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volume 7, issue 4, article 127, 2006.

Received 08 May, 2006;

accepted 10 August, 2006.

Communicated by:B. Yang

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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

c

2000Victoria University ISSN (electronic): 1443-5756 136-06

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A Refinement of van der Corput’s Inequality Da-Wei Niu, Jian Cao and Feng Qi

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J. Ineq. Pure and Appl. Math. 7(4) Art. 127, 2006

http://jipam.vu.edu.au

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

1. Introduction

Leta_n≥0forn∈Nsuch that0<P∞

n=1(n+ 1)a_n<∞, andS_n=Pn m=1

1 m, the harmonic number. Then van der Corput’s inequality [5] states that

(1.1)

∞

X

n=1 n

Y

k=1

a^1/k_k

!_Sn¹

< e^1+γ

∞

X

n=1

(n+ 1)a_n,

whereγ = 0.57721566. . . stands for Euler-Mascheroni’s constant. The factor e^1+γin (1.1) is the best possible.

In 2003, Hu in [3] gave a strengthened version of (1.1) by

(1.2)

∞

X

n=1 n

Y

k=1

a^1/k_k

!_Sn¹

< e^1+γ

∞

X

n=1

n− lnn 4

a_n.

Recently, Yang in [7] obtained a better result than Hu’s inequality (1.2) as

(1.3)

∞

X

n=1 n

Y

k=1

a^1/k_k

!_Sn¹

< e^1+γ

∞

X

n=1

n− lnn 3

a_n.

Moreover, he also extended (1.1) in [7] as follows

(1.4)

∞

X

n=1 n

Y

k=1

a^1/(k+β)_k

!_Sn(β)¹

< e^1+γ(β)

∞

X

n=1

n+ 1

2+β

a_n,

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whereβ ∈(−1,∞),S_n(β) =Pn k=1

1 k+β, and γ(β) = lim

n→∞

" _n X

k=1

1

k+β −ln(n+β)

# .

Applyingβ = 0in (1.4) leads to

(1.5)

∞

X

n=1 n

Y

k=1

a^1/k_k

!_Sn¹

< e^1+γ

∞

X

n=1

n+ 1

2

a_n,

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)

∞

X

n=1 n

Y

k=1

a^1/k_k

!_Sn¹

< e^1+γ

∞

X

n=1

n

1− lnn 3n−1/4

a_n.

The purpose of this note is to refine further inequality (1.6). Our main result is the following.

Theorem 1.1. Forn ∈ N, letS_n = Pn m=1

1

m, the harmonic number. Ifa_n ≥ 0 forn∈Nand

0<

∞

X

n=1

n

1− lnn 2n+ lnn+ 11/6

an <∞,

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then

(1.7)

∞

X

n=1 n

Y

k=1

a^1/k_k

!_Sn¹

< e^1+γ

∞

X

n=1

e⁻

6(6n+1)γ−9 (6n+1)(12n+11)n

1− lnn 2n+ lnn+ 11/6

an,

whereγ = 0.57721566. . . is Euler-Mascheroni’s constant.

Remark 1. Let

A_n=e^1+γe⁻

6(6n+1)γ−9 (6n+1)(12n+11)n

1− lnn 2n+ lnn+ 11/6

forn∈N. Numerical computation showsA₁ = 4.40. . . < e^1+γ

1− _3−1/4^{ln 1}

= 4.84. . . and A₂ = 7.99. . . < e^1+γ

2− _6−1/4^{2 ln 2}

= 8.51. . .. and A₃ = 11.95. . . < e^1+γ

3− _9−1/4^{3 ln 2}

= 12.70. . .. When n ≥ 4, inequality 2n +

11

6 + lnn <3n− ¹₄ is valid, which can be rearranged as 1− lnn

2n+ lnn+ 11/6 ≤1− lnn 3n−1/4. This implies that inequality (1.7) is a refinement of (1.6).

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2. Lemmas

In order to prove our main result, some lemmas are necessary.

Lemma 2.1 ([4]). Forn∈N,

(2.1) 1

2n+ 1/(1−γ)−2 < S_n−lnn−γ < 1 2n+ 1/3. The constants _1−γ¹ −2and ¹₃ in (2.1) are the best possible.

Lemma 2.2 ([2,6]). Ifx >0, then

(2.2)

1 + 1

x x

< e

1− 1

2x+ 11/6

. Lemma 2.3. Forn∈N,

(2.3) B_n,

(n+ 1)S_n+1 nSn

nSn

< e^1+γe⁻

6(6n+1)γ−9 (6n+1)(12n+11)n

1− lnn 2n+ lnn+ 11/6

.

Proof. By virtue of Lemma2.2, it follows that

(n+ 1)S_n+1 nS_n

_Sn+1^nSn

=

1 + S_n+ 1 nS_n

_Sn+1^nSn

< e

1− S_n+ 1

2nS_n+ 11(S_n+ 1)/6

< e

1− 1

2n+ 11/6

.

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Applying Lemma2.1yields B_n <

e

1− 1

2n+ 11/6

Sn+1

(2.4)

< e¹⁺^2n+1/3¹ ^+γ+lnⁿ

1− 1

2n+ 11/6

1+_2n+1/3¹ +γ+lnn

. Taking advantage of inequalities 1− ¹_x−x

> eforx > 1ande^−x ≤ _1+x¹ for x >−1leads to

(2.5)

1− 1

2n+ 11/6 lnn

≤exp

− lnn 2n+ 11/6

≤ 2n+ 11/6 2n+ 11/6 + lnn and

1− 1

2n+ 11/6

_2n+1/3¹ +1+γ

exp

1 2n+ 1/3

(2.6)

<exp

1

2n+ 1/3− 1 +γ

2n+ 11/6 − 1

(2n+ 11/6)(2n+ 1/3)

= exp

− 6(6n+ 1)γ−9 (6n+ 1)(12n+ 11)

.

Combination of (2.4), (2.5), (2.6) gives (2.7) B_n< e^1+γ−

6(6n+1)γ−9 (6n+1)(12n+11)

n− nlnn

2n+ lnn+ 11/6

. Lemma2.3is proved.

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3. Proof of Theorem 1.1

Forn∈Nand1≤k ≤n, let

(3.1) c_k = [(k+ 1)S_k+1]^kS^k (kS_k)^kS^k−1 with assumptionS0 = 0, then

(3.2)

n

Y

k=1

c^1/k_k

!− ¹

Sn

= 1

(n+ 1)Sn+1

.

By using the discrete weighted arithmetic-geometric mean inequality and interchanging the order of summations,

∞

X

n=1 n

Y

k=1

a^1/k_k

!_Sn¹

=

∞

X

n=1

" _n Y

k=1

(c_ka_k)^1/k

#_Sn¹ _n Y

k=1

c^1/k_k

!−_Sn¹

(3.3)

≤

∞

X

n=1 n

Y

k=1

c^1/k_k

!−_Sn¹

1 Sn

n

X

k=1

c_ka_k k

≤

∞

X

n=1

1 (n+ 1)S_n+1S_n

n

X

k=1

c_ka_k k

=

∞

X

k=1

ckak

k

∞

X

n=k

1 (n+ 1)S_nS_n+1

=

∞

X

k=1

c_ka_k k

∞

X

n=k

1

S_n − 1 S_n+1

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=

∞

X

k=1

c_ka_k kSk

=

∞

X

k=1

(k+ 1)S_k+1 kS_k

kSk

a_k=

∞

X

n=1

B_na_n.

Substituting (2.7) into (3.3) leads to (1.7). The proof of Theorem1.1 is com- plete.

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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 ANDF. 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)

[4] F. QI, R.-Q. CUI, CH.-P. CHEN,ANDB.-N. GUO, Some completely mono- tonic functions involving polygamma functions and an application, J. Math.

Anal. Appl., 310(1) (2005), 303–308.

[5] J.G. VAN DER CORPUT, Generalization of Carleman’s inequality, Proc.

Akad. Wet. Amsterdam, 39 (1936), 906–911.

[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 in- equality, Chinese Quart. J. Math., 22 (2007), in press.