Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 1, Issue 2, Article 22, 2000
ON A STRENGTHENED HARDY-HILBERT INEQUALITY
BICHENG YANG
D EPARTMENT OF M ATHEMATICS
G UANGDONG E DUCATION C OLLEGE , G UANGZHOU
G UANGDONG 510303, P EOPLE ’ S R EPUBLIC OF C HINA .
bcyang@163.net
Received 8 May, 2000; accepted 10 June 2000
Communicated by L. Debnath
A BSTRACT. In this paper, a new inequality for the weight coefficient W (n, r) of the form
1
∞
X
n + 21 r
1
W (n, r) =
m + n + 1 m + 12
m=0
1
π
(r > 1, n ∈ N0 = N ∪ {0})
π −
1− 1
sin r
13 (n + 1) (2n + 1) r
is proved. This is followed by a strengthened version of the more accurate Hardy-Hilbert inequality.
<
Key words and phrases: Hardy-Hilbert inequality, Weight Coefficient, Hölder’s inequality.
2000 Mathematics Subject Classification. 26D15.
1. I NTRODUCTION
P
P∞
p
q
a
<
∞,
0
<
If p > 1, p1 + 1q = 1, an , bn ≥ 0, and 0 < ∞
n
n=1−λ
n=1−λ bn < ∞ (λ = 0, 1),
then the Hardy-Hilbert inequality is
! p1
! 1q
∞
∞
∞
∞
X
X
X
X
am b n
π
apn
bqn
,
(1.1)
<
π
m
+
n
+
λ
sin
m=1−λ n=1−λ
n=1−λ
n=1−λ
p
where the constant π/ sin(π/p) is best possible (see [3, Chapter 9]). Inequality (1.1) is important in analysis and it’s applications (see [4, Chapter 5]). In recent years, Yang and Gao
[2, 7], have given a strengthened version of (1.1) for λ = 0 as

  1q
  p1 


∞
∞
∞
∞



X
X X am b n
X
1 − γ q
π
1 − γ p
π


,
−
bn
(1.2)
−
an
<
1
1

  n=1 sin π
m + n  n=1 sin π
nq
np
m=1 n=1
p
ISSN (electronic): 1443-5756
c 2000 Victoria University. All rights reserved.
012-00
p
2
B ICHENG YANG
where γ = 0.5772+ , is Euler’s constant. Later, Yang and Debnath [6] proved a distinctly
strengthened version of (1.1) for λ = 0 as
(1.3)

  1q
  p1 


∞ X
∞
∞
∞




X
X
X
π
1
1
am b n
q
p



 π −
−
a
<
b
,
1
1
n
n
−1
−1
π

 
m + n  n=1 sin π
q + n p
p + n q
2n
2n
sin
n=1
m=1 n=1
p
p
which is not comparable with (1.2).
Inequality (1.1) for λ = 1 is called the more accurate Hardy-Hilbert’s inequality, which has
been strengthened as
(1.4)
∞
∞ X
X
am b n
m+n+1
m=0 n=0

  1q

  p1 

∞
∞



X
X
 π − ln 2 − γ 1  bqn
 π − ln 2 − γ 1  apn
,
<
π
π

 

(2n + 1)1+ q
(2n + 1)1+ p
n=0 sin p
n=0 sin p
by introducing an inequality of the weight coefficient W (n, r) in the form (see [5, equation
(2.9)]):
1
∞
X
n + 12 r
1
π
ln 2 − γ
−
(1.5) W (n, r) =
<
(r > 1, n ∈ N0 ) .
1
π
2− r1
m
+
n
+
1
m
+
sin
(2n
+
1)
2
r
m=0
In this paper we will give another strengthened version of (1.1) for λ = 1, which is not
comparable with (1.4). We need some preparatory works.
2. S OME L EMMAS
6
(2q)
Lemma
(x) > 0 (q = 0, 1, 2, 3), f (i−1) (∞) = 0 (i = 1, 2, . . . , 6),
P∞ 2.1. If f ∈ C [0, ∞), f
and n=0 f (n) < ∞, then we have
∞
X
1
(−1)k f (k) > f (2) (see [1, equation (4.4)]),
2
k=2
Z ∞
Z ∞
∞
X
1
f (m) =
f (t) dt + f (0) +
B̄1 f 0 (t) dt,
2
0
0
m=0
Z
Z
∞
1 0
1 ∞
0
B̄1 f (t) dt = − f (0) + δ2 , δ2 =
B̄3 f 000 (t) dt < 0,
12
6 0
0
(2.1)
(2.2)
(2.3)
where B̄i (t) (i = 1, 3) are Bernoulli functions (see [5, equations (1.7)-(1.9)]).
Setting the weight coefficient W (n, r) in the form:
1
∞
X
n + 12 r
1
π
θ (n, r)
−
(2.4) W (n, r) =
(r > 1, n ∈ N0 ) ,
=
1
π
2− r1
m
+
n
+
1
m
+
sin
(2n
+
1)
2
r
m=0
then we find
1
(2.5)
r
∞
X
1
1
π
2
2− r1
(2n + 1)
.
− (2n + 1)
θ (n, r) =
m + n + 1 2m + 1
sin πr
m=0
J. Ineq. Pure and Appl. Math., 1(2) Art. 22, 2000
http://jipam.vu.edu.au/
H ARDY-H ILBERT I NEQUALITY
If we define the function f (x) as f (x) =
1
(x+n+1)
1
2x+1
r1
3
, x ∈ [0, ∞), then we have f (0) =
1
,
(n+1)
r1
1+ r1
1
1
2
1
f (x) = −
−
,
r (x + n + 1) 2x + 1
(x + n + 1)2 2x + 1
1
2
f 0 (0) = −
,
2 −
r (n + 1)
(n + 1)
0
and
Z
∞
Z
1
f (x) dx =
1
(2n + 1) r
0
∞
1
1
(y + 1)
r1
1
dy
y
" 2n+1
#
∞
X
1
π
(−1)ν
−
=
.
1
π
ν+1− r1
1
(2n + 1) r sin r
1
−
+
ν
(2n
+
1)
ν=0
r
By (2.2) and (2.5), we have
(2.6)
∞
(2n + 1)2 X
(−1)ν
+
θ (n, r) = −
2 (n + 1) ν=0 1 − 1r + ν (2n + 1)ν−1
"
#
Z ∞
(2n + 1)2
2 (2n + 1)2
dt.
+
B̄1 (t)
1 +
1
0
(t + n + 1)2 (2t + 1) r
r (t + n + 1) (2t + 1)1+ r
By Lemma 4 of [5, p. 1106] and (2.4), we have
Lemma 2.2. For r > 1, n ∈ N0 , we have θ (n, r) > θ (n, ∞), and
W (n, r) <
(2.7)
π
θ (n, ∞)
−
(r > 1, n ∈ N0 ) ,
1
π
sin r
(2n + 1)2− r
where
∞
(2n + 1)2 X
(−1)ν
+
(2.8) θ (n, ∞) = −
+
2 (n + 1) ν=0 (1 + ν) (2n + 1)ν−1
Z
0
∞
"
#
(2n + 1)2
B̄1 (t)
dt.
(t + n + 1)2
Since by (2.3) and (2.1),we have
000
Z ∞
Z
1
1
1 ∞
1
B̄1 (t)
dt = −
−
B̄3 (t)
dt
(t + n + 1)
(t + n + 1)2
12 (n + 1)2 3! 0
0
1
>−
12 (n + 1)2
and
∞
X
ν=0
∞
(−1)ν
1 X
(−1)ν
=
(2n
+
1)
−
+
2 ν=2 (1 + ν) (2n + 1)ν−1
(1 + ν) (2n + 1)ν−1
1
1
> (2n + 1) − +
.
2 6 (2n + 1)
Then by (2.8), we find
(2.9)
θ (n, ∞) >
1
1
1
1
−
−
.
2 +
6 6 (n + 1) 12 (n + 1)
6 (2n + 1)
J. Ineq. Pure and Appl. Math., 1(2) Art. 22, 2000
http://jipam.vu.edu.au/
4
B ICHENG YANG
Lemma 2.3. For r > 1, n ∈ N0 , we have
W (n, r) <
(2.10)
π
1
−
1 .
π
sin r
13 (n + 1) (2n + 1)1− r
Proof. Define the function g(x) as
g(x) =
1
1
1
1
−
+
+
, x ∈ [0, ∞).
12 6 (2x + 1) 12 (x + 1) 12 (2x + 1)2
Then by (2.8), we have θ (n, ∞) >
2n+1
g
n+1
(n). Since g(1) > 0.0787 >
1
,
13
and for x ∈ [1, ∞),
1
1
4x2 + 2x − 1
1
−
−
=
> 0,
3 (2x + 1)2 12 (x + 1)2 3 (2x + 1)3
12 (x + 1)2 (2x + 1)3
g 0 (x) =
2n+1
2n+1
then for n ≥ 1, we have θ (n, ∞) > (n+1)
g (1) > 13(n+1)
. Hence by (2.7), inequality (2.10) is
1
valid for n ≥ 1. Since ln 2 − γ = 0.1159+ > 13 , then by (1.5), we find
W (0, r) <
(2.11)
ln 2 − γ
π
1
π
−
−
1 <
1 .
π
π
sin r
sin r
(2 × 0 + 1)2− r
13 (0 + 1) (2 × 0 + 1)1− r
It follows that (2.10) is valid for r > 1,and n ∈ N0 . This proves the lemma.
3. T HEOREM
Theorem 3.1. If p > 1, p1 +
then
(3.1)
∞ X
∞
X
m=0 n=0
1
q
R EMARKS
P
P∞ q
p
= 1, an , bn ≥ 0, 0 < ∞
a
<
∞,
and
0
<
n
n=0
n=0 bn < ∞,
AND

  p1

∞


X
am b n
1
p

 π −
a
<
1
n

m + n + 1  n=0 sin π
13 (n + 1) (2n + 1) p
p


  1q
∞
X

1
q

 π −
b
,
×
1
n


13 (n + 1) (2n + 1) q
sin π
n=0
(3.2)
∞
X
m=0
∞
X
n=0
an
m+n+1
p−1

!p
<
sin
π 
π
p
p
∞
X

n=0


sin
π
−
π
p
1
13 (n + 1) (2n + 1)
1
p
 apn .
Proof. By Hölder’s inequality, we have
J. Ineq. Pure and Appl. Math., 1(2) Art. 22, 2000
http://jipam.vu.edu.au/
H ARDY-H ILBERT I NEQUALITY
∞ X
∞
X
m=0 n=0
5
"
1 #"
1#
∞ X
∞
X
m + 12 pq
n + 12 pq
am
am b n
bn
=
1
1
m + n + 1 m=0 n=0 (m + n + 1) p1 n + 12
(m + n + 1) q m + 2
) p1
( ∞ ∞ "
1#
1 q
XX
m+ 2
1
apm
≤
1
(m
+
n
+
1)
n
+
2
m=0 n=0
) 1q
( ∞ ∞ "
1#
1 p
XX
n
+
1
2
bqn
×
1
(m + n + 1) m + 2
m=0 n=0
( ∞ ∞ "
) p1
1#
1 q
XX
m+ 2
1
=
apm
1
(m
+
n
+
1)
n
+
2
m=0 n=0
(∞ ∞ "
) 1q
1#
1 p
XX
n
+
1
2
×
bqn
1
(m + n + 1) m + 2
n=0 m=0
( ∞
) p1 ( ∞
) 1q
X
X
=
W (m, q) apm
W (n, p) bqn
.
m=0
n=0
Since sin(π/q) = sin(π/p), by (2.10) for r = p, q, we have (3.1).
By (2.10), we have W (n, p) < π/ sin(π/p). Then by Hölder’s inequality, we obtain
"
1 #"
1#
∞
∞
X
X
n + 12 pq
m + 12 pq
an
an
1
=
1
1
1
m
+
n
+
1
m + 12
p
(m + n + 1) q n + 2
n=0
n=0 (m + n + 1)
1
(∞ "
) p1 ( ∞
1#
1 )q
1 q
1 p
X
X
n+ 2
m+ 2
1
1
≤
apn
1
(m + n + 1) m + 2
(m + n + 1) n + 12
n=0
n=0
(∞ "
1 # ) p1
X
1
n + 12 q p
1
=
an
{W (m, p)} q
1
(m + n + 1) m + 2
n=0
1
(∞ "
) p1 
1#
q

1 q
X
n+ 2
1
π 
p
<
an
.
1
π 

(m
+
n
+
1)
m
+
sin
2
n=0
p
Then we find
∞
X
∞
X
m=0
n=0
an
m+n+1
!p
<
∞ X
∞
X
m=0 n=0

=
sin
J. Ineq. Pure and Appl. Math., 1(2) Art. 22, 2000
1
(m + n + 1)
p−1
"
π 
π
p
sin
π 
π
p
∞ X
∞
X
n=0 m=0
p−1

=
"
∞
X
n + 12
m + 12
1q #
 pq

apn 
1
(m + n + 1)
sin
π 
n + 12
m + 12
π
p
1q #
apn
W (n, q) apn .
n=0
http://jipam.vu.edu.au/
6
B ICHENG YANG
Hence by (2.10) for r = q, we have (3.2). The theorem is proved.
Remark 3.1. Inequality (3.1) is a definite improvement over (1.1) for λ = 1.
1
Remark 3.2. Since for n ≥ 2, 13 (2 − γ) < 2 − n+1
, then
π
π
ln 2 − γ
1
−
−
(r > 1, n ≥ 2) .
1 >
1
π
π
2−
sin r
sin r
(2n + 1) r
13 (n + 1) (2n + 1)1− r
In view of (2.11) and (3.3), it follows that (3.1) and (1.4) represent two distinct versions of
strengthened inequalities. However, they are not comparable.
Remark 3.3. Inequality (3.2) reduces to
p
!p 
∞
∞
∞
X
X
an
π  X p

(3.4)
<
an .
m+n+1
sin π
m=0
n=0
n=0
(3.3)
p
This is an equivalent form of the more accurate Hardy-Hilbert’s inequality (see [3, Chapter
9]).
R EFERENCES
[1] R.P. BOAS, Estimating remainders, Math. Mag., 51(2) (1978), 83–89.
[2] M. GAO AND B. YANG, On the extended Hilbert’s inequality, Proc. Amer. Math. Soc., 126(3)
(1998), 751–759.
[3] G.H. HARDY, J.E. LITTLEWOOD
1934.
AND
G. PÓLYA, Inequalities, Cambridge University Press,
[4] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequalities Involving Functions and their
Integrals and Derivatives, Kluwer Academic Publishers, 1991.
[5] B. YANG, On a strengthened version of the more accurate Hardy-Hilbert’s inequality, Acta Math.
Sinica, 42(6) (1999), 1103–1110.
[6] B. YANG AND L. DEBNATH, On new strengthened Hardy-Hilbert’s inequality, Int. J. Math. Math.
Sci., 21(2) (1998), 403–408.
[7] B. YANG
156–164.
AND
M. GAO, On a best value of Hardy-Hilbert’s inequality, Adv. Math., 26(2) (1997),
J. Ineq. Pure and Appl. Math., 1(2) Art. 22, 2000
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