273
Acta Math. Univ. Comenianae
Vol. LXXVI, 2(2007), pp. 273–278
ON THREE INEQUALITIES SIMILAR
TO HARDY-HILBERT’S INTEGRAL INEQUALITY
W. T. SULAIMAN
Abstract. The following three inequalities, which are similar to Hardy-Hilbert
integral inequality are proved.
Z∞Z∞ β/q α/p
x
y
F (x)G(y)
λ
p1−1/p q 1−1/q
dxdy ≤
λ
λ
1/p
1/q
max{x , y }
(p − 1)(q − 1)
(α + 1) (β + 1)
0
0
11/q
11/p 0 ∞
0∞
Z
Z
q
p
@ g (t)dtA
.
· @ f (t)dtA
0
0
where
Zy
Zx
F (x) =
f (t)dt,
G(y) =
0
Z∞Z∞
0
g(t)dt.
0
x1/p y 1/q F (x)G(y)
pq
dxdy < B 1/p (p, p)B 1/q (q, q)
(x + y)4
(p − 1)(q − 1)
0
11/q
11/p 0 ∞
0∞
Z
Z
@ g q (t)dtA
.
· @ f p (t)dtA
0
0
Z∞Z∞Z∞ 1/p 1/q 1/r p
x y z
xyzF (x)G(y)H(z)
dxdydz
(x + y + z)8
0
0
0
0∞
11/p 0 ∞
11/q 0 ∞
11/r
Z
Z
Z
p/2
q/2
r/2
@
A
@
A
@
< Kp Kq Kr
.
f
(t)dt
g (t)dt
h (t)dtA
0
0
0
where
Zz
H(z) =
s
h(t)dt,
Kp =
“p p”
p/2
B 1/p
,
B 1/p (p, p).
p/2 − 1
2 2
0
Received June 13, 2006.
2000 Mathematics Subject Classification. Primary 26D15.
274
W. T. SULAIMAN
1. Introduction
Let f, g ≥ 0 satisfy
Z∞
0 < f 2 (t)dt < ∞
Z∞
and
0<
0
g 2 (t)dt < ∞,
0
then
Z∞ Z∞
(1)
0

f (x)g(y)
dx dy < π 
x+y
0
Z∞
f 2 (t)dt
0
Z∞
1/2
g 2 (t)dt
,
0
where the constant factor π is the best possible (cf. Hardy et al. [3]). Inequality (1)
is well known as Hilbert’s integral inequality. This inequality had been extended
by Hardy [1] as follows
If p > 1, p1 + 1q = 1, f, g ≥ 0 satisfy
Z∞
Z∞
p
f (t)dt < ∞
0<
and
0<
0
g q (t)dt < ∞,
0
then
Z∞ Z∞
(2)
0

f (x)g(y)
π

dx dy <
x+y
sin(π/p)
0
Z∞
1/p  ∞
1/q
Z
f p (t)dt  g q (t)dt ,
0
0
π
sin(π/p)
where the constant factor
is the best possible. Inequality (2) is called
Hardy-Hilbert’s integral inequality and is important in analysis and application
(cf. Mitrinovic et al. [4]).
B. Yang gave the following extension of (2) as follows:
Theorem ([5]). If λ > 2 − min{p, q}, f, g ≥ 0, satisfy
Z∞
Z∞
1−λ p
0< t
f (t)dt < ∞
and
0 < t1−λ g q (t)dt < ∞,
0
0
then
Z∞ Z∞
(3)
0
0
1/q
∞
1/p  ∞
Z
Z
f (x)g(y)
dx dy < kλ (p)  t1−λ f p (t)dt  t1−λ g q (t)dt ,
(x + y)λ
0
0
where the constant factor kλ (p) = B
beta function.
p+λ−2 q+λ−2
, q
p
is the best possible, B is the
The object of this paper is that to give some new inequalities similar to that of
Hardy-Hilbert’s inequality.
We need the following result for our aim
ON INEQUALITIES SIMILAR TO HARDY-HILBERT’S INEQUALITY
275
Theorem A ([2]). Let f be a nonnegative integrable function. Define
Zx
f (t)dt.
F (x) =
a
Then
Z∞ F (x)
x
p
dx <
p
p−1
0
p Z∞
f p (x)dx,
p > 1.
0
2. New results
We state and prove the following:
1
p
Rx
Ry
Theorem 1. Let f, g ≥ 0, F (x) = 0 f (t)dt, G(y) = 0 g(t)dt, p > 1,
+ 1q = 1, p = λ − α − 1 > 1, q = λ − β − 1 > 1, α, β > −1 then
Z∞ Z∞
0
xβ/q y α/p F (x)G(y)
p1−1/p q 1−1/q
λ
dx
dy
≤
max{xλ , y λ }
(α + 1)1/p (β + 1)1/q (p − 1)(q − 1)
0
∞
1/p  ∞
1/q
Z
Z
·  f p (t)dt  g q (t)dt .
0
0
Proof. We have
Z∞ Z∞ β/q α/p
x y F (x)G(y)
dx dy
max{xλ , y λ }
0
0
Z∞ Z∞
Z∞ Z∞
y α/q F (x)
·
xβ/q G(y)
dx dy
1/q
(max{xλ , y λ })
0 0
0 0
∞∞
1/p  ∞ ∞
1/q
Z Z
Z Z
β q
α p
x
G
(y)
y
F
(x)
dx dy  
dx dy 
≤
max{xλ , y λ }
max{xλ , y λ }
1/p
(max{xλ , y λ })
0
0
0
0
= M 1/p N 1/q .
Observe that
Z∞
Z∞
p
M =
F (x)dx
0
=
yα
dx dy =
max{xλ , y λ }
0
λ
(α + 1)(λ − α − 1)
Z∞
 x

Z α
Z∞ α
y
y
F p (x)dx 
dy +
dy 
xλ
yλ
0
Z∞
0
F p (x)
λ
dx =
xλ−α−1
(α + 1)p
x
0
Z∞ 0
F (x)
x
p
dx
276
W. T. SULAIMAN
<
λ
pp−1
(α + 1) (p − 1)p
Z∞
f p (x)dx,
0
in view of Theorem A.
Similarly, It can be shown that
Z∞
λ
q q−1
N<
(β + 1) (q − 1)q
g q (x)dx.
0
Combining these inequalities to obtain
Z∞ Z∞
0
p1−1/p q 1−1/q
xβ/q y α/p F (x)G(y)
λ
dx dy ≤
λ
λ
1/p
1/q
max{x , y }
(α + 1) (β + 1) (p − 1)(q − 1)
0

Z∞
· 
1/p 
f p (t)dt
Z∞

0
1/q
g q (t)dt
.
0
1
p
Theorem 2. Let f, g ≥ 0, F (x) =
+ 1q = 1, then, we have
Z∞ Z∞
0
Rx
0
f (t)dt, G(y) =
Ry
0
g(t)dt, p > 1,
x1/p y 1/q F (x)G(y)
pq
dxdy < B 1/p (p, p)B 1/q (q, q)
(x + y)4
(p − 1)(q − 1)
0

Z∞
· 
1/p 
f p (t)dt
0
Z∞

1/q
g q (t)dt
.
0
Proof.
Z∞ Z∞
0
x1/p y 1/q F (x)G(y)
dxdy
(x + y)4
0
Z∞ Z∞
=
0
y 1/q F (x) x1/p G(y)
·
dx dy
(x + y)2 (x + y)2
0
∞∞
1/p  ∞ ∞
1/q
Z Z p/q p
Z Z q/p q
y F (x)
x G (y)
≤ 
dx dy  
dx dy 
(x + y)2p
(x + y)2q
= P
0 0
1/p 1/q
Q
0
.
0
277
ON INEQUALITIES SIMILAR TO HARDY-HILBERT’S INEQUALITY
Now, we consider
p
p
Z∞ Z∞ y p−1 1
Z∞ Z∞
F (x)
F (x)
up−1
x
x
P =
dx
dy
=
dx
du
2p
x
x
(1 + u)2p
1 + xy
0
0
< B(p, p)
p
p−1
0
p Z∞
0
f p (x)dx,
0
by Theorem A.
Similarly,
Q < B(q, q)
q
q−1
q R∞
g q (y)dy.
0
Therefore, we have
Z∞ Z∞ 1/p 1/q
x y F (x)G(y)
pq
dxdy < B 1/p (p, p)B 1/q (q, q)
(x + y)4
(p − 1)(q − 1)
0
0
∞
1/p  ∞
1/q
Z
Z
·  f p (t)dt  g q (t)dt .
0
0
This completes the proof of the theorem.
Theorem 3. Let f, g, h ≥ 0, p, q, r > 2 p1 + 1q + 1r = 1,
Z x
Z y
Z z
F (x) =
f (t)dt,
G(y) =
g(t)dt,
H(z) =
h(t)dt.
0
1
0
Then
Z∞ Z∞ Z∞ 1/p 1/q 1/r p
xyzF (x)G(y)H(z)
x y z
dxdydz
(x + y + z)8
0
0
0

Z∞
< K p Kq Kr 
where
1/p 
f p/2 (x)dx
0
Kp =
0
0

1/q 
g q/2 (y)dy 
0
s
Proof.
Z∞ Z∞ Z∞
Z∞
Z∞

1/r
hr/2 (z)dz 
0
p p
p/2
B 1/p
,
B 1/p (p, p).
p/2 − 1
2 2
p
x1/p y 1/q z 1/r xyzF (x)G(y)H(z)
dxdydz
(x + y + z)8
0
Z∞ Z∞ Z∞
=
0
0
0
p
p
F (x) z (1/2−1/q) x(1/p+1/r) G(y)
·
(x + y + z)2

p
x(1/2−1/r) y (1/p+1/q) H(z) 
·
dx dy dz
(x + y + z)2
y (1/2−1/p) z (1/q+1/r)
(x + y + z)2
.
278
W. T. SULAIMAN
1/p
∞∞∞
Z Z Z (p/2−1) p−1 p/2
y
z
F (x)
dx dy dz 
≤ 
(x + y + z)2p
0
0
0
∞∞∞
1/q
Z Z Z (q/2−1) q−1 q/2
z
x
G
(y)
·
dx dy dz 
(x + y + z)2q
0
0
0
∞∞∞
1/r
Z Z Z (r/2−1) r−1 r/2
x
y
H
(z)
·
dx dy dz 
(x + y + z)2r
0 0 0
1/p 1/q
= A
C 1/r .
B
Observe that
Z∞
A=
F
p/2
(x)
xp/2
Z∞
dx
0
0
Z∞ =
y p/2−1 1
x
px
1 + xy
F (x)
x
Z∞
p/2
dx
0
Z∞
dy
z
x+y
0
up/2−1
du
(1 + u)p
Z∞
p−1
1+
z
x+y
1
x+y
2p
dz
v p−1
dv
(1 + v)2p
0
0
p/2
Z∞
Z∞ p p
F (x)
p
=B
,
dx < Kp f p/2 (x)dx,
B(p, p)
2 2
x
0
in view of Theorem A.
Similarly,
B<
Kqq
Z∞
g
p/2
0
(y)dy,
C<
0
Krr
Z∞
hr/2 (z)dz.
0
The proof is complete.
References
1. Hardy G. H., Note on a theorem of Hilbert concerning series of positive terms, Proc. Math.
Soc. 23(2) (1925), Records of Proc. XLV-XLVI.
2.
, Note on a theorem of Hilbert, Math. Z., 6 (1920) 314–317.
3. Hardy G. H., Littlewood J. E. and Polya G., Inequalities, Cambridge University Press,
Cambridge, 1952.
4. Mitrinovic D. S., Pecaric J. E. and Fink A. M., Inequalities involving functions and their
integrals and derivatives, Kluwer Academic Publishers, Boston, 1991.
5. Yang B., On Hardy-Hilbert’s integral inequality, J. Math. Anal. Appl. 261 (2001) 295–306.
W. T. Sulaiman, Department of Mathematics, College of Computer Sciences & Mathematics,
University of Mosul, Mosul, Iraq, e-mail: waadsulaiman@hotmail.com