Advances in Pure Mathematics, 2011, 1, 84-89
doi:10.4236/apm.2011.13019 Published Online May 2011 (http://www.scirp.org/journal/apm)
Copyright © 2011 SciRes. APM
A Hilbert-Type Inequality with Some Parameters and the
Integral in Whole Plane
Zitian Xi e1, Zheng Zeng2
1Department of Mat hem at ic s, Zhaoqing University , Zhaoqing, Chi na
2 Department of Mathematics, Shaoguan University, Shaoguan, China
E-mail: gdzqxzt@163.com, zz@sgu.edu.cn
Received February 13, 2011; revised March 7, 2011; accepted March 10, 2011
Abstract
In this paper, by introducing some parameters and estimating the weight coefficient, we give a new Hilbert’s
inequality with the integral in whole plane and with a non-homogeneous and the equivalent form is given as
well. The best constant factor is calculated by the way of Complex Analysis.
Keywords: Hilbert-Type Integral Inequality, Weight Function, Holder’s Inequality
1. Introduction
If

,
f
x

g
x are non-negative functions such that

2
0
0dfxx

and

2
0
0dgxx

, then
  

12
000 0
dd πdd
fxgy
x
yfxxgxx
xy
 
  (1.1)
where the constant factor π is the best possible. Ine-
quality (1.1) is well-known as Hilbert’s integral inequal-
ity, which has been extended by Hardy-Riesz as [2]:
If 11
1, 1ppq

,

,
f
x

0gx, such that

0
0d
p
fxx

and

0
0d
q
gxx

, then we
have the following Hardy-Hilbert’s integral inequality:
 

00
11
00
dd
πdd
sin π
pq
pq
fxgyxy
xy
f
xx gxx
p




(1.2)
where the constant factor

π
sin πp also is the best
possible.
In recent years, by introducing some parameters and
estimating the way of weight function, inequalities (1.1)
and (1.2) have many generalizations and variants (1.1)
has been strengthened by Yang and others. (including
double series inequalities) [3-15].
In 2005 Yang gave a Hilbert-type Inequality [3] as
follows: If 1,p 111
pq
,

,
f
x
0gx,
2min ,1pq
,

1
0
0d
p
xfxx

, and

1
0
0d
q
xgxx

then


00
11
11
00
dd
dd
p
q
pq
fxgyxy
xy
K
x fxxxgxx






(1.3)
where the constant factor
22
,1 ,1
pq
KB B
pq


 



also is
the best possible.
In 2008 Xie gave a new Hilbert-type Inequality [4] as
follows:
If
 
11
1,1, ,0pfxgx
pq
 

12
0
0d
pp
xfxx

and

12
0
0d
qq
xgxx

,
then



222
00
11
12 12
00
dd
dd
p
q
pp qq
fxgy xy
xayxayxay
K
xf xxxgxx






where the constant factor

π
Kabbcca

Z. T. XIE ET AL.
Copyright © 2011 SciRes. APM
85
also is the best possible.
In this paper, by using the way of weight function and
the technic of real analysis and by the way of complex
analysis, a new Hilbert-type inequality with the integral
in whole plane is given.
In the following, we always suppose that: 0,p
111
pq

, 1,p

1, 2, 0πr


.
2. Some Lemmas
Lemma 2.1 If
2
12
0
2
22
0
12cos
:ln d,
12cos
12cos
:ln d,
12cos
r
r
uu
ku u
uu
uu
ku u
uu




then
 
 

 

1
2
11
4πsin sin
22
:,
1sinπ
1
4πsinsin 1
22
:1sinπ
rr
krr
rr
krr
 
 
 






(2.1)
and
 

2
2
12
12cos
:ln d
12cos
11π
4πsin cos
22
π
1sin
2
ruu
ku u
uu
kk
rr
r
r





 
(2.2)
Proof We have



2
0
12
0
1
2
0
:ln2cos1d
1ln2 cos1
1
cos
2d
12cos1
2
:1
r
r
r
A
xxxx
xxx
r
xx
x
rxx
B
r




setting
 
1
2
cos
2cos1
r
zz
fz zz

, 12
,
ii
zeze
 ,
then
 


12
2π
11
11 22
2π
12 21
2πRe ,Re ,
1
cos cos
2π
1
πcos 1
sin π
ri
rr
ri
i
Bsfzsfz
e
zz zz
i
zz zz
e
r
r








We find


2πcos[ 1]
2
11sinπ
r
AB
rrr
 

.
then


2
12
0
12cos
ln d
12cos
11
4πsinsin
22
1sinπ
ruu
ku u
uu
rr
rr



 
and


 

2
22
0
2
2
0
12cos
ln d
12cos
12cosπ
ln d
12cosπ
1
4πsinsin 1π
22
1sinπ
r
r
uu
ku u
uu
uu
uu
uu
rr
rr
 




 







  

2
2
2
2
0
2
0
2
12
12cos
ln d
12cos
12cos
ln d
12cos
12cos
ln d
12cos
11π
4πsin cos
22
π
1sin
2
r
r
r
uu
ku u
uu
uu
uu
uu
uu
uu
uu
kk
rr
r
r








 

 
The lemma is proved.
Lemma 2.2 Define the weight functions as follow:


122
22
122
22
12 cos
:ln d,
12 cos
12 cos
:lnd
12 cos
r
r
r
r
xxyx y
wx y
xyx y
y
yxyx y
wx x
xyx y
x








then
Z. T. XIE ET AL.
Copyright © 2011 SciRes. APM
86
  

11π
4πsin cos
22
π
1sin
2
wx wx k
rr
r
r


 
(2.3)
Proof We only prove that
wx
k
for
,0x .




122
0
22
122
22
0
122
0
22
122
22
0
12
12 cos
ln d
12 cos
12 cos
ln d
12 cos
12 cos
ln d
12 cos
12 cos
ln d
12 cos
r
r
r
r
r
r
r
r
xxyx y
wx y
xyx y
y
xxyx y
y
xyx y
y
xxyx y
y
xyx y
y
xxyx y
y
yxyxy
ww















Setting uxy, then


122
0
122
2
1
2
0
12 cos
ln d
12 cos
12cos
ln d
12cos
r
r
r
xxyx y
wy
xyx y
y
uu
uuk
uu







Similarly, setting uxy ,
2
22
2
0
12cos
ln d
12cos
ruu
wu uk
uu



And
12
wxk kk
, we have (2.3).
Lemma 2.3 For 0
, and

22
max,1,2rpq








, define both functions,
f
and
g
as follow:



 
2
2
, if 1,,
0, if 1,1,
, if ,1 ;
rp
rp
xx
fx x
xx





and



2
0, if 1,,
, if 1,1,
0, if ,1;
rq
x
gx xx
x




then



11
11
:2dd 1
pp
pq
Ixxfxxxgx x


 




(2.4)





22
22
12 cos
:2 lndd
12 cos
1, for0
xyx y
I
fxgy xy
xyx y
ko





 

(2.5)
Proof Easily,


11
1
12 12
10
:2dd 1;
pp
Ixxxxxxx

 

Let yY
, using


,
f
xfxgxgx , and




22
22
22
22
12 cos
ln d
12 cos
12 cos
ln d
12 cos
xyx y
f
xgy y
xyx y
xYx Y
f
xgY Y
xYx Y






we have that


22
22
12 cos
ln d
12 cos
xyx y
f
xgy y
xyx y



is an even function, then




22
22
0
22 22
0 1
2
222
22 22
11 10
1
12 cos
2ln dd
12 cos
12 cos12 cos
2lnddlndd
12 cos12 cos
rq
rp rp rq
xyx y
Ifxgy yx
xyx y
xy xyxy xy
x
yyxxyyx
xyx yxyx y
I






 

  






 

 
 
 


 
 

 

 
2.I
Setting uxy then
22 2
1
22 122
122 2
10 10
2
1
12 2122
2
10 11
12 cos12cos
2lndd2lndd
12 cos12cos
12cos
2lndd
12cos
x
rp rqrq
x
rq rq
xyx yuu
I
xyyxxu ux
xyx yuu
uu
xu uxxu
uu
 
 



 


 

 


 







 
 2
2
12cos
lnd d
12cos
uu
ux
uu





Z. T. XIE ET AL.
Copyright © 2011 SciRes. APM
87
22
12212
22
01
22
122
22
01
2
2
2
0
12cos 12cos
lnd2lnd d
12cos 12cos
12cos 12cos
lnd lnd
12cos 12cos
12cos
lnd (
12cos
rq rq
u
rq rp
rp
uu uu
uuu xxu
uu uu
uu uu
uuuu
uu uu
uu
uu
uu







 
 

 

 
 

 







2
122
2
0
12cos
)ln d
12cos
2
12
4πsinsin 1
22
212
1sin π
2
qpr uu
uuu u
uu
rprp
rr
pp

 






 
 


  

 






 




There

0
lim 0

, and we have 12
I
k (for
0
).
Similarly 21
I
k (for 0
). The lemma is
proved.
Lemma 2.4 If

1
0d
pr p
xfxx


, we have


22
1
22
1
12 cos
:lndd
12 cos
d
p
pr p
pr
pp
xyx y
J
yfx xy
xyx y
kxfxx


 








(2.6)
Proof By lemma 2.2, we find




22
22
22
22
1
22
22
22
2
12 cos
ln d
12 cos
12 cos
ln d
12 cos
12 cos
ln d
12 cos
12 cos
ln 12 cos
p
p
rq rp
rp rq
rp
p
r
xyx y
fx x
xyx y
xy
xyx y
f
xx
xyx yyx
x
xyx yfxx
xyx yy
xyx y
xyx y

























1
1
2
1
22
1
1
22
d
12 cos
lnd ,
12 cos
p
rq
r
rp
pr p
pp
r
yx
x
x
xyx y
ky fxx
xyx yy














1
22
1
22
1
22
1
22
1
12 cos
lnd d
12 cos
12 cos
lnd d
12 cos
d
rp
pp
r
rp
pp
r
pr
pp
J
x
xyx y
kfxxy
xyx yy
x
xyx y
kyfxx
xyx yy
kxfxx



















(2.7)
3. Main Results
Theorem If both functions,
f
x and
g
x are
nonnegative measurable functions, and satisfy

1
0d
pr p
x
fxx


, and

1
0d
qr q
x
gxx


,
then



22
*
22
11
11
12 cos
:ln dd
12 cos
dd
pp
pr qr
pq
xyx y
I
fxgy xy
xyx y
kxfxx xgxx

 


 




(3.1)
and


22
1
22
1
12 cos
lnd d
12 cos
d
p
pr p
pr
pp
xyx y
J
yfx xy
xyx y
kxfxx


 








(3.2)
Inequalities (3.1)and (3.2) are equivalent, and the con-
stant factors in the two forms are all the best possible.
Proof If (2.7) takes the form of equality for some
,0 0,y
, then there exist constants
M
and
N, such that they are not all zero, and



11rp rq
p
rr
xy
Mfx
yx

a.e. in
,, ,
Hence, there exists a constant C, such that

rp rq
p
M
xfxNyC
a.e. in
,, .
We claim that 0M
. In fact, if 0M
, then

1rpp
x
fxCMx
a.e. in
,  which con-
tradicts the fact that

1
0d
pr p
xfxx


. In
the same way, we claim that 0N. This is too a con-
tradiction and hence by (2.7), we have (3.2).
By Holder’s inequality with weight and (3.2), we have,
Z. T. XIE ET AL.
Copyright © 2011 SciRes. APM
88
 
 
22
11
*
22
1
11
12 cos
lnd d
12 cos
d
rq rq
q
qrpq
xyx y
I
yfx xygyy
xyx y
Jygyy

 
 








(3.3)
Using (3.2), we have (3.1).
Setting
 
1
22
1
22
12 cos
ln d
12 cos
p
pr pxyx y
gy yfxx
xyx y








then 1()d
qr q
J
ygyy

by (2.7) we have J
.
if 0J then (3.2) is proved; if 0J, by (3.1),
we obtain


1*
11
11
0()d
dd,
qr q
pq
pr qr
pq
ygyyJI
kxfxxxgxx



 




11
11
1
dd
p
p
qr pr
qp p
x
gxxJkxf xx


 


Inequalities (3.1) and (3.2) are equivalent.
If the constant factor k in (3.1) is not the best possi-
ble, then there exists a positive h (with hk), such
that



22
22
11
11
12 cos
lnd d
12 cos
dd
p
q
pr qr
pq
xyx y
f
xg yxy
xyx y
hxfxx xgxx

 


 




(3.4)
For 0
, by (3.4), using lemma 2.3, we have
 


11
11
1
dd
p
q
pq
pr qr
Iko
hxfxx xgxx


 


(3.5)
Hence we find,

1ko h
, for 0
, it follows
that kh, which contradicts the fact that hk
.
Hence the constant h in (3.1) is the best possible. As
(3.1) and (3.2) are equivalent, if the constant factor in
(3.2) is not the best possible, then by using (3.2), we can
get a contradiction that the constant factor in (3.1) is not
the best possible.
Thus we complete the prove of the theorem.
Remark For π
4
, π
3
in (3.1), we have the
following particular result:



22
22
11
11
12
lnd d
1
dd
pq
prqr
pq
xyx y
fxgy xy
xyx y
kxfxx xgxx

 


 




(3.6)
Where the constant factor

1π51π
4πsin cos
2424
π
1sin2
rr
kr
r

also is the best possible.
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