Journal of Inequalities in Pure and
Applied Mathematics
A REFINEMENT OF VAN DER CORPUT’S INEQUALITY
DA-WEI NIU, JIAN CAO, AND FENG QI
School of Mathematics and Informatics
Henan Polytechnic University
Jiaozuo City, Henan Province
454010, China
EMail: nnddww@hotmail.com
EMail: 21caojian@163.com
Research Institute of Mathematical Inequality Theory
Henan Polytechnic University
Jiaozuo City, Henan Province
454010, China
EMail: nnddww@hotmail.com
URL: http://rgmia.vu.edu.au/qi.html
volume 7, issue 4, article 127,
2006.
Received 08 May, 2006;
accepted 10 August, 2006.
Communicated by: B. Yang
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
136-06
Quit
Abstract
In this note, a refinement of van der Corput’s inequality is given.
2000 Mathematics Subject Classification: Primary 26D15.
Key words: Refinement, van der Corput’s inequality, Harmonic number.
The authors were supported in part by the Science Foundation of the Project for
Fostering Innovation Talents at Universities of Henan Province, China
A Refinement of van der
Corput’s Inequality
Da-Wei Niu, Jian Cao and Feng Qi
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
6
8
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 10
J. Ineq. Pure and Appl. Math. 7(4) Art. 127, 2006
http://jipam.vu.edu.au
1.
Introduction
P
Pn
Let an ≥ 0 for n ∈ N such that 0 < ∞
n=1 (n + 1)an < ∞, and Sn =
m=1
the harmonic number. Then van der Corput’s inequality [5] states that
n
∞
Y
X
(1.1)
n=1
! S1
n
1/k
ak
<e
1+γ
∞
X
1
,
m
(n + 1)an ,
n=1
k=1
where γ = 0.57721566 . . . stands for Euler-Mascheroni’s constant. The factor
e1+γ in (1.1) is the best possible.
In 2003, Hu in [3] gave a strengthened version of (1.1) by
∞
n
X
Y
(1.2)
n=1
! S1
n
1/k
ak
<e
1+γ
∞ X
n=1
k=1
ln n
n−
4
an .
(1.3)
n=1
! S1
n
1/k
ak
<e
1+γ
∞ X
n=1
k=1
ln n
n−
3
an .
(1.4)
n=1
k=1
! S 1(β)
n
1/(k+β)
ak
<e
1+γ(β)
∞ X
n=1
Title Page
JJ
J
II
I
Go Back
Close
Moreover, he also extended (1.1) in [7] as follows
∞
n
X
Y
Da-Wei Niu, Jian Cao and Feng Qi
Contents
Recently, Yang in [7] obtained a better result than Hu’s inequality (1.2) as
∞
n
X
Y
A Refinement of van der
Corput’s Inequality
Quit
1
n + + β an ,
2
Page 3 of 10
J. Ineq. Pure and Appl. Math. 7(4) Art. 127, 2006
http://jipam.vu.edu.au
where β ∈ (−1, ∞), Sn (β) =
Pn
1
k=1 k+β ,
"
γ(β) = lim
n→∞
n
X
k=1
and
#
1
− ln(n + β) .
k+β
Applying β = 0 in (1.4) leads to
n
∞
X
Y
(1.5)
n=1
! S1
n
1/k
ak
<e
1+γ
∞ X
n=1
k=1
1
n+
2
an ,
A Refinement of van der
Corput’s Inequality
which improved inequality (1.1) clearly, but is not more accurate than (1.2) and
(1.3).
In [1], among other things, the authors established a sharper inequality than
(1.1), (1.2), (1.3) and (1.5) as follows
(1.6)
∞
n
X
Y
n=1
k=1
! S1
n
1/k
ak
<e
1+γ
∞
X
n 1−
n=1
ln n
3n − 1/4
an .
The purpose of this note is to refine further inequality (1.6). Our main result
is the following.
P
Theorem 1.1. For n ∈ N, let Sn = nm=1 m1 , the harmonic number. If an ≥ 0
for n ∈ N and
∞
X
ln n
0<
n 1−
an < ∞,
2n + ln n + 11/6
n=1
Da-Wei Niu, Jian Cao and Feng Qi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 10
J. Ineq. Pure and Appl. Math. 7(4) Art. 127, 2006
http://jipam.vu.edu.au
then
(1.7)
∞
n
X
Y
n=1
! S1
n
1/k
ak
k=1
<e
1+γ
∞
X
6(6n+1)γ−9
− (6n+1)(12n+11)
e
n=1
ln n
n 1−
2n + ln n + 11/6
an ,
where γ = 0.57721566 . . . is Euler-Mascheroni’s constant.
A Refinement of van der
Corput’s Inequality
Remark 1. Let
Da-Wei Niu, Jian Cao and Feng Qi
ln n
2n + ln n + 11/6
ln 1
1+γ
for n ∈ N. Numerical computation shows A1 = 4.40 . . . < e
1 − 3−1/4 =
2 ln 2
4.84 . . . and A2 = 7.99 . . . < e1+γ 2 − 6−1/4
= 8.51 . . . . and A3 =
3 ln 2
11.95 . . . < e1+γ 3 − 9−1/4
= 12.70 . . . . When n ≥ 4, inequality 2n +
6(6n+1)γ−9
An = e1+γ e− (6n+1)(12n+11) n 1 −
11
6
+ ln n < 3n −
1
4
is valid, which can be rearranged as
Title Page
Contents
JJ
J
II
I
Go Back
Close
ln n
ln n
1−
≤1−
.
2n + ln n + 11/6
3n − 1/4
Quit
Page 5 of 10
This implies that inequality (1.7) is a refinement of (1.6).
J. Ineq. Pure and Appl. Math. 7(4) Art. 127, 2006
http://jipam.vu.edu.au
2.
Lemmas
In order to prove our main result, some lemmas are necessary.
Lemma 2.1 ([4]). For n ∈ N,
1
1
< Sn − ln n − γ <
.
2n + 1/(1 − γ) − 2
2n + 1/3
(2.1)
1
1−γ
The constants
− 2 and
1
3
in (2.1) are the best possible.
A Refinement of van der
Corput’s Inequality
Lemma 2.2 ([2, 6]). If x > 0, then
x
1
1
<e 1−
(2.2)
1+
.
x
2x + 11/6
Da-Wei Niu, Jian Cao and Feng Qi
Title Page
Lemma 2.3. For n ∈ N,
(2.3) Bn ,
Contents
(n + 1)Sn+1
nSn
nSn
6(6n+1)γ−9
<e
1+γ − (6n+1)(12n+11)
e
ln n
n 1−
2n + ln n + 11/6
JJ
J
.
Proof. By virtue of Lemma 2.2, it follows that
(n + 1)Sn+1
nSn
SnS+1
n
n
nSn
Sn + 1 Sn +1
= 1+
nSn
Sn + 1
1
<e 1−
<e 1−
.
2nSn + 11(Sn + 1)/6
2n + 11/6
II
I
Go Back
Close
Quit
Page 6 of 10
J. Ineq. Pure and Appl. Math. 7(4) Art. 127, 2006
http://jipam.vu.edu.au
Applying Lemma 2.1 yields
Sn +1
1
(2.4)
Bn < e 1 −
2n + 11/6
1
1+ 2n+1/3
+γ+ln n
1
1
1+ 2n+1/3
+γ+ln n
<e
1−
.
2n + 11/6
−x
1
Taking advantage of inequalities 1 − x1
> e for x > 1 and e−x ≤ 1+x
for
x > −1 leads to
ln n
ln n
1
2n + 11/6
≤ exp −
≤
(2.5)
1−
2n + 11/6
2n + 11/6
2n + 11/6 + ln n
and
(2.6)
1−
exp
Combination of (2.4), (2.5), (2.6) gives
6(6n+1)γ−9
1+γ− (6n+1)(12n+11)
(2.7)
Bn < e
n−
Lemma 2.3 is proved.
Da-Wei Niu, Jian Cao and Feng Qi
Title Page
1
2n+1/3
+1+γ
1
2n + 1/3
1
1+γ
1
< exp
−
−
2n + 1/3 2n + 11/6 (2n + 11/6)(2n + 1/3)
6(6n + 1)γ − 9
.
= exp −
(6n + 1)(12n + 11)
1
2n + 11/6
A Refinement of van der
Corput’s Inequality
Contents
JJ
J
II
I
Go Back
Close
Quit
n ln n
2n + ln n + 11/6
.
Page 7 of 10
J. Ineq. Pure and Appl. Math. 7(4) Art. 127, 2006
http://jipam.vu.edu.au
3.
Proof of Theorem 1.1
For n ∈ N and 1 ≤ k ≤ n, let
[(k + 1)Sk+1 ]kSk
ck =
(kSk )kSk−1
(3.1)
with assumption S0 = 0, then
n
Y
(3.2)
!− S1
n
1/k
=
ck
k=1
1
.
(n + 1)Sn+1
A Refinement of van der
Corput’s Inequality
By using the discrete weighted arithmetic-geometric mean inequality and
interchanging the order of summations,
! S1
" n
# S1
!− S1
∞
n
∞
n
n
n
n
X
Y
Y
X
Y
1/k
1/k
(3.3)
=
(ck ak )1/k
ck
ak
n=1
n=1
k=1
≤
∞
n
X
Y
n=1
≤
k=1
∞
X
n=1
∞
X
k=1
k=1
!− S1
n
1/k
ck
n
1 X c k ak
Sn k=1 k
n
X
1
c k ak
(n + 1)Sn+1 Sn k=1 k
∞
c k ak X
1
=
k n=k (n + 1)Sn Sn+1
k=1
∞
∞ X
c k ak X 1
1
−
=
k n=k Sn Sn+1
k=1
Da-Wei Niu, Jian Cao and Feng Qi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 8 of 10
J. Ineq. Pure and Appl. Math. 7(4) Art. 127, 2006
http://jipam.vu.edu.au
=
=
∞
X
c k ak
kSk
k=1
∞ X
k=1
(k + 1)Sk+1
kSk
kSk
ak =
∞
X
Bn an .
n=1
Substituting (2.7) into (3.3) leads to (1.7). The proof of Theorem 1.1 is complete.
A Refinement of van der
Corput’s Inequality
Da-Wei Niu, Jian Cao and Feng Qi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 9 of 10
J. Ineq. Pure and Appl. Math. 7(4) Art. 127, 2006
http://jipam.vu.edu.au
References
[1] J. CAO, D.-W. NIU AND F. QI, An extension and a refinement of van der
Corput’s inequality, Int. J. Math. Math. Sci., 2006 (2006), Article ID 70786,
10 pages.
[2] CH.-P. CHEN AND F. QI, On further sharpening of Carleman’s inequality,
Dàxué Shùxué (College Mathematics), 21(2) (2005), 88–90. (Chinese)
[3] K. HU, On van der Corput’s inequality, J. Math. (Wuhan), 23(1) (2003),
126–128. (Chinese)
A Refinement of van der
Corput’s Inequality
[4] F. QI, R.-Q. CUI, CH.-P. CHEN, AND B.-N. GUO, Some completely monotonic functions involving polygamma functions and an application, J. Math.
Anal. Appl., 310(1) (2005), 303–308.
Da-Wei Niu, Jian Cao and Feng Qi
[5] J.G. VAN DER CORPUT, Generalization of Carleman’s inequality, Proc.
Akad. Wet. Amsterdam, 39 (1936), 906–911.
Contents
[6] Z.-T. XIE AND Y.-B. ZHONG, A best approximation for constant e and
an improvment to Hardy’s inequality, J. Math. Anal. Appl., 252 (2000),
994–998.
[7] B.-CH. YANG, On an extension and a refinement of van der Corput’s inequality, Chinese Quart. J. Math., 22 (2007), in press.
Title Page
JJ
J
II
I
Go Back
Close
Quit
Page 10 of 10
J. Ineq. Pure and Appl. Math. 7(4) Art. 127, 2006
http://jipam.vu.edu.au