λ apn ,
Aϕλ (p) B ϕλ (q) n=1
p
where the constant factor kλ (p)/Aϕλ (p) B ϕλ (q) is the best possible. The inequalities (1.5) and (1.6) are equivalent.
Four Inequalities Similar To
Hardy-Hilbert’s Integral
Inequality
W.T. Sulaiman
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2.
New Inequalities
The aim of this paper is to give the following results:
Theorem 2.1. Let ln f , ln g be convex for nonnegative functions f and g such
that f (0) = g(0) = 0, f (∞) = g(∞) = ∞, f 0 (s) ≥ 0, g 0 (s) ≥ 0, s ∈
{xp , y q } . Let λ > max {p, q} , p > 1, p1 + 1q = 1. Let
Z
0<
∞ −p2 /q 2
t
p
[f 0 (t)] q
0
Z
0<
∞ −q 2 /p2
t
0
0
[g(tq )]2−λ+q/p
q
0
then we have
Z ∞Z
(2.1)
[f (tp )]2−λ+p/q
[g 0 (t)] p
dt < ∞,
dt < ∞,
Four Inequalities Similar To
Hardy-Hilbert’s Integral
Inequality
W.T. Sulaiman
Title Page
∞
f (xy)g(xy)
dxdy
(f (xp ) + g(y q ))λ
1
1
1
p (p, λ − p) B q (q, λ − q)
≤ √
B
√
p p q q
! p1
Z ∞ −p2 /q2
t
[f (tp )]2−λ+p/q
×
dt
p
[f 0 (t)] q
0
! 1q
Z ∞ −q2 /p2
t
[g(tq )]2−λ+q/p
×
dt
.
q
[g 0 (t)] p
0
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Proof. Since ln f is convex and xy ≤
ln f (xy)
f (xy) = e
ln f
≤e
Therefore, we have
Z ∞Z ∞
f (xy)g(xy)
0
(f (xp ) + g(y q ))λ
0
1
∞
Z
∞
Z
q
xp
+ yq
p
xp
p
+
≤e
yq
,
q
then
ln f (xp )
ln f (y q )
+
p
q
0
1
f p (xp )g q (y q ) [g (y
1
q )] p
1
[f 0 (xp )] q
q−1
p
p−1
x q
y
1
∞Z
∞
≤
0
×
λ
q
1
dxdy
Four Inequalities Similar To
Hardy-Hilbert’s Integral
Inequality
W.T. Sulaiman
f (xp )g p/q (y q )g 0 (y q )y q−1
0
= M pN q,
p−1
q
q−1
y p
x
(f (xp ) + g(y q )) q
! p1
dxdy
p
x(p−1)p/q [f 0 (xp )] q (f (xp ) + g(y q ))λ
Z ∞Z ∞
f (y q )g q/p (xp )f 0 (xp )xp−1
0
1
1
p )] q
1
[g 0 (y q )] p
λ
Z
0
1
f q (y q )g p (xp ) [f (x
(f (xp ) + g(y q )) p
0
1
dxdy
≤
0
1
= f p (xp )f q (y q ).
0
y (q−1)q/p [g 0 (y q )] p (f (xp ) + g(y q ))λ
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! 1q
dxdy
say.
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1
M=
q
Z
∞
x
(1−p)p/q
2−λ+p/q
p
[f (x )]
p
0
[f 0 (x)] q
Z
dx
0
∞
g(y q )
f (xp )
pq
1+
q−1
g 0 (y q ) qy
f (xp )
dy
λ
q
g(y )
f (xp )
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J. Ineq. Pure and Appl. Math. 7(2) Art. 76, 2006
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1
=
q
∞
Z
2 /q 2
x−p
[f (xp )]2−λ+p/q
Z
∞
up/q
dx
du
p
(1 + u)λ
[f 0 (x)] q
0
Z ∞ −p2 /q2
1
x
[f (xp )]2−λ+p/q
= B (p, λ − p)
dx.
p
q
[f 0 (x)] q
0
0
Similarly,
1
N = B (q, λ − q)
p
Therefore
Z ∞Z ∞
0
0
Z
∞
y −q
2 /p2
[g(y q )]2−λ+q/p
q
0
[g 0 (y)] p
dy.
Four Inequalities Similar To
Hardy-Hilbert’s Integral
Inequality
W.T. Sulaiman
f (xy)g(xy)
dxdy
(f (xp ) + g(y q ))λ
1
1
1
p (p, λ − p) B q (q, λ − q)
≤ √
B
√
q p p q
! p1
Z ∞ −p2 /q2
p 2−λ+p/q
t
[f (t )]
×
t
dt
p
[f 0 (t)] q
0
! 1q
Z ∞ −q2 /p2
q 2−λ+q/p
t
[g(t )]
.
dt
×
q
[g 0 (t)] p
0
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Theorem 2.2. Let f, g, h be nonnegative functions, h(x, y) is homogeneous of
order n such that h(0, 1) = h(1, 0) = 0, h(∞, 1) = h(1, ∞) = ∞. Let p > 1,
J. Ineq. Pure and Appl. Math. 7(2) Art. 76, 2006
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1
p
+
1
q
= 1,
0 < 1 + µ − r < λ,
where hx = dh/dx, 0 <
Z
(2.2)
∞
Z
∞
r∈
R∞
0
n
p q
,
q p
o
,
hx (x, y) ≥ 0,
tp/q f p (t)dt < ∞, 0 <
R∞
0
hy (x, y) ≥ 0,
tq/p g q (t)dt < ∞, then
f (x)g(y)hµ (x, y)
dxdy
(1 + h(x, y))λ
1
1 p1
p
p
q
q
q
≤ B
1 + µ− , λ − 1 − µ+
B
1 + µ− , λ − 1 − µ−
n
q
q
p
p
1q
p1 Z ∞
Z ∞
q
p
q
p
p−1
q−1
t g (t)dt .
×
t f (t)dt
0
0
0
Proof. Observe that
Z ∞Z ∞
f (x)g(y)hµ (x, y)
0
0
λ
0
1
q
=M N ,
W.T. Sulaiman
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dxdy
(1 + h(x, y))
! p1
Z ∞Z ∞ p
µ
f (x)h (x, y)hy (x, y)
≤
dxdy
p/q
hx (x, y) (1 + h(x, y))λ
0
0
! 1q
Z ∞Z ∞ q
µ
g (y)h (x, y)hx (x, y)
×
dxdy
q/p
hy (x, y) (1 + h(x, y))λ
0
0
1
p
Four Inequalities Similar To
Hardy-Hilbert’s Integral
Inequality
say.
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Then
∞
Z
Z
p
M=
∞
f (x)dx
0
0
hµ (x, y)hy (x, y)
p/q
hx (x, y) (1 + h(x, y))λ
dy.
Let y = xv, dy = xdv and hence
dh(x, xv)
dh(1, v)
dh(1, v)
= xn
= xn−1
= xn−1 hv (1, v),
dy
dy
dv
dh(x, xv)
d n
hx (x, y) =
=
x h(1, v) = nxn−1 h(1, v),
dx
dx
hy (x, y) =
therefore
1
M=
Z
∞
x
p/q−1 p
Z
f (x)dx
∞
[xn h(1, v)]µ−p/q xn hv (1, v)
(1 + xn h(1, v))λ
Z ∞
p
1
p
= p/q B 1 + µ− , λ − 1 − µ+
xp/q−1 f p (x)dx.
n
q
q
0
np/q
0
W.T. Sulaiman
dv
0
q
q
N = q/p B 1 + µ − , λ − 1 − µ +
n
p
p
This implies
Z
0
∞Z
0
∞
Z
0
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1
Four Inequalities Similar To
Hardy-Hilbert’s Integral
Inequality
∞
y q/p−1 g q (y)dy.
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f (x)g(y)hµ (x, y)
λ
(1 + h(x, y))
dxdy
Page 10 of 18
J. Ineq. Pure and Appl. Math. 7(2) Art. 76, 2006
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1 1
≤ Bp
n
1
p
p
q
q
q
B
1 + µ − , λ − 1 − µ+
1 + µ − , λ − 1 − µ+
q
q
p
p
Z ∞
p1 Z ∞
1q
p/q−1 p
q/p−1 q
×
t
f (t)dt
t
g (t)dt .
0
0
The following lemma is needed for the coming result.
Lemma 2.3. Let s ≥ 1, 0 < 1 + µ ≤ min {α, λ} and define
Z s
tµ
−α
f (s) = s
dt,
λ
0 (1 + t)
Four Inequalities Similar To
Hardy-Hilbert’s Integral
Inequality
W.T. Sulaiman
then f (s) ≤ f (1).
Title Page
Proof. We have
Contents
0
sµ
Z
s
tµ
dt (−α) s−α−1
λ
λ
(1 + s)
0 (1 + t)
µ−α
−α−1 Z s
s
αs
≤
−
tµ dt
λ
(1 + s)
(1 + s)λ 0
sµ−α
α
=
1−
≤ 0.
1+µ
(1 + s)λ
f (s) = s
−α
+
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This shows that f is nonincreasing and hence f (s) ≤ f (1).
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Rs
TheoremR2.4. Let f, g, F, G be nonnegative functions such that F (s) = 0 f (t)dt,
s
G(s) = 0 g(t)dt, let, p > 1, p1 + 1q = 1, (1 − λ/2)r ≤ λ/2 + α ≤ 2α,
o
n
r ∈ pq , pq ,
Z
x
(x − t)t(1−λ/2)p/q−λ/2−α F p−1 (t)f (t)dt < ∞,
0<
Z0 x
0<
(x − t)t(1−λ/2)q/p−λ/2−α Gq−1 (t)g(t)dt < ∞,
0
Four Inequalities Similar To
Hardy-Hilbert’s Integral
Inequality
then
Z
x
Z
(2.3)
0
0
x
√
√
p p q q
λ λ
α
dsdt ≤
x B
,
2
2 2
(s + t)λ
Z x
p1
×
(x − t)t(1−λ/2)p/q−λ/2−α F p−1 (t)f (t)dt
F (s)G(t)
W.T. Sulaiman
Title Page
Contents
0
Z
×
x
(1−λ/2)q/p−λ/2−α
(x − t)t
0
Proof. Observe that
Z xZ x
F (s)G(t)
dsdt
(s + t)λ
0
0
1/q λ/2−1
λ/2−1
Z x Z x F (s) t1/p
G(t) st1/p
s1/q
=
·
dsdt
(s + t)λ/p
(s + t)λ/q
0
0
G
q−1
1q
(t)g(t)dt .
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Z
x
Z
≤
0
1
0
x
! p1
F p (s)tλ/2−1
(s + t)λ s(λ/2−1)p/q
1
= M pN q,
Z
x
x
Z
dsdt
0
0
! 1q
Gq (t)sλ/2−1
(s + t)λ t(λ/2−1)q/p
dsdt
say.
Then
Z
x
s(1−λ/2)p/q−λ/2 F p (s)ds
M=
0
Z
0
x
=
s
(1−λ/2)p/q−λ/2
p
F (s)ds
0
Z
x
Z
x
t λ/2−1 1
s
s
t λ
1+ s
dt
x α x −α Z
s
x α Z
s
1
x/s
0
uλ/2−1
(1 + u)λ
Four Inequalities Similar To
Hardy-Hilbert’s Integral
Inequality
du
u
s(1−λ/2)p/q−λ/2 F p (s)ds
du
λ
s
0
0 (1 + u)
Z x
xα
λ λ
=
B
,
s(1−λ/2)p/q−λ/2−α F p (s)ds,
2
2 2
0
≤
by virtue of the lemma.
As
Z
p
F (s) = p
W.T. Sulaiman
λ/2−1
s
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F p−1 (u)f (u)du,
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0
then
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p
M = xα B
2
λ λ
,
2 2
Z
0
x
s(1−λ/2)p/q−λ/2−α ds
Z
s
F p−1 (u)f (u)du
Page 13 of 18
0
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Z xZ s
p α
λ λ
= x B
,
u(1−λ/2)p/q−λ/2−α F p−1 (u)f (u)dsdu
2
2 2
Z0 x 0
λ λ
p α
= x B
,
(x − s)s(1−λ/2)p/q−λ/2−α F p−1 (s)f (s)ds.
2
2 2
0
Similarly,
q
N = xα B
2
λ λ
,
2 2
Z
x
(x − t)t(1−λ/2)q/p−λ/2−α Gq−1 (t)g(t)dt.
0
Four Inequalities Similar To
Hardy-Hilbert’s Integral
Inequality
Therefore, we have
x
Z
Z
x
W.T. Sulaiman
F (s)G(t)
dsdt
(s + t)λ
Z x
p1
√
√
p p q q
λ
λ
≤
xα B
,
(x − t)t(1−λ/2)p/q−λ/2−α F p−1 (t)f (t)dt
2
2 2
0
Z x
1q
(1−λ/2)q/p−λ/2−α q−1
×
(x − t)t
G (t)g(t)dt .
0
0
0
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Theorem 2.5. Let f, g be nonnegative functions, f is submultiplicative and g is
concave nonincreasing, f 0 (x), g 0 (y) ≥ 0, f (0) = g(0) = 0, f (∞) = g(∞) =
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∞, p > 1,
1
p
+
1
q
= 1, 0 < a + 1 < λ, 0 < b + 1 < λ,
Z
∞
0<
[f (x)]µp−bp/q [g(x)]1+a−λ
p
[f 0 (x)] q
0
Z
∞
0<
[g(y)]µq−aq/p [f (y)]1+b−λ
q
[g 0 (y)] p
0
dx < ∞,
dy < ∞,
then
Z
∞Z
∞
(2.4)
0
0
f µ (xy)
dxdy
g λ/2 (xy)
λ/2
≤2
Proof. Since
√
Four Inequalities Similar To
Hardy-Hilbert’s Integral
Inequality
W.T. Sulaiman
1
q
1
p
B (a + 1, λ − a − 1) B (b + 1, λ − b − 1)
! p1
Z ∞
[f (t)]µp [g(t)]1+a−λ−bp/q
×
dt
p
[g 0 (t)] q
0
! 1q
Z ∞
[f (t)]µq [g(t)]1+b−λ−aq/p
×
dt
.
q
[g 0 (t)] p
0
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xy ≤
x+y
,
2
then
2 2
√
x+y
g(x) + g(y)
√
2
2
g(xy) = g ( xy) ≥ (g ( xy)) ≥ g
≥
,
2
2
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J. Ineq. Pure and Appl. Math. 7(2) Art. 76, 2006
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and hence
Z ∞Z ∞
0
0
f µ (xy)
dxdy
g λ/2 (xy)
Z ∞Z
λ/2
≤2
0
∞
f µ (x)f µ (y)
(g(x) + g(y))λ
0
a/p
= 2λ/2
∞
Z
1/p
0
b/q
[g (y)]
f µ (x) [g(y)]
[g(x)]b/q [g 0 (x)]1/q
·
[g(x)]
f µ (y) [g(y)]
a/p ·
[g 0 (x)]1/q
[g 0 (y)]1/p
λ
(g(x) + g(y)) q
! p1
Z ∞Z ∞
f µp (x)g a (y)g 0 (y)
dxdy
p
[g(x)]bp/q [g 0 (x)] q (g(x) + g(y))λ
0
0
! 1q
Z ∞Z ∞
f µq (y)g b (x)g 0 (x)
×
dxdy
q
[g(y)]aq/p [g 0 (y)] p (g(x) + g(y))λ
0
0
≤ 2λ/2
=2
∞
λ
0
λ/2
Z
dxdy
(g(x) + g(y)) p
0
1
p
1
q
M N ,
say.
M=
∞
µp
1+a−λ−bp/q
[f (x)] [g(x)]
∞
Z
dx
p
0
W.T. Sulaiman
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Then
Z
Four Inequalities Similar To
Hardy-Hilbert’s Integral
Inequality
[g 0 (x)] q
0
Z
= B (a + 1, λ − a − 1)
∞
ηp
g(y)
g(x)
1+
g 0 (y)
g(x)
g(y)
g(x)
Go Back
λ dy
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1+a−λ−bp/q
[f (x)] [g(x)]
p
0
a
II
I
[g 0 (x)] q
dx.
Page 16 of 18
J. Ineq. Pure and Appl. Math. 7(2) Art. 76, 2006
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Similarly, we can show that
Z
N = B (b + 1, λ − b − 1)
∞
[f (y)]µq [g(y)]1+b−λ−aq/p
q
0
[g 0 (y)] p
dy.
The result follows.
Four Inequalities Similar To
Hardy-Hilbert’s Integral
Inequality
W.T. Sulaiman
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References
[1] G.H. HARDY, J.E. LITTLEWOOD AND G. POLYA, Inequalities, Cambridge Univ. Press. Cambridge, 1952.
[2] B. YANG AND L. DEBNATH, On the extended Hardy-Hilbert’s inequality,
J. Math. Anal. Appl., 272 (2002), 187–199.
[3] M. GAO, TAN LI AND L. DEBNATH, Some improvements on Hilbert’s
integral inequality, J. Math. Anal. Appl., 229 (1999), 682–689.
[4] M. GAO, A note on Hilbert double series theorem, Hunan Ann. Math., 12(12) (1992), 142–147.
[5] HU KE, On Hilbert’s inequality, Chinese Ann. of Math., 13B(1) (1992),
35–39.
[6] W. RUDIN, Principles of Mathematical Analysis, Third Edition, McGrawHill Book Co. 1976.
Four Inequalities Similar To
Hardy-Hilbert’s Integral
Inequality
W.T. Sulaiman
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J. Ineq. Pure and Appl. Math. 7(2) Art. 76, 2006
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