JJ II

volume 7, issue 5, article 180, 2006.

Received 16 April, 2006;

accepted 20 June, 2006.

Communicated by:B. Yang

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Journal of Inequalities in Pure and Applied Mathematics

ON A NEW STRENGTHENED VERSION OF A HARDY-HILBERT TYPE INEQUALITY AND APPLICATIONS

WEIHONG WANG AND DONGMEI XIN

Department of Mathematics,

Guangdong Education College, Guangzhou, Guangdong 510303, People’s Republic of China EMail:[email protected]

c

2000Victoria University ISSN (electronic): 1443-5756 116-06

On a New Strengthened Version of a Hardy-Hilbert Type Inequality and Applications Weihong Wang and Dongmei Xin

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Abstract

By improving an inequality of the weight coefficient, we give a new strengthened version of Hardy-Hilbert’s type inequality. As its applications, we build some strengthened versions of the equivalent form and some particular results.

2000 Mathematics Subject Classification:26D15.

Key words: Hardy-Hilbert’s inequality, Weight coefficient, Hölder’s inequality.

Contents

1 Introduction. . . 3 2 Some Lemmas. . . 5 3 Main Results and Applications . . . 9

References

On a New Strengthened Version of a Hardy-Hilbert Type Inequality and Applications Weihong Wang and Dongmei Xin

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1. Introduction

Ifp >1, ¹_p +¹_q = 1, an, bn ≥0 (n∈N⁰ =N∪ {0}),such that0<P∞

n=0a^p_n <

∞and0 < P∞

n=0b^q_n < ∞, then we have the famous Hardy-Hilbert inequality as follows [1]:

(1.1)

∞

X

n=0

∞

X

m=0

a_mb_n

m+n+ 1 < π sin

π p

∞

X

n=0

a^p_n

!¹_{p ∞} X

n=0

b^q_n

!¹_q ,

where the constant factor _sin(^ππ

p) is the best possible.

Inequality (1.1) is important in analysis and its applications. In recent years, [2] – [5] considered the strengthened version, generalizations and improvements of inequality (1.1) and Pachpatte [6] built some inequalities similar to inequality (1.1).

Under the same condition with (1.1), we still have Mulholand’s inequality (cf. [7]):

(1.2)

∞

X

n=2

∞

X

n=2

a_mb_n

mnlnmn < π sin

π p

( _∞

X

n=2

1 na^p_n

)¹_p( _∞ X

n=2

1 nb^q_n

)¹_q ,

where the constant factor _sin(^ππ

p) is the best possible.

For the double series:

∞

X

n=1

∞

X

m=1

a_mb_n

m^sn^t(lnm+ lnn+ lnα) =

∞

X

n=1

∞

X

m=1

a_mb_n

m^sn^tlnαm (s, t ∈R, α≥e^7/6),

On a New Strengthened Version of a Hardy-Hilbert Type Inequality and Applications Weihong Wang and Dongmei Xin

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in 2003, Yang [8] built an inequality of the weight coefficient as follows:

∞

X

m=1

1 mlnαmn

ln√ αn ln√

αm ¹_r

< π

sinπ(1−1/r) (r >1, α≥e^7/6), then he gave

(1.3)

∞

X

n=1

∞

X

m=1

ambn

m^sn^t(lnαmn)

< π sin

π p

∞

X

n=1

n^1q−sa_np!¹_{q ∞} X

n=1

n^1p−tb_nq!¹_p ,

(1.4)

∞

X

n=1

1 n

∞

X

m=1

a_m m^slnαmn

!p

<



 π sin

π p





p ∞

X

n=1

n^1q−sa_np

,

where the constant factors ^π

sin(^πp) and

π sin(^πp)

p

are the best possible.

In this paper, by using the refined Euler-Maclaurin formula, we have some strengthened versions of inequalities (1.3) and (1.4).

On a New Strengthened Version of a Hardy-Hilbert Type Inequality and Applications Weihong Wang and Dongmei Xin

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2. Some Lemmas

If f⁽⁴⁾ ∈ C[1,∞),R∞

1 f(x)dx < ∞, and (−1)ⁿf⁽ⁿ⁾(x) > 0, f⁽ⁿ⁾(∞) = 0 (n = 0,1,2,3,4),then we have the following inequality (see [9]):

(2.1)

∞

X

m=1

f(m)<

Z ∞ 1

f(x)dx+ 1

2f(1)− 1 12f⁰(1).

Lemma 2.1. Settingr >1, n ∈Nandα ≥e^7/6, define the functionω(r, n, α) as:

ω(r, n, α) =

∞

X

m=1

1 mlnαmn

ln√ αn ln√

αm ¹_r

.

The we have

(2.2) ω(r, n, α)< π

sinπ(1−1/r)− 3

8(r−1)(2 lnn+ 1)^1−1/r.

Proof. For fixedx∈[1,∞), settingf(x) = _xlnαnx¹ ·

ln√ αn ln√

αx

¹_r

,we have

f(1) = 1 lnαn ·

ln√ αn ln√

α ¹_r

= ln√ αn ln√

α·lnαn·

ln√ α ln√

αn ¹⁻¹_r

,

f⁰(1) =− 1

lnαn+ 1

ln²αn+ 1 rln√

α·lnαn

·

ln√ αn ln√

α ¹⁻¹_r

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=−

ln√ αn ln√

αn·lnαn + ln√ αn rln²√

α·lnαn + ln√ αn ln√

α·ln²αn

·

ln√ α ln√

αn ¹⁻¹_r

,

Z ∞ 1

f(x)dx= Z ∞

1

1 xlnαnx ·

ln√ αn ln√

αx ¹_r

dx

= Z ∞

ln√ α ln√

αn

1 1 +u ·

1 u

¹_r du

= π

sin(π/r)− Z ^ln

√α ln√

αn

0

1 1 +u ·

1 u

¹_r du.

Since Z ^ln

√α ln√

αn

0

1 1 +u ·

1 u

¹_r

du= r r−1

Z ^ln

√α ln√

αn

0

1

1 +u ·du^1−1r

= rln√ αn (r−1) lnαn·

ln√ α ln√

αn ¹⁻¹_r

+ r

r−1 Z ^ln

√α ln√

αn

0

u^1−1r · 1 (1 +u)²du

= rln√ αn (r−1) lnαn·

ln√ α ln√

αn ¹⁻¹_r

+ r²

(r−1)(2r−1) Z ^ln

√α ln√

αn

0

1

(1 +u)²du^2−1r

> rln√ αn (r−1) lnαn·

ln√ α ln√

αn 1−¹_r

+ r²

(r−1)(2r−1)

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· ln√

α lnα n

2

· ln√

α lnα n

²⁻_r¹

=

ln√ α ln√

αn

1−¹_r

rln√ αn

(r−1) lnαn+ r²

(r−1)(2r−1) · ln√

α·ln√ αn ln²αn

,

in view of (2.1) and the above result, we have

ω(r, n, α) =

∞

X

m=1

f(m)

<

Z ∞ 1

f(x)dx+1

2f(1)− 1 12f⁰(1)

< π sin

π p

−

ln√ α ln√

αn ¹⁻¹_r

·

rln√ αn

(r−1) lnαn+ r²

(r−1)(2r−1)· ln√

α·ln√ αn ln²αn

+ ln√ αn 2 ln√

α·lnαn·

ln√ α ln√

αn ¹⁻¹_r

+ 1 12

ln√ αn ln√

α·lnαn

+ ln√

αn rln²√

α·lnαn + ln√

αn ln√

α·ln²αn ·

ln√ α ln√

αn ¹⁻¹_r

= π

sin

π p

−

ln√ α ln√

αn ¹⁻¹_r

·

r

r−1− 7

6 lnα − 1 3rln²α

· ln√ αn lnαn

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+

r²lnα

2(r−1)(2r−1) − 1 6 lnα

· ln√ αn ln²αn

.

Forn∈N, r >1, α≥e^7/6, since r

1−r − 7

6 lnα − 1 3rln²α

· ln√ αn lnαn ≥

1 + 1

r−1− 7

6· ⁷₆ − 1 3r·(⁷₆)²

· 1 2

= 3

8(r−1),

r²lnα

2(r−1)(2r−1) − 1 6 lnα

· ln√ αn ln²αn

>

lnα 4

1 + 3

2(r−1)

− 1 6 lnα

· ln√ αn ln²αn

>

7 24

1 + 3

2(r−1)

− 1 7

· ln√ αn ln²αn >0, and

ln√ α ln√

αn ¹⁻¹_r

> 1

(2 lnn+ 1)^1−1/r, we have

ω(r, n, α)< π

sinπ(1−1/r)− 3

8(r−1)(2 lnn+ 1)^1−1/r. The lemma is proved.

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3. Main Results and Applications

Theorem 3.1. Ifp >1, ¹_p +¹_q = 1, α≥e^7/6, s, t∈R, a_nb_nare two sequences of non-negative real numbers, such that0 < P∞

n=1

n^1q−sa_np

<∞ and0 <

P∞ n=1

n^1p−tb_nq

<∞,then we have

(3.1)

∞

X

n=1

∞

X

m=1

a_mb_n m^sn^tlnαmn

<





∞

X

n=1



 π sin

π p



− 3(p−1) 8(2 lnn+ 1)^1p

n^1q−sa_np





1 p

×





∞

X

n=1



 π sin

π p

− 3(q−1) 8(2 lnn+ 1)^1q





n^1p−tb_nq





1 q

.

In particular,

(a) fors= ¹_q, t= ¹_p, we have

(3.2)

∞

X

n=1

∞

X

m=1

a_mb_n m^1pn^1qlnαmn

<





∞

X

n=1



 π sin

π p

− 3(p−1) 8(2 lnn+ 1)^1p



a^p_n





1 p

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×







 π sin

π p

− 3(p−1) 8(2 lnn+ 1)^1q



b^q_n





1 q

;

(b) fors=t= 1, we have

(3.3)

∞

X

n=1

∞

X

m=1

a_mb_n mnlnαmn

<





∞

X

n=1



 π sin

π p

− 3(p−1) 8(2 lnn+ 1)^1p



 a^p_n

n





1 p

×







 π sin

π p

− 3(p−1) 8(2 lnn+ 1)^1q



 b^q_n

n





1 q

(c) fors=t= 0, we have

(3.4)

∞

X

n=1

∞

X

m=1

a_mb_n lnαmn

<





∞

X

n=1



 π sin

π p

− 3(p−1) 8(2 lnn+ 1)^1p



n^p−1a^p_n





1 p

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×







 π sin

π p

− 3(p−1) 8(2 lnn+ 1)^1q



n^q−1b^q_n





1 q

.

Proof. By Hölder’s inequality, we have

∞

X

n=1

∞

X

m=1

ambn

m^sn^tlnαmn

=

∞

X

n=1

∞

X

m=1

"

a_m (lnαmn)^1p

ln√ αm ln√

αn _pq¹

m^1q−s n^1p

!#

×

"

b_n (lnαmn)^1q

ln√ αn ln√

αm _pq¹

n^1p−t m^1q

!#

≤

∞

X

m=1

∞

X

n=1

"

a^p_m lnαmn

ln√ αm ln√

αn ¹_q

m^p(1q−s) n

!#¹_p

×

∞

X

n=1

∞

X

m=1

"

b^q_n lnαmn

ln√ αn ln√

αm ¹_p

n^q(1p−t) m

!#¹_q

=

" _∞ X

m=1

ω(q, m, α)

m^1q−sam

p#¹_p" _∞ X

n=1

ω(p, n, α)

n^1p−tbn

q#¹_q .

In view of (2.2), we have (3.1). The theorem is proved.

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Theorem 3.2. If p > 1, ¹_p + ¹_q = 1, α ≥ e^7/6, s ∈ R, a_n is sequence of non-negative real numbers, such that 0 < P∞

n=1

n^1q−sa_np

< ∞, then we have

(3.5)

∞

X

n=1

1 n

∞

X

m=1

a_m m^slnαmn

!p

<

"

π sin(^π_p)

#p−1 ∞

X

m=1

π

sin(^π_p) − 3(p−1) 8(2 lnm+ 1)^1p

!

m^1q−sa_mp

.

In particular,

(a) fors= ¹_q, we have

(3.6)

∞

X

n=1

1 n

∞

X

m=1

a_m m^1q lnαmn

!p

<

"

π sin(^π_p)

#p−1 ∞

X

m=1

π

sin(^π_p)− 3(p−1) 8(2 lnm+ 1)^1p

!

(a_m)^p;

(b) fors= 1, we have

(3.7)

∞

X

n=1

1 n

∞

X

m=1

a_m mlnαmn

!p

<

"

π sin(^π_p)

#p−1 ∞

X

m=1

π

sin(^π_p)− 3(p−1) 8(2 lnm+ 1)^1p

!a^p_m m;

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(c) fors= 0, we have

(3.8)

∞

X

n=1

1 n

∞

X

m=1

a_m lnαmn

!p

<

"

π sin(^π_p)

#p−1 ∞

X

m=1

π

sin(^π_p)− 3(p−1) 8(2 lnm+ 1)^1p

!

(m^p−1a_m)^p.

Proof. It is obvious that for anym ∈N₀, ω(r, m, α)< _{sinπ(1−1/r)}^π .By Cauchy’s inequality, we obtain

∞

X

m=1

a_m m^slnαmn

!p

=

" _∞ X

m=1

1 (lnαmn)^1p

ln√ αm ln√

αn _pq¹

m^1q−s a_m

· 1 (lnαmn)^1q

ln√ αn ln√

αm _pq¹

1 m^1q

#^p

≤

∞

X

m=1

1 (lnαmn)

ln√ αm ln√

αn ¹_q

m^p(1q−s)a^p_m

·

∞

X

m=1

"

1 (lnαmn)

ln√ αn ln√

αm ¹_p

1 m

#^(p−1)

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=

∞

X

m=1

1 (lnαmn)

ln√ αm ln√

αn ¹_q

m^p(1q−s)a^p_m·[ω(p, n, α)]^(p−1)

<

"

π sin^π_p

#(p−1) ∞

X

m=1

1 (lnαmn)

ln√ αm ln√

αn ¹_q

m^p(1q−s)a^p_m.

Hence, by (2.2) we find

∞

X

n=1

1 n

∞

X

m=1

a_m m^slnαmn

!p

<

"

π sin^π_p

#(p−1) ∞

X

n=1

1 n

∞

X

m=1

1 (lnαmn)

ln√ αm ln√

αn ¹_q

m^p(1q−s)a^p_m

=

"

π sin^π_p

#(p−1) ∞

X

m=1

∞

X

n=1

1 nlnαmn

ln√ αm ln√

αn ¹_q

m^p(1q−s)a^p_m

=

"

π sin^π_p

#(p−1) ∞

X

m=1

ω(q, m, α)

m^1q−sa_mp

<

"

π sin^π_p

#(p−1) ∞

X

m=1

π

sin(^π_p)− 3(p−1) 8(2 lnm+ 1)^1p

!

m^1q−sa_mp

.

The theorem is proved.

Remark 1. Obviously, inequalities (3.1) and (3.5) are separately strengthened versions of inequalities (1.3) and (1.4).

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References

[1] F.H. HARDYAND J.E. LITTLEWOOD, Inequalities, London, Cambridge Univ. Press, 1952.

[2] B.C. YANG AND L. DEBNATH, On a new generalization of Hardy- Hilbert’s inequality and its applications., J. Math. Anal. Appl., 233 (1999), 484–497.

[3] B.C. YANG, On a strengthened version of the more accurate Hardy- Hilbert’s inequality, Acta Mathematical Sinica, 43(6) (1999), 1003–1110 (in Chinese).

[4] B.C. YANG, On a strengthened Hardy-Hilbert’s inequality, J. Ineq. Pure and Appl. Math., 1(2) (2000), Art. 22. [ONLINE:http://jipam.vu.

edu.au/article.php?sid=116].

[5] JICHANG KUANG AND L. DEBNATH, On new generalizations of Hilbert’s inequality and their applications, J. Math. Anal. Appl., 245 (2000), 248–265.

[6] B.G. PACHPATTE, On some inequalities similar to Hilbert’s inequality, J.

Math. Anal., 226 (1998), 166–179.

[7] H.P. MULHOLAND, A further generalization of Hilbert’s double series theorem, J. London Math. Soc., 6 (1931), 100–106.

[8] B.C. YANG, A new Hardy-Hilbert’s type inequality and Applications, Acta Mathematica Sinica, 46(6) (2003), 1079–1086 (in Chinese).

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[9] JICHANG KUANG AND L. DEBNATH, On new generalization of Hilbert’s inequality and their applications, J. Math. Anal. Appl., 245 (2000), 248–265.