On a Strengthened Hardy-hilbert Inequality_bicheng Yang
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Transcript of On a Strengthened Hardy-hilbert Inequality_bicheng Yang
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volume 1, issue 2, article 22,2000.
Received 8 May, 2000;accepted 10 June 2000.
Communicated by: L. Debnath
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Journal of Inequalities in Pure andApplied Mathematics
ON A STRENGTHENED HARDY-HILBERT INEQUALITY
BICHENG YANGDepartment of MathematicsGuangdong Education CollegeGuangzhouGuangdong 510303PEOPLES REPUBLIC OF CHINAEMail : [email protected]
c2000 Victoria UniversityISSN (electronic): 1443-5756012-00
Please quote this number (012-00) in correspondence regarding this paper with the Editorial Office.
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On a StrengthenedHardy-Hilbert Inequality
Bicheng Yang
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J. Ineq. Pure and Appl. Math. 1(2) Art. 22, 2000http://jipam.vu.edu.au
Abstract
In this paper, a new inequality for the weight coefficient W (n, r) of the form
W (n, r) =
m=0
1m+ n+ 1
(n+ 12m+ 12
) 1r
1, n N0 = N {0})
is proved. This is followed by a strengthened version of the more accurateHardy-Hilbert inequality.
2000 Mathematics Subject Classification: 26D15Key words: Hardy-Hilbert inequality, Weight Coefficient, Hlders inequality.
Contents1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Some Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Theorem and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10References
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On a StrengthenedHardy-Hilbert Inequality
Bicheng Yang
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J. Ineq. Pure and Appl. Math. 1(2) Art. 22, 2000http://jipam.vu.edu.au
1. IntroductionIf p > 1,1
p+ 1
q= 1, an, bn 0, and 0 1, n N0, we have (n, r) > (n,), and
(2.7) W (n, r) < pisin(pir
) (n,)(2n+ 1)2
1r
(r > 1, n N0) ,
where(2.8) (n,) = (2n+ 1)
2
2 (n+ 1)+
=0
(1)(1 + ) (2n+ 1)1
+
0
B1 (t)
[(2n+ 1)2
(t+ n+ 1)2
]dt.
Since by (2.3) and (2.1),we have 0
B1 (t)1
(t+ n+ 1)2dt = 1
12 (n+ 1)2 1
3!
0
B3 (t)
[1
(t+ n+ 1)
]dt
> 112 (n+ 1)2
and=0
(1)(1 + ) (2n+ 1)1
= (2n+ 1) 12+
=2
(1)(1 + ) (2n+ 1)1
> (2n+ 1) 12+
1
6 (2n+ 1).
Then by (2.8), we find
(2.9) (n,) > 16 1
6 (n+ 1) 1
12 (n+ 1)2+
1
6 (2n+ 1).
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On a StrengthenedHardy-Hilbert Inequality
Bicheng Yang
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J. Ineq. Pure and Appl. Math. 1(2) Art. 22, 2000http://jipam.vu.edu.au
Lemma 2.3. For r > 1, n N0, we have
(2.10) W (n, r) < pisin(pir
) 113 (n+ 1) (2n+ 1)1
1r
.
Proof. Define the function g(x) as
g(x) =1
12 1
6 (2x+ 1)+
1
12 (x+ 1)+
1
12 (2x+ 1)2, x [0,).
Then by (2.8), we have (n,) > 2n+1n+1
g (n). Since g(1) > 0.0787 > 113
, andfor x [1,),
g (x) =1
3 (2x+ 1)2 112 (x+ 1)2
13 (2x+ 1)3
=4x2 + 2x 1
12 (x+ 1)2 (2x+ 1)3> 0,
then for n 1, we have (n,) > 2n+1(n+1)
g (1) > 2n+113(n+1)
. Hence by (2.7),inequality (2.10) is valid for n 1. Since ln 2 = 0.1159+ > 1
13, then by
(1.5), we find(2.11)W (0, r) 1,1
p+ 1
q= 1, an, bn 0, 0
