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On New Generalizations of Hilbert's Inequality
Journal of Mathematical Analysis and Applications 248, 29᎐40 Ž2000. doi:10.1006rjmaa.2000.6860, available online at http:rrwww.idealibrary.com on
On New Generalizations of Hilbert’s Inequality Yang Bicheng Department of Mathematics, Guangdong Education College, Guangzhou, Guangdong 510303, People’s Republic of China E-mail: [email protected] Submitted by L. Debnath Received February 25, 2000
By introducing three parameters A, B, and , we give some generalizations of Hilbert’s integral inequality and its equivalent form with the best constant factors. We also consider their associated double series forms. 䊚 2000 Academic Press Key Words: Hilbert’s inequality; weight function;  function; weight coefficient.
1. INTRODUCTION If f and g are real functions, such that 0 - H0⬁ f 2 Ž t . dt - ⬁ and 0 dt - ⬁, then
H0⬁ g 2 Ž t .
⬁
⬁
H0 H0
f Ž x. g Ž y. xqy
dx dy -
⬁
žH
f 2 Ž t . dt
⬁
H0
0
1r2
g 2 Ž t . dt
/
,
Ž 1.1.
where the constant factor is best possible. Its associated double series form is as follows: If a n4 and bn4 are sequences of real numbers such that 0 - Ý⬁ns 1 a2n - ⬁ and 0 - Ý⬁ns1 bn2 - ⬁, then ⬁
⬁
Ý Ý ns1 ms1
a m bn mqn
-
⬁
žÝ
ns1
1r2
⬁
a2n
Ý ns1
bn2
/
,
Ž 1.2.
where the constant factor is still best possible. Hilbert’s inequalities Ž1.1. and Ž1.2. are important in our analysis and its applications Žcf. w1, Chap. 9x.. In recent years, Hu w2x and Gao w3x gave two distinct improvements of Ž1.1., and Gao w4x gave Ž1.2. a strengthened version. By introducing a 29 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
30
YANG BICHENG
parameter g Ž0, 1x and estimating the weight function, Yang w5x gave Ž1.1. a generalization as ⬁
⬁
H0 H0
f Ž x. g Ž y.
Ž x q y.
, 2 2
dx dy - B
ž /ž
⬁
H0
t 1y f 2 Ž t . dt
⬁
H0 t
1r2 1y 2
g Ž t . dt
/
,
Ž 1.3. where B Ž p, q . is the  function. But w5x hasn’t proved that the constant factor B Ž 2 , 2 . Ž0 - F 1. in Ž1.3. is best possible. In this paper, we continue this work and make some generalizations of Ž1.3. by introducing three parameters, A, B, and . We also give the associated generalizations of Ž1.2..
2. SOME NEW RESULTS ON HILBERT’S INTEGRAL INEQUALITY First, we define the weight function , A, B Ž x . as
, A , B Ž x . s
⬁
H0
1y r2
x
1
Ž Ax q By .
ž /
x g Ž 0, ⬁ . Ž A, B, ) 0 . .
dy,
y
Ž 2.1. Setting u s Ž By .rŽ Ax . in Ž2.1., we have
, A , B Ž x . s s
Ž Ax .
⬁
H0
1y
B Ž 1 q u. 1
Ž AB .
r2
B
x 1y
1y r2
ž /
du
Au
1
⬁
H0
Ž 1 q u.
uy1q r2 du.
Ž 2.2.
Since the -function B Ž p, q . possesses the explicit formulation Žcf. w5, p. 117 Ž10.x.
B Ž p, q . s
⬁
H0
t py 1
Ž1 q t .
pq q
dt
Ž p, q ) 0 . ,
Ž 2.3.
GENERALIZATIONS OF HILBERT’S INEQUALITY
31
by Ž2.2. we have
, A , B Ž x . s
1
Ž AB .
1y , x , 2 2
x g Ž 0, ⬁ . Ž A, B, ) 0 . .
ž /
B
r2
Ž 2.4. If A G B ) 0, ) 0, and 0 - - r2, we ha¨ e
LEMMA 2.1. ⬁
H1
xy1 y
H0Br Ax Ž
1
.
Ž 1 q u.
uŽ y2y .r2 du dx s O Ž 1 .
Ž ª 0q . . Ž 2.5.
Proof. Since A G B ) 0, for x G 1, we have 0 - BrŽ Ax . F 1, and 1
Br Ž Ax .
H0
Ž 1 q u.
uŽ y2y .r2 du
Br Ž Ax .
H0
uŽ y2y r2.r2 du s
4
B
r4
ž / Ax
.
Then we find that 0-
⬁
H1 4
xy1 y
H0Br Ax Ž
r4
B
ž /
⬁
H1
A
1
.
Ž 1 q u.
xy1 y r4 dx s
uŽ y2y .r2 du dx 2
4
r4
B
ž /ž /
.
A
Hence Ž2.5. is valid. The lemma is proved. THEOREM 2.1. Let f and g be real functions, ) 0, such that 0 H0⬁ t 1y f 2 Ž t . dt - ⬁ and 0 - H0⬁ t 1y g 2 Ž t . dt - ⬁. If A, B ) 0, we ha¨ e ⬁
⬁
H0 H0
f Ž x. g Ž y.
Ž Ax q By .
dx dy
B , r2 2 2 Ž AB . 1
⬁
H0
y y1
⬁
H0
⬁
ž / žH
dx
f Ž t . dt
dy -
1
Ž AB .
⬁
H0
0
2
f Ž x.
Ž Ax q By .
t
1y 2
B
1r2
t
1y 2
, 2 2
g Ž t . dt
ž /
2
⬁
H0
/
; Ž 2.6.
t 1y f 2 Ž t . dt.
Ž 2.7.
32
YANG BICHENG
Inequalities Ž2.6. and Ž2.7. are equi¨ alent. Both the constant factor Ž1r Ž AB . r2 . B Ž 2 , 2 . in Ž2.6. and the constant factor Ž1rŽ AB . .? B Ž 2 , 2 .@ 2 in Ž2.7. are best possible. In particular, for s 1, if A, B ) 0, we ha¨ e
Ži. ⬁
f Ž x. g Ž y.
⬁
H0 H0
dx dy -
Ž Ax q By . ⬁
1r2
žH
H0
0
2
dy -
dx
Ž Ax q By .
⬁
f 2 Ž t . dt
2
⬁
H Ž AB . 0
1r2
g 2 Ž t . dt
/
, Ž 2.8.
f 2 Ž t . dt ;
Ž 2.9.
for s 2, if A, B ) 0, we ha¨ e
Žii. ⬁
⬁
Ž AB .
f Ž x.
⬁
H0 H0
f Ž x. g Ž y.
⬁
H0 H0
Ž Ax q By .
2
1
dx dy -
Ž AB .
ž
⬁
H0
1 t
f 2 Ž t . dt
⬁
H0
1 t
1r2
g 2 Ž t . dt
/
,
Ž 2.10. ⬁
H0 y H0
Ž Ax q By .
⬁
Ž Ax q Ay1 y .
-B
y
dy -
dx
f Ž x. g Ž y.
⬁
H0 H0
H0
2
1
⬁
Ž AB .
2
H0
1 t
f 2 Ž t . dt ;
Ž 2.11.
for ) 0, if A ) 0 Ž B s 1rA., we ha¨ e
Žiii.
⬁
2
f Ž x.
⬁
y1
⬁
H0
, 2 2
t 1y f 2 Ž t . dt 2
y.
dx
⬁
H0
0
f Ž x.
Ž Ax q A
dx dy
⬁
ž / žH y1
dy - B
1r2
t 1y g 2 Ž t . dt
, 2 2
ž /
2
⬁
H0
/
,
Ž 2.12.
t 1y f 2 Ž t . dt,
Ž 2.13. where the constant factors in Ž2.8. ᎐ Ž2.13. are still best possible.
GENERALIZATIONS OF HILBERT’S INEQUALITY
33
Proof. By Cauchy’s inequality, we have ⬁
f Ž x.
⬁
H0 H0
Ž Ax q By .
F
½
⬁
r2
Ž Ax q By .
y
Ž Ax q By .
⬁
x
Ž Ax q By .
r2
Ž1y r2 .r2
ž /
dx dy
x
dx dy
y
y
r2
y
1y r2
ž /
g2 Ž y.
⬁
H0 H0
g Ž y.
ž /
f 2Ž x.
⬁
H0 H0 =
Ž1y r2 .r2
x
1r2
1y r2
ž /
dx dy
x
5
.
Ž 2.14.
If Ž2.14. takes the form of the equality, then there exist constants c and d such that Žcf. w7, p. 29x.
c
f 2Ž x.
Ž Ax q By .
x
1y r2
ž / y
g2 Ž y.
sd
Ž Ax q By .
y
1y r2
ž / x
a.e. in Ž 0, ⬁ . = Ž 0, ⬁ . . Then we have cf 2 Ž x . x 2y s dg 2 Ž y . y 2y a.e. in Ž0, ⬁. = Ž0, ⬁.. Hence we have cf 2 Ž x . x 2y s dg 2 Ž y . y 2y s constant a.e. in Ž 0, ⬁ . = Ž 0, ⬁ . , which contradicts the facts that 0 - H0⬁ t 1y f 2 Ž t . dt - ⬁ and 0 H0⬁ t 1y g 2 Ž t . dt - ⬁. Hence Ž2.14. takes the form of strict inequality. By Ž2.1., we have ⬁
⬁
H0 H0
-
f Ž x. g Ž y.
Ž Ax q By .
½
⬁
H0
dx dy
, A , B Ž x . f 2 Ž x . dx
⬁
H0
1r2
, B , A Ž y . g 2 Ž y . dy
5
. Ž 2.15.
In view of Ž2.4., we have Ž2.6.. Since H0⬁ t 1y f 2 Ž t . dt ) 0, then there exists a constant T0 ) 0 such that < .< for any T ) T0 , H0T t 1y f 2 Ž t . dt ) 0. Setting g Ž y, T . s y y1H0T Ž AxfqŽ x By . dx,
34
YANG BICHENG
y g Ž0, T xŽT ) T0 ., by Ž2.6., we have 0-
s
-
2
T 1y 2
y
H0
s
g Ž y, T . dy
T
½H
y y1
H0
0
T
H0 H0
Ž Ax q By .
1
Ž AB .
, 2 2
dx
dy
5
dx dy
2
ž /
B
2
f Ž x . g Ž y, T .
T
Ž Ax q By .
2
2
f Ž x.
T
T 1y 2
x
H0
T 1y 2
f Ž x . dt
H0
y
g Ž y, T . dy.
Ž 2.16.
Thus we find T
H0
y
F s
2
f Ž x.
T
y1
H0
Ž Ax q By .
dx
dy
T 1y 2
y
H0
T
y
H0
g Ž y, T . dy
y1
1
-
Ž AB .
2
f Ž x.
T
H0
Ž Ax q By .
B
, 2 2
2
ž /
dx
dy
T 1y 2
H0
x
f Ž x . dx.
Ž 2.17.
Since 0 - H0⬁ t 1y f 2 Ž t . dt - ⬁, by Ž2.17., it follows that 0 H0⬁ y 1y g 2 Ž y, ⬁. dy - ⬁. Hence by Ž2.6., when T ª ⬁, neither Ž2.16. nor Ž2.17. takes equality. We have Ž2.7.. On the other hand, if Ž2.7. is valid, by Cauchy’s inequality we have ⬁
⬁
H0 H0
s
F
f Ž x. g Ž y.
Ž Ax q By . ⬁
dx dy
y Ž y1.r2
H0
⬁
½H
0
y y1
⬁
H0
⬁
H0
f Ž x.
Ž Ax q By .
y Ž1y .r2 g Ž y . dy
dx
1r2
2
f Ž x.
Ž Ax q By .
dx
dy
⬁
H0
y 1y g 2 Ž y . dy
5
. Ž 2.18.
Whence by Ž2.7. we have Ž2.6.. It follows that Ž2.6. and Ž2.7. are equivalent.
GENERALIZATIONS OF HILBERT’S INEQUALITY
35
For 0 - - 12 , setting f Ž t . as f Ž t . s t Ž y2y .r2 , for t g w1, ⬁.; f Ž t . s 0, for t g Ž0, 1., we find H0⬁ t 1y f2 Ž t . dt s 1 . If there exists A, B, and ) 0 such that the constant factor Ž1rŽ AB . r2 . B Ž 2 , 2 . in Ž2.6. is not best possible, without loss of generality suppose that A G B; then there exists a positive number K - Ž1rŽ AB . r2 . B Ž 2 , 2 . such that Ž2.6. is valid by changing Ž1rŽ AB . r2 . B Ž 2 , 2 . to K. Particularly, we have ⬁
f Ž x . f Ž y .
⬁
H0 H0
Ž Ax q By .
⬁
dx dy - K
t 1y f2 Ž t . dt s Kr .
H0
Ž 2.19.
Since 1
⬁
H0
Ž 1 q u.
uŽ y2y .r2 du s B
, q o Ž 1. 2 2
ž /
Ž ª 0q . ,
setting u s Ž By .rŽ Ax ., by Ž2.5., we find ⬁
⬁
H0 H0
s s s
f Ž x . f Ž y .
Ž Ax q By . ⬁
H1
dx dy
1
Ž AB .
⬁
r2
H1
r2
½
1
Ž AB .
H1
⬁
H1
⬁
s
Ž AB . 1
1
r2
1
⬁
xy1 y
1
⬁
H0
Ž 1 q u.
H0Br Ax Ž
.
B
r2
B
uŽ y2y .r2 du dx
uŽ y2y .r2 du dx 1
Ž 1 q u.
, q o Ž 1. y O Ž 1. 2 2
½ ž /
1
Ž AB .
dy dx
HBrŽ Ax . Ž 1 q u .
xy1 y
H1
1
Ž Ax q By .
xy1 y
y s
y Ž y2y .r2
⬁
x Ž y2y .r2
, q o Ž 1. . 2 2
ž /
uŽ y2y .r2 du dx
5
5 Ž 2.20.
Since for Ž) 0. small enough we have wŽ1rŽ AB . r2 . B Ž 2 , 2 . q oŽ1.x ) K, by Ž2.20., in this case we obtain H0⬁H0⬁Ž f Ž x . f Ž y .rŽ Ax q By . . dx dy ) Kr , which contradicts Ž2.19.. Hence the constant factor Ž1rŽ AB . r2 . B Ž 2 , 2 . in Ž2.6. is best possible. Since Ž2.6. and Ž2.7. are equivalent, we may conclude
36
YANG BICHENG
that the constant factor in Ž2.7. is still best possible; otherwise, using Ž2.18., we may get a contradiction. Since B Ž 12 , 12 . s , and B Ž1, 1. s 1, by Ž2.6. and Ž2.7. we have Ž2.8. ᎐ Ž2.11.. The theorem is proved. Remark 1. For A s 1, ) 0, inequality Ž2.12. changes to Ž1.3. and Ž2.13. changes to the equivalent form of Ž1.3. as
⬁
H0
y
y1
H0
2
f Ž x.
⬁
Ž x q y.
dy - B
dx
, 2 2
2
ž /
⬁
H0
t 1y f 2 Ž t . dt. Ž 2.21.
Inequalities Ž2.12. and Ž2.6. are generalizations of Ž1.1. and Ž1.3..
3. SOME NEW RESULTS ON THE ASSOCIATED DOUBLE SERIES FORM Define the weight coefficient , A, B Ž m. as ⬁
, A , B Ž m . s
m
1
Ý ns1
Ž Am q Bn .
1y r2
ž / n
Ž m g N . Ž A, B, ) 0 . . Ž 3.1.
For g Ž0, 2x, 1 y 2 G 0, by Ž2.4. we have
, A , B Ž m . s
⬁ ms1
-
⬁
H0
m
1
Ý
B Ž AmrB q n .
B Ž AmrB q y .
n
1y r2
ž / ž / .
1
Ž AB
1y r2
ž /
m
1
s , A , B Ž m . s
r2
y
B
dy
, m1y 2 2
ŽmgN.. Ž 3.2.
THEOREM 3.1. Let 0 - F 2 and a n4 and bn4 be sequences of real numbers such that 0 - Ý⬁ns 1 n1y a2n - ⬁ and 0 - Ý⬁ns1 n1y bn2 - ⬁. If A, B
GENERALIZATIONS OF HILBERT’S INEQUALITY
37
) 0, we ha¨ e ⬁
⬁
a m bn
Ý Ý
Ž Am q Bn .
ns1 ms1
1
-
Ž AB .
B r2
, 2 2
ž /ž
1r2
⬁
⬁
ns1
ns1
Ý n1y a2n Ý n1y bn2
/
,
Ž 3.3. ⬁
⬁
ns1
2
am
Ý n y1 Ý ms1
Ž Am q Bn .
1
-
Ž AB .
B
, 2 2
⬁
2
Ý n1y a2n . Ž 3.4.
ž /
ns1
The inequalities Ž3.3. and Ž3.4. are equi¨ alent. Both the constant factor Ž1rŽ AB . r2 . B Ž 2 , 2 . in Ž3.3. and the constant factor Ž1rŽ AB . .w B Ž 2 , 2 .x2 in Ž3.4. are best possible. In particular, for s 1, if A, B ) 0, we ha¨ e
Ži.
⬁
⬁
a m bn
-
Ý Ý Ž Am q Bn .
ns1 ms1 ⬁
⬁
Ž AB .
⬁
⬁
Ý Ý ns1 ms1
a m bn
Ž Am q Bn .
⬁
⬁
Ýn Ý ns1
⬁
Ý ns1
2
/
,
Ž 3.5.
⬁
Ž AB .
ms1
-
2
⬁
1
Ž AB .
žÝ
Ž Am q Bn .
2
1
ns1
2
am
a m bn
Ý Ý ns1 ms1
ns1
bn2
Ý a2n ;
Ž 3.6.
ns1
-
n
⬁
a2n
Ý
1
Ž AB .
ns1
n
⬁
1
Ý
2
1r2
1
ns1
n
bn2
/
,
a2n ;
Ž 3.7. Ž 3.8.
for 0 - F 2, if A ) 0 Ž B s 1rA., we ha¨ e
Žiii.
Ý
žÝ
ns1
-
1r2
⬁
a2n
for s 2, if A, B ) 0, we ha¨ e
Žii.
⬁
1r2
2
am
Ý Ý ns1 ms1 Ž Am q Bn .
⬁
⬁
Ž Am q Ay1 n .
n y1
⬁
Ý ms1
-B
, 2 2
⬁
ž /ž Ý
n1y a2n
2
, 2 2
am y1
Ž Am q A n .
ns1
- B
1r2
⬁
n1y bn2
Ý ns1
ž /
2
⬁
/
, Ž 3.9.
Ý n1y a2n , Ž 3.10. ns1
where the constant factors in Ž3.5. ᎐ Ž3.10. are still best possible.
38
YANG BICHENG
Proof. By Cauchy’s inequality, we have ⬁
⬁
a m bn
Ý Ý
Ž Am q Bn .
ns1 ms1
s
⬁
⬁
am
Ý Ý
bn
= ⬁
⬁
Ý Ý
½
⬁
Ý Ý =
⬁
⬁
½Ý
ms1
m
1r2
5
Ž1y r2 .
ž /
n
n
Ž Am q Bn .
, B , A Ž m . a2m
ž / m
bn2
Ý Ý
Ž1y r2 .
n
Ž Bn q Am .
ns1 ms1
s
n
a2m
ms1 ns1 ⬁
ž /
Ž Am q Bn .
⬁
Ž1y r2 .
m
bn2
ns1 ms1
s
m
Ž Am q Bn .
ms1 ms1
=
Ž1y r2 .r2
a2m
½Ý Ý ⬁
n
ž /
r2
⬁
ž /
r2
n
Ž Am q Bn .
Ž1y r2 .r2
m
Ž Am q Bn .
ns1 ms1
F
Ž1y r2 .
ž / m
1r2
5
1r2
⬁
, A , B Ž n . bn2
Ý ns1
5
.
By Ž3.2., we have Ž3.3.. By the argument used to derive Ž2.7. and the equivalence of Ž2.6. and Ž2.7. we have Ž3.4., and we can conclude that Ž3.3. and Ž3.4. are equivalent. For 0 - - r2, setting a ˜n as a˜n s nŽ y2y .r2 Ž n g N ., then we obtain 1
s
1
⬁
H1
s1q
t
1q
dt -
ns2
Ý
⬁
n1y a ˜2n s
Ý
ns1
⬁
Ý
⬁
1 1q
n
-1q
ns1
1
⬁
H1
t
1q
1 1q
n
dt s 1 q
and ⬁
1
ns1
Ý n1y a˜2n s
q O Ž 1.
Ž ª 0q . .
1
GENERALIZATIONS OF HILBERT’S INEQUALITY
39
If there exist A G B ) 0 and 0 - F 2 such that the constant factor Ž1rŽ AB . r2 . B Ž 2 , 2 . in Ž3.3. is not best possible, then there exists a positive number K - Ž1rŽ AB . r2 . B Ž 2 , 2 . such that Ž3.3. is valid by changing Ž1rŽ AB . r2 . B Ž 2 , 2 . to K. Particularly, we have ⬁
⬁
a ˜m a˜n
Ý Ý ns1 ms1
Ž Am q Bn .
-K
⬁
Ý n1y a˜2n s K ns1
s
1
1
ž
q o Ž 1.
/
Ž K q o Ž 1. . .
Ž 3.11.
Since by Ž2.20. we have ⬁
⬁
Ý Ý ns1 ms1
a ˜m a˜n
Ž Am q Bn .
s
⬁
⬁
Ž mn .
Ý Ý ns1 ms1
) s
⬁
Ž Am q Bn .
H1
x Ž y2y .r2
1
1
Ž AB .
Ž y2y .r2
r2
B
y Ž y2y .r2
⬁
H1
Ž Ax q By .
dy dx
, q o Ž 1. , 2 2
ž /
Ž 3.12.
it is obvious that inequality Ž3.12. contradicts Ž3.11., for Ž) 0. small enough. Hence the constant factor Ž1rŽ AB . r2 . B Ž 2 , 2 . in Ž3.3. is best possible. We may show that the constant factor in Ž3.4. is best possible by the equivalence of Ž3.3. and Ž3.4.. The theorem is proved. Remark 2. For A s 1, 0 - F 2, inequalities Ž3.9. and Ž3.10. change to the following two new equivalent inequalities with the best constant factors: ⬁
⬁
Ý Ý ns1 ms1
a m bn
Ž m q n.
⬁
Ýn ns1
y1
⬁
Ý ms1
-B
, 2 2
⬁
ž /ž Ý 2
am
Ž m q n.
ns1
- B
1r2
⬁
n1y a2n
n1y bn2
Ý ns1
, 2 2
ž /
2
/
; Ž 3.13.
⬁
Ý n1y a2n . Ž 3.14. ns1
Inequalities Ž3.3., Ž3.9., and Ž3.13. are generalizations of Ž1.2.. Inequality
40
YANG BICHENG
Ž3.13. is similar to the new inequality Žcf. w8x. ⬁
⬁
Ý Ý ns0 ms0
-B
a m bn
Ž m q n q 1. , 2 2
⬁
ž /½ Ý ž ns0
nq
1 2
1y
/
⬁
a2n
Ý ns0
ž
nq
1 2
1r2
1y
/
bn2
5
Ž 0 - F 2 . . Ž 3.15. REFERENCES 1. G. H. Hardy, J. E. Littlewood, and G. Polya, ‘‘Inequalities,’’ Cambridge Univ. Press, Cambridge, 1952. 2. Hu Ke, On Hilbert’s inequality, Chinese Ann. of Math. 13B, No. 1 Ž1992., 35᎐39. 3. M. Gao, Tan Li, and L. Debnath, Some improvements on Hilbert’s integral inequality, J. Math. Anal. Appl. 229 Ž1999., 682᎐689. 4. M. Gao, A note on Hilbert double series theorem, Hunan Ann. Math. 12, Nos. 1᎐2 Ž1992., 142᎐147. 5. B. Yang, On Hilbert’s integral inequality, J. Math. Anal. Appl. 220 Ž1998., 778᎐785. 6. Z. Wang and D. Guo, ‘‘An Introduction to Special Functions,’’ Science Press, Beijing, 1979. 7. J. Kuang, ‘‘Applied Inequalities,’’ Hunan Education Press, Changsha, 1992. 8. B. Yang and L. Debnath, On a new generalization of Hardy᎐Hilbert’s inequality and its applications, J. Math. Anal. Appl. 233 Ž1999., 484᎐497.
On New Generalizations of Hilbert’s Inequality Yang Bicheng Department of Mathematics, Guangdong Education College, Guangzhou, Guangdong 510303, People’s Republic of China E-mail: [email protected] Submitted by L. Debnath Received February 25, 2000
By introducing three parameters A, B, and , we give some generalizations of Hilbert’s integral inequality and its equivalent form with the best constant factors. We also consider their associated double series forms. 䊚 2000 Academic Press Key Words: Hilbert’s inequality; weight function;  function; weight coefficient.
1. INTRODUCTION If f and g are real functions, such that 0 - H0⬁ f 2 Ž t . dt - ⬁ and 0 dt - ⬁, then
H0⬁ g 2 Ž t .
⬁
⬁
H0 H0
f Ž x. g Ž y. xqy
dx dy -
⬁
žH
f 2 Ž t . dt
⬁
H0
0
1r2
g 2 Ž t . dt
/
,
Ž 1.1.
where the constant factor is best possible. Its associated double series form is as follows: If a n4 and bn4 are sequences of real numbers such that 0 - Ý⬁ns 1 a2n - ⬁ and 0 - Ý⬁ns1 bn2 - ⬁, then ⬁
⬁
Ý Ý ns1 ms1
a m bn mqn
-
⬁
žÝ
ns1
1r2
⬁
a2n
Ý ns1
bn2
/
,
Ž 1.2.
where the constant factor is still best possible. Hilbert’s inequalities Ž1.1. and Ž1.2. are important in our analysis and its applications Žcf. w1, Chap. 9x.. In recent years, Hu w2x and Gao w3x gave two distinct improvements of Ž1.1., and Gao w4x gave Ž1.2. a strengthened version. By introducing a 29 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
30
YANG BICHENG
parameter g Ž0, 1x and estimating the weight function, Yang w5x gave Ž1.1. a generalization as ⬁
⬁
H0 H0
f Ž x. g Ž y.
Ž x q y.
, 2 2
dx dy - B
ž /ž
⬁
H0
t 1y f 2 Ž t . dt
⬁
H0 t
1r2 1y 2
g Ž t . dt
/
,
Ž 1.3. where B Ž p, q . is the  function. But w5x hasn’t proved that the constant factor B Ž 2 , 2 . Ž0 - F 1. in Ž1.3. is best possible. In this paper, we continue this work and make some generalizations of Ž1.3. by introducing three parameters, A, B, and . We also give the associated generalizations of Ž1.2..
2. SOME NEW RESULTS ON HILBERT’S INTEGRAL INEQUALITY First, we define the weight function , A, B Ž x . as
, A , B Ž x . s
⬁
H0
1y r2
x
1
Ž Ax q By .
ž /
x g Ž 0, ⬁ . Ž A, B, ) 0 . .
dy,
y
Ž 2.1. Setting u s Ž By .rŽ Ax . in Ž2.1., we have
, A , B Ž x . s s
Ž Ax .
⬁
H0
1y
B Ž 1 q u. 1
Ž AB .
r2
B
x 1y
1y r2
ž /
du
Au
1
⬁
H0
Ž 1 q u.
uy1q r2 du.
Ž 2.2.
Since the -function B Ž p, q . possesses the explicit formulation Žcf. w5, p. 117 Ž10.x.
B Ž p, q . s
⬁
H0
t py 1
Ž1 q t .
pq q
dt
Ž p, q ) 0 . ,
Ž 2.3.
GENERALIZATIONS OF HILBERT’S INEQUALITY
31
by Ž2.2. we have
, A , B Ž x . s
1
Ž AB .
1y , x , 2 2
x g Ž 0, ⬁ . Ž A, B, ) 0 . .
ž /
B
r2
Ž 2.4. If A G B ) 0, ) 0, and 0 - - r2, we ha¨ e
LEMMA 2.1. ⬁
H1
xy1 y
H0Br Ax Ž
1
.
Ž 1 q u.
uŽ y2y .r2 du dx s O Ž 1 .
Ž ª 0q . . Ž 2.5.
Proof. Since A G B ) 0, for x G 1, we have 0 - BrŽ Ax . F 1, and 1
Br Ž Ax .
H0
Ž 1 q u.
uŽ y2y .r2 du
Br Ž Ax .
H0
uŽ y2y r2.r2 du s
4
B
r4
ž / Ax
.
Then we find that 0-
⬁
H1 4
xy1 y
H0Br Ax Ž
r4
B
ž /
⬁
H1
A
1
.
Ž 1 q u.
xy1 y r4 dx s
uŽ y2y .r2 du dx 2
4
r4
B
ž /ž /
.
A
Hence Ž2.5. is valid. The lemma is proved. THEOREM 2.1. Let f and g be real functions, ) 0, such that 0 H0⬁ t 1y f 2 Ž t . dt - ⬁ and 0 - H0⬁ t 1y g 2 Ž t . dt - ⬁. If A, B ) 0, we ha¨ e ⬁
⬁
H0 H0
f Ž x. g Ž y.
Ž Ax q By .
dx dy
B , r2 2 2 Ž AB . 1
⬁
H0
y y1
⬁
H0
⬁
ž / žH
dx
f Ž t . dt
dy -
1
Ž AB .
⬁
H0
0
2
f Ž x.
Ž Ax q By .
t
1y 2
B
1r2
t
1y 2
, 2 2
g Ž t . dt
ž /
2
⬁
H0
/
; Ž 2.6.
t 1y f 2 Ž t . dt.
Ž 2.7.
32
YANG BICHENG
Inequalities Ž2.6. and Ž2.7. are equi¨ alent. Both the constant factor Ž1r Ž AB . r2 . B Ž 2 , 2 . in Ž2.6. and the constant factor Ž1rŽ AB . .? B Ž 2 , 2 .@ 2 in Ž2.7. are best possible. In particular, for s 1, if A, B ) 0, we ha¨ e
Ži. ⬁
f Ž x. g Ž y.
⬁
H0 H0
dx dy -
Ž Ax q By . ⬁
1r2
žH
H0
0
2
dy -
dx
Ž Ax q By .
⬁
f 2 Ž t . dt
2
⬁
H Ž AB . 0
1r2
g 2 Ž t . dt
/
, Ž 2.8.
f 2 Ž t . dt ;
Ž 2.9.
for s 2, if A, B ) 0, we ha¨ e
Žii. ⬁
⬁
Ž AB .
f Ž x.
⬁
H0 H0
f Ž x. g Ž y.
⬁
H0 H0
Ž Ax q By .
2
1
dx dy -
Ž AB .
ž
⬁
H0
1 t
f 2 Ž t . dt
⬁
H0
1 t
1r2
g 2 Ž t . dt
/
,
Ž 2.10. ⬁
H0 y H0
Ž Ax q By .
⬁
Ž Ax q Ay1 y .
-B
y
dy -
dx
f Ž x. g Ž y.
⬁
H0 H0
H0
2
1
⬁
Ž AB .
2
H0
1 t
f 2 Ž t . dt ;
Ž 2.11.
for ) 0, if A ) 0 Ž B s 1rA., we ha¨ e
Žiii.
⬁
2
f Ž x.
⬁
y1
⬁
H0
, 2 2
t 1y f 2 Ž t . dt 2
y.
dx
⬁
H0
0
f Ž x.
Ž Ax q A
dx dy
⬁
ž / žH y1
dy - B
1r2
t 1y g 2 Ž t . dt
, 2 2
ž /
2
⬁
H0
/
,
Ž 2.12.
t 1y f 2 Ž t . dt,
Ž 2.13. where the constant factors in Ž2.8. ᎐ Ž2.13. are still best possible.
GENERALIZATIONS OF HILBERT’S INEQUALITY
33
Proof. By Cauchy’s inequality, we have ⬁
f Ž x.
⬁
H0 H0
Ž Ax q By .
F
½
⬁
r2
Ž Ax q By .
y
Ž Ax q By .
⬁
x
Ž Ax q By .
r2
Ž1y r2 .r2
ž /
dx dy
x
dx dy
y
y
r2
y
1y r2
ž /
g2 Ž y.
⬁
H0 H0
g Ž y.
ž /
f 2Ž x.
⬁
H0 H0 =
Ž1y r2 .r2
x
1r2
1y r2
ž /
dx dy
x
5
.
Ž 2.14.
If Ž2.14. takes the form of the equality, then there exist constants c and d such that Žcf. w7, p. 29x.
c
f 2Ž x.
Ž Ax q By .
x
1y r2
ž / y
g2 Ž y.
sd
Ž Ax q By .
y
1y r2
ž / x
a.e. in Ž 0, ⬁ . = Ž 0, ⬁ . . Then we have cf 2 Ž x . x 2y s dg 2 Ž y . y 2y a.e. in Ž0, ⬁. = Ž0, ⬁.. Hence we have cf 2 Ž x . x 2y s dg 2 Ž y . y 2y s constant a.e. in Ž 0, ⬁ . = Ž 0, ⬁ . , which contradicts the facts that 0 - H0⬁ t 1y f 2 Ž t . dt - ⬁ and 0 H0⬁ t 1y g 2 Ž t . dt - ⬁. Hence Ž2.14. takes the form of strict inequality. By Ž2.1., we have ⬁
⬁
H0 H0
-
f Ž x. g Ž y.
Ž Ax q By .
½
⬁
H0
dx dy
, A , B Ž x . f 2 Ž x . dx
⬁
H0
1r2
, B , A Ž y . g 2 Ž y . dy
5
. Ž 2.15.
In view of Ž2.4., we have Ž2.6.. Since H0⬁ t 1y f 2 Ž t . dt ) 0, then there exists a constant T0 ) 0 such that < .< for any T ) T0 , H0T t 1y f 2 Ž t . dt ) 0. Setting g Ž y, T . s y y1H0T Ž AxfqŽ x By . dx,
34
YANG BICHENG
y g Ž0, T xŽT ) T0 ., by Ž2.6., we have 0-
s
-
2
T 1y 2
y
H0
s
g Ž y, T . dy
T
½H
y y1
H0
0
T
H0 H0
Ž Ax q By .
1
Ž AB .
, 2 2
dx
dy
5
dx dy
2
ž /
B
2
f Ž x . g Ž y, T .
T
Ž Ax q By .
2
2
f Ž x.
T
T 1y 2
x
H0
T 1y 2
f Ž x . dt
H0
y
g Ž y, T . dy.
Ž 2.16.
Thus we find T
H0
y
F s
2
f Ž x.
T
y1
H0
Ž Ax q By .
dx
dy
T 1y 2
y
H0
T
y
H0
g Ž y, T . dy
y1
1
-
Ž AB .
2
f Ž x.
T
H0
Ž Ax q By .
B
, 2 2
2
ž /
dx
dy
T 1y 2
H0
x
f Ž x . dx.
Ž 2.17.
Since 0 - H0⬁ t 1y f 2 Ž t . dt - ⬁, by Ž2.17., it follows that 0 H0⬁ y 1y g 2 Ž y, ⬁. dy - ⬁. Hence by Ž2.6., when T ª ⬁, neither Ž2.16. nor Ž2.17. takes equality. We have Ž2.7.. On the other hand, if Ž2.7. is valid, by Cauchy’s inequality we have ⬁
⬁
H0 H0
s
F
f Ž x. g Ž y.
Ž Ax q By . ⬁
dx dy
y Ž y1.r2
H0
⬁
½H
0
y y1
⬁
H0
⬁
H0
f Ž x.
Ž Ax q By .
y Ž1y .r2 g Ž y . dy
dx
1r2
2
f Ž x.
Ž Ax q By .
dx
dy
⬁
H0
y 1y g 2 Ž y . dy
5
. Ž 2.18.
Whence by Ž2.7. we have Ž2.6.. It follows that Ž2.6. and Ž2.7. are equivalent.
GENERALIZATIONS OF HILBERT’S INEQUALITY
35
For 0 - - 12 , setting f Ž t . as f Ž t . s t Ž y2y .r2 , for t g w1, ⬁.; f Ž t . s 0, for t g Ž0, 1., we find H0⬁ t 1y f2 Ž t . dt s 1 . If there exists A, B, and ) 0 such that the constant factor Ž1rŽ AB . r2 . B Ž 2 , 2 . in Ž2.6. is not best possible, without loss of generality suppose that A G B; then there exists a positive number K - Ž1rŽ AB . r2 . B Ž 2 , 2 . such that Ž2.6. is valid by changing Ž1rŽ AB . r2 . B Ž 2 , 2 . to K. Particularly, we have ⬁
f Ž x . f Ž y .
⬁
H0 H0
Ž Ax q By .
⬁
dx dy - K
t 1y f2 Ž t . dt s Kr .
H0
Ž 2.19.
Since 1
⬁
H0
Ž 1 q u.
uŽ y2y .r2 du s B
, q o Ž 1. 2 2
ž /
Ž ª 0q . ,
setting u s Ž By .rŽ Ax ., by Ž2.5., we find ⬁
⬁
H0 H0
s s s
f Ž x . f Ž y .
Ž Ax q By . ⬁
H1
dx dy
1
Ž AB .
⬁
r2
H1
r2
½
1
Ž AB .
H1
⬁
H1
⬁
s
Ž AB . 1
1
r2
1
⬁
xy1 y
1
⬁
H0
Ž 1 q u.
H0Br Ax Ž
.
B
r2
B
uŽ y2y .r2 du dx
uŽ y2y .r2 du dx 1
Ž 1 q u.
, q o Ž 1. y O Ž 1. 2 2
½ ž /
1
Ž AB .
dy dx
HBrŽ Ax . Ž 1 q u .
xy1 y
H1
1
Ž Ax q By .
xy1 y
y s
y Ž y2y .r2
⬁
x Ž y2y .r2
, q o Ž 1. . 2 2
ž /
uŽ y2y .r2 du dx
5
5 Ž 2.20.
Since for Ž) 0. small enough we have wŽ1rŽ AB . r2 . B Ž 2 , 2 . q oŽ1.x ) K, by Ž2.20., in this case we obtain H0⬁H0⬁Ž f Ž x . f Ž y .rŽ Ax q By . . dx dy ) Kr , which contradicts Ž2.19.. Hence the constant factor Ž1rŽ AB . r2 . B Ž 2 , 2 . in Ž2.6. is best possible. Since Ž2.6. and Ž2.7. are equivalent, we may conclude
36
YANG BICHENG
that the constant factor in Ž2.7. is still best possible; otherwise, using Ž2.18., we may get a contradiction. Since B Ž 12 , 12 . s , and B Ž1, 1. s 1, by Ž2.6. and Ž2.7. we have Ž2.8. ᎐ Ž2.11.. The theorem is proved. Remark 1. For A s 1, ) 0, inequality Ž2.12. changes to Ž1.3. and Ž2.13. changes to the equivalent form of Ž1.3. as
⬁
H0
y
y1
H0
2
f Ž x.
⬁
Ž x q y.
dy - B
dx
, 2 2
2
ž /
⬁
H0
t 1y f 2 Ž t . dt. Ž 2.21.
Inequalities Ž2.12. and Ž2.6. are generalizations of Ž1.1. and Ž1.3..
3. SOME NEW RESULTS ON THE ASSOCIATED DOUBLE SERIES FORM Define the weight coefficient , A, B Ž m. as ⬁
, A , B Ž m . s
m
1
Ý ns1
Ž Am q Bn .
1y r2
ž / n
Ž m g N . Ž A, B, ) 0 . . Ž 3.1.
For g Ž0, 2x, 1 y 2 G 0, by Ž2.4. we have
, A , B Ž m . s
⬁ ms1
-
⬁
H0
m
1
Ý
B Ž AmrB q n .
B Ž AmrB q y .
n
1y r2
ž / ž / .
1
Ž AB
1y r2
ž /
m
1
s , A , B Ž m . s
r2
y
B
dy
, m1y 2 2
ŽmgN.. Ž 3.2.
THEOREM 3.1. Let 0 - F 2 and a n4 and bn4 be sequences of real numbers such that 0 - Ý⬁ns 1 n1y a2n - ⬁ and 0 - Ý⬁ns1 n1y bn2 - ⬁. If A, B
GENERALIZATIONS OF HILBERT’S INEQUALITY
37
) 0, we ha¨ e ⬁
⬁
a m bn
Ý Ý
Ž Am q Bn .
ns1 ms1
1
-
Ž AB .
B r2
, 2 2
ž /ž
1r2
⬁
⬁
ns1
ns1
Ý n1y a2n Ý n1y bn2
/
,
Ž 3.3. ⬁
⬁
ns1
2
am
Ý n y1 Ý ms1
Ž Am q Bn .
1
-
Ž AB .
B
, 2 2
⬁
2
Ý n1y a2n . Ž 3.4.
ž /
ns1
The inequalities Ž3.3. and Ž3.4. are equi¨ alent. Both the constant factor Ž1rŽ AB . r2 . B Ž 2 , 2 . in Ž3.3. and the constant factor Ž1rŽ AB . .w B Ž 2 , 2 .x2 in Ž3.4. are best possible. In particular, for s 1, if A, B ) 0, we ha¨ e
Ži.
⬁
⬁
a m bn
-
Ý Ý Ž Am q Bn .
ns1 ms1 ⬁
⬁
Ž AB .
⬁
⬁
Ý Ý ns1 ms1
a m bn
Ž Am q Bn .
⬁
⬁
Ýn Ý ns1
⬁
Ý ns1
2
/
,
Ž 3.5.
⬁
Ž AB .
ms1
-
2
⬁
1
Ž AB .
žÝ
Ž Am q Bn .
2
1
ns1
2
am
a m bn
Ý Ý ns1 ms1
ns1
bn2
Ý a2n ;
Ž 3.6.
ns1
-
n
⬁
a2n
Ý
1
Ž AB .
ns1
n
⬁
1
Ý
2
1r2
1
ns1
n
bn2
/
,
a2n ;
Ž 3.7. Ž 3.8.
for 0 - F 2, if A ) 0 Ž B s 1rA., we ha¨ e
Žiii.
Ý
žÝ
ns1
-
1r2
⬁
a2n
for s 2, if A, B ) 0, we ha¨ e
Žii.
⬁
1r2
2
am
Ý Ý ns1 ms1 Ž Am q Bn .
⬁
⬁
Ž Am q Ay1 n .
n y1
⬁
Ý ms1
-B
, 2 2
⬁
ž /ž Ý
n1y a2n
2
, 2 2
am y1
Ž Am q A n .
ns1
- B
1r2
⬁
n1y bn2
Ý ns1
ž /
2
⬁
/
, Ž 3.9.
Ý n1y a2n , Ž 3.10. ns1
where the constant factors in Ž3.5. ᎐ Ž3.10. are still best possible.
38
YANG BICHENG
Proof. By Cauchy’s inequality, we have ⬁
⬁
a m bn
Ý Ý
Ž Am q Bn .
ns1 ms1
s
⬁
⬁
am
Ý Ý
bn
= ⬁
⬁
Ý Ý
½
⬁
Ý Ý =
⬁
⬁
½Ý
ms1
m
1r2
5
Ž1y r2 .
ž /
n
n
Ž Am q Bn .
, B , A Ž m . a2m
ž / m
bn2
Ý Ý
Ž1y r2 .
n
Ž Bn q Am .
ns1 ms1
s
n
a2m
ms1 ns1 ⬁
ž /
Ž Am q Bn .
⬁
Ž1y r2 .
m
bn2
ns1 ms1
s
m
Ž Am q Bn .
ms1 ms1
=
Ž1y r2 .r2
a2m
½Ý Ý ⬁
n
ž /
r2
⬁
ž /
r2
n
Ž Am q Bn .
Ž1y r2 .r2
m
Ž Am q Bn .
ns1 ms1
F
Ž1y r2 .
ž / m
1r2
5
1r2
⬁
, A , B Ž n . bn2
Ý ns1
5
.
By Ž3.2., we have Ž3.3.. By the argument used to derive Ž2.7. and the equivalence of Ž2.6. and Ž2.7. we have Ž3.4., and we can conclude that Ž3.3. and Ž3.4. are equivalent. For 0 - - r2, setting a ˜n as a˜n s nŽ y2y .r2 Ž n g N ., then we obtain 1
s
1
⬁
H1
s1q
t
1q
dt -
ns2
Ý
⬁
n1y a ˜2n s
Ý
ns1
⬁
Ý
⬁
1 1q
n
-1q
ns1
1
⬁
H1
t
1q
1 1q
n
dt s 1 q
and ⬁
1
ns1
Ý n1y a˜2n s
q O Ž 1.
Ž ª 0q . .
1
GENERALIZATIONS OF HILBERT’S INEQUALITY
39
If there exist A G B ) 0 and 0 - F 2 such that the constant factor Ž1rŽ AB . r2 . B Ž 2 , 2 . in Ž3.3. is not best possible, then there exists a positive number K - Ž1rŽ AB . r2 . B Ž 2 , 2 . such that Ž3.3. is valid by changing Ž1rŽ AB . r2 . B Ž 2 , 2 . to K. Particularly, we have ⬁
⬁
a ˜m a˜n
Ý Ý ns1 ms1
Ž Am q Bn .
-K
⬁
Ý n1y a˜2n s K ns1
s
1
1
ž
q o Ž 1.
/
Ž K q o Ž 1. . .
Ž 3.11.
Since by Ž2.20. we have ⬁
⬁
Ý Ý ns1 ms1
a ˜m a˜n
Ž Am q Bn .
s
⬁
⬁
Ž mn .
Ý Ý ns1 ms1
) s
⬁
Ž Am q Bn .
H1
x Ž y2y .r2
1
1
Ž AB .
Ž y2y .r2
r2
B
y Ž y2y .r2
⬁
H1
Ž Ax q By .
dy dx
, q o Ž 1. , 2 2
ž /
Ž 3.12.
it is obvious that inequality Ž3.12. contradicts Ž3.11., for Ž) 0. small enough. Hence the constant factor Ž1rŽ AB . r2 . B Ž 2 , 2 . in Ž3.3. is best possible. We may show that the constant factor in Ž3.4. is best possible by the equivalence of Ž3.3. and Ž3.4.. The theorem is proved. Remark 2. For A s 1, 0 - F 2, inequalities Ž3.9. and Ž3.10. change to the following two new equivalent inequalities with the best constant factors: ⬁
⬁
Ý Ý ns1 ms1
a m bn
Ž m q n.
⬁
Ýn ns1
y1
⬁
Ý ms1
-B
, 2 2
⬁
ž /ž Ý 2
am
Ž m q n.
ns1
- B
1r2
⬁
n1y a2n
n1y bn2
Ý ns1
, 2 2
ž /
2
/
; Ž 3.13.
⬁
Ý n1y a2n . Ž 3.14. ns1
Inequalities Ž3.3., Ž3.9., and Ž3.13. are generalizations of Ž1.2.. Inequality
40
YANG BICHENG
Ž3.13. is similar to the new inequality Žcf. w8x. ⬁
⬁
Ý Ý ns0 ms0
-B
a m bn
Ž m q n q 1. , 2 2
⬁
ž /½ Ý ž ns0
nq
1 2
1y
/
⬁
a2n
Ý ns0
ž
nq
1 2
1r2
1y
/
bn2
5
Ž 0 - F 2 . . Ž 3.15. REFERENCES 1. G. H. Hardy, J. E. Littlewood, and G. Polya, ‘‘Inequalities,’’ Cambridge Univ. Press, Cambridge, 1952. 2. Hu Ke, On Hilbert’s inequality, Chinese Ann. of Math. 13B, No. 1 Ž1992., 35᎐39. 3. M. Gao, Tan Li, and L. Debnath, Some improvements on Hilbert’s integral inequality, J. Math. Anal. Appl. 229 Ž1999., 682᎐689. 4. M. Gao, A note on Hilbert double series theorem, Hunan Ann. Math. 12, Nos. 1᎐2 Ž1992., 142᎐147. 5. B. Yang, On Hilbert’s integral inequality, J. Math. Anal. Appl. 220 Ž1998., 778᎐785. 6. Z. Wang and D. Guo, ‘‘An Introduction to Special Functions,’’ Science Press, Beijing, 1979. 7. J. Kuang, ‘‘Applied Inequalities,’’ Hunan Education Press, Changsha, 1992. 8. B. Yang and L. Debnath, On a new generalization of Hardy᎐Hilbert’s inequality and its applications, J. Math. Anal. Appl. 233 Ž1999., 484᎐497.