AbstractThis paper deals with a generalization of the Hardy–Hilbert inequality with best constant factor which involves the β function. As an application, we obtain a new equivalent form of the Hardy–Hilbert inequality

Bicheng, Yang

Debnath, Lokenath

This paper deals with a generalization of the Hardy-Hilbert inequality with best constant factor which involves the β function. As an application, we obtain a new equivalent form of the Hardy-Hilbert inequality. © 1999 Academic Press

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On A New Generalization Of Hardy-Hilbert\u27S Inequality And Its Applications

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Ž .Journal of Mathematical Analysis and Applications 233, 484]497 1999
Article ID jmaa.1999.6295, available online at http:rrwww.idealibrary.com on
On a New Generalization of Hardy]Hilbert's Inequality
and Its Applications
Yang Bicheng
Department of Mathematics, Guangdong Education College, Guangzhou, Guangdong
510303, The People’s Republic of China
and
Lokenath Debnath
Department of Mathematics, Uni¤ersity of Central Florida, Orlando, Florida 32816
Submitted by William F. Ames
Received November 2, 1998
This paper deals with a generalization of the Hardy]Hilbert inequality with best
constant factor which involves the b function. As an application, we obtain a new
equivalent form of the Hardy]Hilbert inequality. Q 1999 Academic Press
Key Words: Hardy]Hilbert's inequality; weight coefficient; b function.
1. INTRODUCTION
1 1 ‘ p ‘ qIf a , b G 0, p ) 1, q s 1, and 0 - Ý a - ‘, 0 - Ý b - ‘,n n ns0 n ns0 np q
Ž w x.then the famous Hardy]Hilbert's inequality Hardy et al. 1 may be
written in the following form:
1rp 1rq‘ ‘ ‘ ‘a b pn m p q- a b , 1.1Ž .Ý Ý Ý Ýn n½ 5 ½ 5m q n q 1 sin prpŽ . Ž .ms0 ns0 ns0 ns0
pŽ .where the constant factor prsin is best possible.p
Ž .In particular, when p s q s 2, inequality 1.1 reduces to the standard
Hilbert's inequality
1r2‘ ‘ ‘ ‘a bn m 2 2- p a b . 1.2Ž .Ý Ý Ý Ýn n½ 5m q n q 1Ž .ms0 ns0 ns0 ns0
484
0022-247Xr99 $30.00
Copyright Q 1999 by Academic Press
All rights of reproduction in any form reserved.
HARDY]HILBERT'S INEQUALITY 485
Ž . Ž .Both inequalities 1.1 and 1.2 play an important role in analysis and its
w x w xapplications. Recently, Yang 2, 3 and Gau 4 gave some improvements of
Ž . Ž . w x1.1 and 1.2 by estimating the weight coefficients. Yang 5 also general-
Ž .ized the integral form of 1.1 by introducing some parameters. We
consider the weight function
Ž .2yl r2‘ 1 x
v x s dy , x g 0, ‘ , 0 - l F 1 .Ž . Ž . Ž .Hl l ž /y0 x q yŽ .
1.3Ž .
w xRecently, Yang 6 gives a generalization of Hilbert's integral inequality as
follows:
‘ ‘ f x g yŽ . Ž .
dx dyH H l
0 0 x q yŽ .
1r2
‘ ‘l l
1yl 2 1yl 2F B , x f x dx x g x dx , 1.4Ž . Ž . Ž .H H½ 5ž /2 2 0 0
Ž . Ž .where B p, q p, q ) 0 is the b function. Following the paper by Yang
w x6 , we introduce a parameter l and consider the weight coefficient
Ž .2yl rr1‘ 1 n q 2
v r , n s , 1.5Ž . Ž .Ýl 1l ž /m qm q n q 1Ž . 2ms0
Žwhere N is the set of nonnegative integers and n g N r ) 1, 2 y r - l0 0
.F 2 .
The main objective of this paper is to formulate a new inequality related
to the double series
‘ ‘ a bn m
1.6Ž .Ý Ý lm q n q 1Ž .ms0 ns0
Ž . ŽŽso that inequality 1.1 can be generalized with a best constant B l q p
. Ž . .y 2 rp, l q q y 2 rq . As an application, we give its equivalent form
and obtain some particular results.
2. SOME RESULTS AND LEMMAS
Ž w x.First, we need the following inequality see Yang and Debnath 7 :
Ž r .Ž . Ž2 ry1.Ž . Ž2 r .Ž . w .If for r s 0, 1, 2, 3, 4, f ‘ s 0, f x - 0, f x G 0, x g 0, ‘ ,
‘ Ž .and H f x dx - ‘, then0
‘ ‘ 1 1
f m F f x dx q f 0 y f 9 0 . 2.1Ž . Ž . Ž . Ž . Ž .Ý H 2 120ms0
YANG AND DEBNATH486
LEMMA 2.1. If n g N , r ) 1, 2 y r - l F 2, and l ) 0, define the0
Ž .function R r, n byl
Ž . Ž .1yly 2yl rr 2yl rr1 1 1Ž .1r 2 nq1
R r , n s n q duŽ . Hl lž / ž /2 u0 1 q uŽ .
3r q 2 y l lŽ .
Ž2yl.r r Ž2yl.r ry ? 2 y ? 2 , 2.2Ž .l lq16 r n q 1 12 n q 1Ž . Ž .
Ž .then R r, n ) 0.l
Proof. Integrating by parts, we have
Ž .2yl rr1 1Ž .1r 2 nq1
duH l ž /u0 1 q uŽ .
1 1Ž .1r 2 nq1 1y Ž2yl.r rs duH l1 y 2 y l rrŽ . 0 1 q uŽ .
Ž .1r 2 nq11 1
1y Ž2yl.r rs u
l1 y 2 y l rrŽ . 1 q uŽ . 0
1Ž .1r 2 nq1 1y Ž2yl.r ry u dH l
0 1 q uŽ .
Ž .ly1q 2yl rry1qŽ2yl.r r2 1 1
s n q
l ž /1 y 2 y l rr 2Ž . n q 1Ž .
l 1Ž .1r 2 nq1 1y Ž2yl.r rq u duH lq11 y 2 y l rrŽ . 0 1 q uŽ .
Ž .ly1q 2yl rry1qŽ2yl.r r2 1 1
s n q
l ž /1 y 2 y l rr 2Ž . n q 1Ž .
Ž .ly1q 2yl rr1y2qŽ2yl.r rl 2 n qŽ .2q ?
lq11 y 2 y l rr 2 y 2 y l rŽ . Ž . n q 1Ž .
1Ž .1r 2 nq1 2y Ž2yl.r rq l q 1 u duŽ .H lq2
0 1 q uŽ .
HARDY]HILBERT'S INEQUALITY 487
Ž .ly1q 2yl rr1y1qŽ2yl.r r2 n qŽ .2
)
l1 y 2 y l rrŽ . n q 1Ž .
Ž .ly1q 2yl rr1l n qŽ .2q ? .
lq11 y 2 y l rrŽ . n q 1Ž .
Ž .Then, by 2.1 , we find
y1 Ž2yl.r r3r q 2 y l 2 2
R r , n ) y qŽ .l l6 r 1 y 2 y l rrŽ . n q 1Ž .
y2 Ž2yl.r r1 1 2 l 2
q y q ? .
lq112 1 y 2 y l rr 2 y 2 y l rrŽ . Ž . n q 1Ž .
2.3Ž .
Ž . Ž . Ž .2Since 0 - l F 2 and 0 F 2 y l rr - 1, inequality 2 2 y l q 2 y l rr
wŽ . x w y1 Ž Ž . .xG 0 is equivalent to y 3r q 2 y l r6 r q 2 r 1 y 2 y l rr G 0,
12ŽŽ . . ŽŽ . .and inequality 2 y l rr y 3 2 y l rr y 1 - 0 is equivalent to y12
w Ž Ž . .x y2 Ž Ž . . Ž .q 1r 1 y 2 y l rt ? 2 r 2 y 2 y l rr ) 0. Hence, by 2.3 , we
Ž .have R r, n ) 0. The lemma is proved.l
1 1  4 Ž .LEMMA 2.2. If p ) 1, q s 1, 2 y min p, q - l F 2, and v r, nlp q
Ž .is defined by 1.5 , we ha¤e
1ylp y 2 q l q y 2 q l 1
v r , n - B , n q , 2.4Ž . Ž .l ž /ž /p q 2
Ž . ŽŽ . Ž . . ‘ Žwhere n g N r s p, q , B p y 2 q l rp, q y 2 q l rq s H 1r 1 q0 0
1l Ž2yl.r p. Ž .u du.u
Ž .Proof. By 2.1 , we have
Ž .2yl rr‘ 1 1
Ý 1l ž /m qm q n q 1Ž . 2ms0
Ž .2yl rr
‘ 1 1 1
Ž2yl.r rF dt q ? 2H 1l lž /t q0 t q n q 1 2 n q 1Ž . Ž .2
l 2 y l
Ž2yl.r r 1qŽ2yl.r rq ? 2 q ? 2
lq1 l12 n q 1 12 r n q 1Ž . Ž .
YANG AND DEBNATH488
Ž .2yl rr
‘ 1 1 3r q 2 y l
Ž2yl.r rs dt q ? 2H 1l lž /t q0 t q n q 1 6 r n q 1Ž . Ž .2
l
Ž2yl.r rq ? 2 . 2.5Ž .lq112 n q 1Ž .
1 1Ž . Ž .Setting u s t q r n q , we find2 2
Ž .2yl rr
‘ 1 1
dtH 1l ž /t q0 t q n q 1Ž . 2
Ž . Ž .1yly 2yl rr 2yl rr
‘1 1 1
s n q duH lž / ž /2 uŽ .1r 2 nq1 1 q uŽ .
Ž . Ž .1yly 2yl rr 2yl rr
‘1 1 1
s n q duH lž / ž /2 u0 1 q uŽ .
Ž .2yl rr1 1Ž .1r 2 nq1y du .H l ž /u0 1 q uŽ .
Ž . Ž . Ž .Then, by 1.5 , 2.5 , and 2.2 , we have
Ž . Ž .2yl rr 1yly 2yl rr1 1
v r , n F n q n qŽ .l ž / ž /½2 2
=
Ž .2yl rr
‘ 1 1
du y R r , n .Ž .H ll ž / 5u0 1 q uŽ .
2.6Ž .
 4Since 2 y min p, q - l F 2, for r s p, q, we have 2 y r - l F 2, and
Ž .l ) 0. By Lemma 1.1 and 2.6 , we have
Ž .1yl 2yl rr
‘1 1 1
v r , n - n q du. 2.7Ž . Ž .Hl lž / ž /2 u0 1 q uŽ .
Ž w x.Since the b function has the relation see 8, p. 117
‘
y1qut
B u , ¤ s dt s B ¤ , u u , ¤ ) 0 ,Ž . Ž . Ž .H uq¤
1 q tŽ .0
HARDY]HILBERT'S INEQUALITY 489
Ž . Ž .and q y 2 q l rq q p y 2 q l rp s l, we have
Ž .2yl rp
‘ 1 1
duH l ž /u0 1 q uŽ .
‘ 1 p y 2 q l q y 2 q l
y1qŽ py2ql.r ps u du s B ,H l ž /p q0 1 q uŽ .
Ž .2yl rq
‘q y 2 q l p y 2 q l 1 1
s B , s du.H l ž /ž /q p u0 1 q uŽ .
Ž . Ž .Hence, by 2.7 , we have 2.4 . This proves the lemma.
1 1  4LEMMA 2.3. If p ) 1, q s 1, l ) 2 y min p, q , x G 0, and « ) 0,p q
then
y«y1 Žyzqly« .r q
‘ 1 uŽ .1r 2 xq1 qx q du dx s O 1 « “ 0 .Ž . Ž .H H lž /20 0 1 q uŽ .
2.8Ž .
Proof. We have
uŽy2qly« .r qŽ .1r 2 xq1
duH l
0 1 q uŽ .
Ž .1r 2 xq1 Žy2qly« .r q- u duH
0
Ž .1q y2qly« rq1 1
s .ž /1 q 1rq y2 q l y « 2 x q 1Ž . Ž .
Ž .  4Since 1 q y2 q l rq ) 0 and l ) 2 y min p, q G 0, there exists n g0
Ž .N, such that lrn - 1, for 0 - « - lrn , 1 q y2 q l y « rq ) 0. Since0 0
1 1 yŽ . Ž . Ž Ž ..for a ) 1 the function g y s y g 0, ‘ is strictly decreasing, wey a
find
Ž .1q y2qly« rq1 1ž /1 q y2 q l y « rq 2 x q 1Ž .
w Ž .x1q y2ql 1y1rn rq01 1
F ;ž /1 q 1rq y2 q l 1 y 1rn 2 x q 1Ž . Ž .0
YANG AND DEBNATH490
y«y1 Žy2qly« .r q
‘ 1 uŽ .1r 2 xq1
0 - x q du dxH H lž /20 0 1 q uŽ .
w Ž .x2q y2ql 1y1rn rq1qŽl r n . 00 ‘2 1
- dxH ž /1 q 1rq y2 q l 1 y 1rn 2 x q 1Ž . Ž .Ž . 00
21
lr n0s 2 .½ 51 q 1rq y2 q l 1 y 1rnŽ . Ž .Ž .0
Ž .Hence relation 2.8 is valid. The lemma is proved.
3. MAIN RESULTS
1 1  4THEOREM 3.1. If a , b G 0, p ) 1, q s 1, 2 y min p, q - l F 2,n n p q
1 1‘ 1yl p ‘ 1yl qŽ . Ž .and 0 - Ý n q a - ‘, 0 - Ý n q b - ‘, thenns0 n ns0 n2 2
‘ ‘ a bn mÝ Ý lm q n q 1Ž .ms0 ns0
p y 2 q l q y 2 q l
- B ,ž /p q
1rp 1rq1yl 1yl‘ ‘1 1
p q= n q a n q b ,Ý Ýn nž / ž /½ 5 ½ 52 2ns0 ns0
3.1Ž .
ŽŽ . Ž . .where the constant factor B p y 2 q l rp, q y 2 q l rq is best possible.
In particular, when l s 2, we ha¤e
1rp 1rq‘ ‘ ‘ ‘a b 1 1n m p q- 2 a b ,Ý Ý Ý Ýn n2 ½ 5 ½ 52n q 1 2n q 1Ž . Ž .m q n q 1Ž .ms0 ns0 ns0 ns0
3.2Ž .
where the constant 2 is still best possible.
Proof. By Holder's inequality, we have
‘ ‘ a bn mÝ Ý lm q n q 1Ž .ms0 ns0
Ž .2yl rp q1‘ ‘ a n qn 2s Ý Ý 1lrp ž /m qm q n q 1Ž . 2ms0 ns0
HARDY]HILBERT'S INEQUALITY 491
Ž .2yl rp q1b m qm 2
= 1lrq ž /n qm q n q 1Ž . 2
1rpŽ .2yl rq1p‘ ‘ a n qn 2F Ý Ý 1l½ 5ž /m qm q n q 1Ž . 2ms0 ns0
1rqŽ .2yl rp1q‘ ‘ b m qm 2
= Ý Ý 1l½ 5ž /n qm q n q 1Ž . 2ms0 ns0
1rpŽ .2yl rq1p‘ ‘ a n qn 2s Ý Ý 1l½ 5ž /m qm q n q 1Ž . 2ns0 ms0
1rqŽ .2yl rp1q‘ ‘ b m qm 2
= Ý Ý 1l½ 5ž /n qm q n q 1Ž . 2ms0 ns0
1rp 1rq‘ ‘
p qs v q , n a v p , m b .Ž . Ž .Ý Ýl n l m½ 5 ½ 5
ns0 ms0
Ž . Ž .Then, by 2.4 , we have 3.1 .
1 Žly2y« .r pŽ . ŽFor 0 - « F lrn - 1, setting a s n q , b s m0 n m2
1 Žly2y« .r q.q , then2
y1y« 1yl y1y«« ‘ ‘‘2 1 1 1
ps x q dx - n q a s n qÝ ÝH nž / ž / ž /« 2 2 20 ns0 ns0
y1y« y1y«‘ ‘1 1
1q« 1q«s 2 q n q - 2 q x q dxÝ Hž / ž /2 20ns1
- 4 q 2«r« ;
1yl 1yl «‘ ‘1 1 2
p q qn q a s n q b s q O 1 « “ 0 .Ž . Ž .Ý Ýn nž / ž /2 2 «ns0 ns0
3.3Ž .
 4 Ž .Since 0 F 2 y min p, q - l F 2, by 2.8 , we have
‘ ‘ a bn mÝ Ý lm q n q 1Ž .ms0 ns0
Ž .ly2y« rqŽ . 1ly2y« rp
‘ ‘1 y qŽ .2
) x q dy dxH H lž /20 0 x q y q 1Ž .
YANG AND DEBNATH492
y1y« Žly2y« .r q
‘ ‘1 u
s x q du dxH H lž /2 Ž .0 1r 2 xq1 1 q uŽ .
y1y« Žly2y« .r q
‘ ‘1 u
s x q du dxH H lž /20 0 1 q uŽ .
y1y« Žly2y« .r q
‘ 1 uŽ .1r 2 xq1y x q du dxH H lž /20 0 1 q uŽ .
«
‘
Žly2y« .r q2 u
s du y O 1 .Ž .H l« 0 1 q uŽ .
Since we have
‘
Žly2y« .r qu p y 2 q l q y 2 q l
lim du s B , ;H l ž /p q«“0q 0 1 q uŽ .
‘
Žly2y« .r qu p y 2 q l q y 2 q l
qdu s B , q o 1 « “ 0 ,Ž . Ž .H l ž /p q0 1 q uŽ .
we find
‘ ‘ a bn mÝ Ý lm q n q 1Ž .ms0 ns0
«2 p y 2 q l q y 2 q l
) B , q o 1 y O 1Ž . Ž .ž /« p q
«2 p y 2 q l q y 2 q l
s B , q o 1 , 3.4Ž . Ž .ž /« p q
ŽŽ . Ž . . Žwhere 0 - K - B p y 2 q l rp, q y 2 q l rq . Putting « ) 0 « F
.lrn - 1 small enough so that0
«2 p y 2 q l q y 2 q l
B , q o 1 ) K ,Ž .« ž /2 q « O 1 p qŽ .
HARDY]HILBERT'S INEQUALITY 493
Ž . Ž .we have, by 3.3 and 3.4 ,
‘ ‘ a bn mÝ Ý lm q n q 1Ž .ms0 ns0
2«
) K q O 1Ž .ž /«
1rp 1rq1yl 1yl‘ ‘1 1
p qs K n q a n q b .Ý Ýn nž / ž /½ 5 ½ 52 2ns0 ns0
ŽŽ . Ž . .It follows that the constant factor B p y 2 q l rp, q y 2 q l rq in
Ž .3.1 is best possible.
Ž . Ž . Ž . Ž .Since k 2 s B 1, 1 s 1, by 3.1 , inequality 3.2 is valid. It is obviousp
Ž .that the constant factor 2 in 3.2 is still best possible. The theorem is
proved.
When p s 2, we have the following:
1‘ 1yl 2 ‘Ž . ŽCOROLLARY 3.1. If 0 - Ý n q a - ‘ and 0 - Ý n qns0 n ns02
1 1yl 2. b - ‘, where 0 - l F 2, thenn2
‘ ‘ a bn mÝ Ý lm q n q 1Ž .ms0 ns0
1r21yl 1yl‘ ‘l l 1 1
2 2- B , n q a n q b , 3.5Ž .Ý Ýn nž / ž / ž /½ 52 2 2 2ns0 ns0
Ž .where the constant factor B lr2, lr2 is best possible.
In particular, for l s 2, we have
1r2‘ ‘ ‘ ‘a b 1 1n m 2 2- 2 a b ,Ý Ý Ý Ýn n2 ½ 52n q 1 2n q 1Ž . Ž .m q n q 1Ž .ms0 ns0 ns0 ns0
3.6Ž .
where the constant 2 is still best possible.
Ž . Ž . Ž .Remark. When l s 1, 3.1 reduces to 1.1 . Inequality 3.1 is a
Ž . Ž .generalization of 1.1 . It is obvious that inequality 3.5 is a generalization
Ž .of 1.2 .
YANG AND DEBNATH494
4. SOME APPLICATIONS
1 1  4THEOREM 4.1. If a G 0, p ) 1, q s 1, 2 y min p, q - l F 2,n p q
1‘ 1yl pŽ .and 0 - Ý n q a - ‘, thenns0 n2
pŽ .Ž .ly1 py1‘ ‘1 an
m qÝ Ý lž /2 m q n q 1Ž .ms0 ns0
p 1yl‘p y 2 q l q y 2 q l 1
p- B , n q a , 4.1Ž .Ý nž /ž /p q 2ns0
w ŽŽ . Ž . .x pwhere the constant factor B p y 2 q l rp, q y 2 q l rq is best possi-
Ž . Ž .ble. Inequality 4.1 is equi¤alent to 3.1 .
In particular,
Ž .i for l s 1, we ha¤e
p p‘ ‘ ‘a pn p- a , 4.2Ž .Ý Ý Ý nž /m q n q 1 sin prpŽ .ms0 ns0 ns0
Ž .ii for l s 2, we ha¤e
ppy1‘ ‘ ‘1 a 1n pm q - 2 a , 4.3Ž .Ý Ý Ý n2ž /2 2n q 1Ž .m q n q 1Ž .ms0 ns0 ns0
where the constant 2 is still best possible.
1‘ 1yl pŽ .Proof. Since 0 - Ý n q a - ‘, there exists k g N such thatns0 n 02
1k 1yl pŽ .for any k G k , 0 - Ý n q a - ‘.0 ns0 n2
We set
py1ly1 k1 an
b k s m qŽ . Ým lž /2 m q n q 1Ž .ns0
) 0 m g N , k G k ,Ž .0
Ž .and use 3.1 to obtain
1ylk 1
q0 - m q b kŽ .Ý mž /2ms0
pŽ .Ž .ly1 py1k k1 ans m qÝ Ý lž /2 m q n q 1Ž .ms0 ns0
HARDY]HILBERT'S INEQUALITY 495
k k a b kŽ .n ms Ý Ý lm q n q 1Ž .ms0 ns0
1rp1ylkp y 2 q l q y 2 q l 1
p- B , n q aÝ nž /½ 5ž /p q 2ns0
=
1rq1ylk 1
qn q b k . 4.4Ž . Ž .Ý nž /½ 52ns0
Hence, we find
1rp1ylk 1
qm q b kŽ .Ý mž /2ms0
1rppŽ .Ž .ly1 py1k k1 ans m qÝ Ý lž /½ 52 m q n q 1Ž .ms0 ns0
1rp1ylkp y 2 q l q y 2 q l 1
p- B , n q a . 4.5Ž .Ý nž /½ 5ž /p q 2ns0
1‘ 1yl qŽ . Ž . Ž .It follows that 0 - Ý m q b ‘ - ‘. Hence, 4.4 is valid asms 0 m2
Ž . Ž . Ž .k “ ‘ by 3.1 ; so is 4.5 . Thus, inequality 4.1 holds. We have proved
Ž . Ž .that inequality 3.1 implies 4.1 .
Ž . Ž .We next show that inequality 4.1 implies the inequality 3.1 :
‘ ‘ a bn mÝ Ý lm q n q 1Ž .ms0 ns0
Ž .1yl rq‘ ‘1 a 1ns m q bÝ Ý mŽ . l1yl rq ž /1 2m q n q 1Ž .m qms0 ns0Ž .2
1rpp‘ ‘1 anF Ý ÝŽ .Ž . l1yl py11½ 5m q n q 1Ž .m qms0 ns0Ž .2
1rqqŽ .1yl rq‘ 1
= m q bÝ mž /½ 52ms0
1rppŽ .Ž .ly1 py1‘ ‘1 ans m qÝ Ý lž /½ 52 m q n q 1Ž .ms0 ns0
YANG AND DEBNATH496
1rq1yl‘ 1
q= m q b . 4.6Ž .Ý mž /½ 52ms0
1‘ 1yl qŽ . Ž . Ž .Since 0 - Ý m q b - ‘, by 4.1 , we obtain 3.1 . It follows thatms 0 m2
Ž . Ž . Ž .3.1 implies 4.1 . Since the constant factor in 3.1 is best possible, we
Ž .may show that the constant factor in 4.1 is also best possible by using
Ž . Ž  4 .4.6 . Otherwise, there exist l 2 y min p, q - l F 2 and K ) 0 such
ŽŽ . Ž . .that K - B p y 2 q l rp, q y 2 q l rq , and
pŽ .Ž . 1ylly1 py1‘ ‘ ‘1 a 1n p pm q - K n q a .Ý Ý Ý nlž / ž /2 2m q n q 1Ž .ms0 ns0 ns0
Ž .Then, by 4.6 , we find
‘ ‘ a bn mÝ Ý lm q n q 1Ž .ms0 ns0
1rp 1rq1yl 1yl‘ ‘1 1
p q- K n q a n q b .Ý Ýn nž / ž /½ 5 ½ 52 2ns0 ns0
1 1ŽŽ . Ž . . Ž .This is a contradiction. Since B p y 1 rp, q y 1 rq s B , sq p
Ž . Ž . Ž . Ž .prsin prp , for l s 1, by 4.1 , we have 4.2 . It is obvious that 4.3 holds
Ž .for l s 2, by 4.1 . This proves the theorem.
1‘ 1yl 2Ž .COROLLARY 4.1. If 0 - l F 2, and 0 - Ý n q a - ‘, thenns0 n2
we ha¤e
2Ž .ly1‘ ‘1 an
m qÝ Ý lž /2 m q n q 1Ž .ms0 ns0
2 1yl‘l l 1
2- B , n q a , 4.7Ž .Ý nž / ž /2 2 2ns0
l l 2w Ž .x Ž .where the constant factor B , is best possible. Inequality 4.7 is2 2
Ž .equi¤alent to 3.5 . In particular,
Ž .i for l s 1, we ha¤e
2‘ ‘ ‘an 2 2- p a ; 4.8Ž .Ý Ý Ý nž /m q n q 1ms0 ns0 ns0
HARDY]HILBERT'S INEQUALITY 497
Ž .ii for l s 2, we ha¤e
2 2‘ ‘ ‘1 a an n
m q - 2 , 4.9Ž .Ý Ý Ý2ž /2 2n q 1Ž .m q n q 1Ž .ms0 ns0 ns0
where the constant 2 is still best possible.
Ž . Ž . Ž .Remark. Inequality 4.2 is equivalent to 1.1 , and inequality 4.1 is a
Ž . Ž .generalization of 4.2 , which is equivalent to 3.1 . Since all the constant
factors involving the b function are best possible, we obtain some new
results.
REFERENCES
1. G. H. Hardy, J. E. Littlewood, and G. Polya, ``Inequalities,'' Cambridge University Press,
Cambridge, MA, 1952.
2. Yang Bicheng, A refinement of the more profound Hardy]Hilbert's inequality, Hunan
Ž .Ann. Math. 17, No. 2 1997 , 35]38.
Ž .3. Yang Bicheng, A refinement of Hilbert's inequality, Huanghuai J. 13, No. 2 1997 , 47]51.
4. Gau Mingzhe, On Hilbert's inequality and its applications, J. Math. Anal. Appl. 212
Ž .1997 , 316]323.
5. Yang Bicheng, On generalizations of Hardy]Hilbert's integral inequalities, Acta Math.
Ž .Sinica 41, No. 4 1998 , 839]844.
Ž .6. Yang Bicheng, On Hilbert's integral inequality, J. Math. Anal. Appl. 220 1998 , 778]785.
7. Yang Bicheng and L. Debnath, On new strengthened Hardy-Hilbert's inequality, Internat.
Ž .J. Math. Math. Sci. 21, No. 2 1998 , 403]408.
8. Wang Zhuxi and Gua, Dunren, ``An Introduction to Special Functions,'' Science Press,
1979.
