π sin πr− 1 13 (n+ 1) (2n+ 1)1−1r (r >1, n∈N0=N∪ {0}) is proved

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Volume 1, Issue 2, Article 22, 2000

ON A STRENGTHENED HARDY-HILBERT INEQUALITY

BICHENG YANG DEPARTMENT OFMATHEMATICS

GUANGDONGEDUCATIONCOLLEGE, GUANGZHOU

GUANGDONG510303, PEOPLE’SREPUBLIC OFCHINA. [email protected]

Received 8 May, 2000; accepted 10 June 2000 Communicated by L. Debnath

ABSTRACT. In this paper, a new inequality for the weight coefficientW(n, r)of the form

W(n, r) =

∞

X

m=0

1 m+n+ 1

n+¹₂ m+¹₂

1 r

< π

sin ^π_r− 1

13 (n+ 1) (2n+ 1)^1−1r

(r >1, n∈N0=N∪ {0})

is proved. This is followed by a strengthened version of the more accurate Hardy-Hilbert in- equality.

Key words and phrases: Hardy-Hilbert inequality, Weight Coefficient, Hölder’s inequality.

2000 Mathematics Subject Classification. 26D15.

1. INTRODUCTION

Ifp >1,¹_p +¹_q = 1,a_n, b_n ≥0, and0<P∞

n=1−λa^p_n <∞,0<P∞

n=1−λb^q_n <∞(λ= 0,1), then the Hardy-Hilbert inequality is

(1.1)

∞

X

m=1−λ

∞

X

n=1−λ

ambn

m+n+λ < π sin

π p

∞

X

n=1−λ

a^p_n

!¹_{p ∞} X

n=1−λ

b^q_n

!¹_q ,

where the constantπ/sin(π/p)is best possible (see [3, Chapter 9]). Inequality (1.1) is im- portant 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) forλ = 0as

(1.2)

∞

X

m=1

∞

X

n=1

ambn

m+n <







∞

X

n=1



 π sin

π p

− 1−γ n^1p



a^p_n







1

p





∞

X

n=1



 π sin

π p

−1−γ n^1q



b^q_n







1 q

,

ISSN (electronic): 1443-5756

c 2000 Victoria University. All rights reserved.

012-00

where γ = 0.5772⁺, is Euler’s constant. Later, Yang and Debnath [6] proved a distinctly strengthened version of (1.1) forλ= 0as

(1.3)

∞

X

m=1

∞

X

n=1

a_mb_n m+n <







∞

X

n=1



 π sin

π p

− 1 2n^p1 +n^−1q



a^p_n







1

p





∞

X

n=1



 π sin

π p

− 1 2n^1q +n^−1p



b^q_n







1 q

, which is not comparable with (1.2).

Inequality (1.1) forλ = 1 is called the more accurate Hardy-Hilbert’s inequality, which has been strengthened as

(1.4)

∞

X

m=0

∞

X

n=0

a_mb_n m+n+ 1

<







∞

X

n=0



 π sin

π p

− ln 2−γ (2n+ 1)^1+1p



a^p_n







1

p





∞

X

n=0



 π sin

π p

− ln 2−γ (2n+ 1)^1+1q



b^q_n







1 q

, by introducing an inequality of the weight coefficient W(n, r) in the form (see [5, equation (2.9)]):

(1.5) W(n, r) =

∞

X

m=0

1 m+n+ 1

n+¹₂ m+ ¹₂

1 r

< π

sin ^π_r − ln 2−γ (2n+ 1)^2−1r

(r >1, n∈N₀). 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. SOME LEMMAS

Lemma 2.1. Iff ∈ C⁶[0,∞), f^(2q)(x) > 0 (q= 0,1,2,3), f⁽ⁱ⁻¹⁾(∞) = 0 (i= 1,2, . . . ,6), andP∞

n=0f(n)<∞, then we have

∞

X

k=2

(−1)^kf(k)> 1

2f(2) (see [1, equation (4.4)]), (2.1)

∞

X

m=0

f(m) = Z ∞

0

f(t)dt+ 1

2f(0) + Z ∞

0

B¯₁f⁰(t)dt, (2.2)

Z ∞ 0

B¯₁f⁰(t)dt= − 1

12f⁰(0) +δ₂, δ₂ = 1 6

Z ∞ 0

B¯₃f⁰⁰⁰(t)dt <0, (2.3)

whereB¯i(t) (i= 1,3)are Bernoulli functions (see [5, equations (1.7)-(1.9)]).

Setting the weight coefficientW(n, r)in the form:

(2.4) W(n, r) =

∞

X

m=0

1 m+n+ 1

n+¹₂ m+ ¹₂

¹r

= π

sin ^π_r − θ(n, r) (2n+ 1)^2−1r

(r >1, n∈N0), then we find

(2.5) θ(n, r) = π

sin ^π_r(2n+ 1)^2−1r −(2n+ 1)²

∞

X

m=0

1 m+n+ 1

1 2m+ 1

¹_r .

If we define the functionf(x)asf(x) = _(x+n+1)¹ _2x+1^{1 1}_r

,x∈[0,∞), then we havef(0) = _(n+1)¹ , f⁰(x) = − 1

(x+n+ 1)² 1

2x+ 1 ¹_r

− 2

r(x+n+ 1) 1

2x+ 1 1+¹_r

, f⁰(0) =− 1

(n+ 1)² − 2 r(n+ 1), and

Z ∞ 0

f(x)dx= 1 (2n+ 1)^1r

Z ∞

1 2n+1

1 (y+ 1)

1 y

¹_r dy

= 1

(2n+ 1)^1r

"

π sin ^π_r −

∞

X

ν=0

(−1)^ν 1− ¹_r +ν

(2n+ 1)^ν+1−1r

# . By (2.2) and (2.5), we have

θ(n, r) =−(2n+ 1)² 2 (n+ 1) +

∞

X

ν=0

(−1)^ν 1−¹_r+ν

(2n+ 1)^ν−1 (2.6)

+ Z ∞

0

B¯₁(t)

"

(2n+ 1)² (t+n+ 1)²(2t+ 1)^1r

+ 2 (2n+ 1)² r(t+n+ 1) (2t+ 1)^1+1r

# dt.

By Lemma 4 of [5, p. 1106] and (2.4), we have

Lemma 2.2. Forr >1,n∈N₀, we haveθ(n, r)> θ(n,∞), and

(2.7) W(n, r)< π

sin ^π_r − θ(n,∞) (2n+ 1)^2−1r

(r >1, n∈N₀), where

(2.8) θ(n,∞) = −(2n+ 1)² 2 (n+ 1) +

∞

X

ν=0

(−1)^ν

(1 +ν) (2n+ 1)^ν−1 + Z ∞

0

B¯₁(t)

"

(2n+ 1)² (t+n+ 1)²

# dt.

Since by (2.3) and (2.1),we have Z ∞

0

B¯₁(t) 1

(t+n+ 1)²dt=− 1

12 (n+ 1)² − 1 3!

Z ∞ 0

B¯₃(t)

1 (t+n+ 1)

000

dt

>− 1 12 (n+ 1)² and

∞

X

ν=0

(−1)^ν

(1 +ν) (2n+ 1)^ν−1 = (2n+ 1)−1 2 +

∞

X

ν=2

(−1)^ν (1 +ν) (2n+ 1)^ν−1

>(2n+ 1)−1

2 + 1

6 (2n+ 1). Then by (2.8), we find

(2.9) θ(n,∞)> 1

6 − 1

6 (n+ 1) − 1

12 (n+ 1)² + 1 6 (2n+ 1).

Lemma 2.3. Forr >1,n∈N₀, we have

(2.10) W(n, r)< π

sin ^π_r − 1

13 (n+ 1) (2n+ 1)^1−1r .

Proof. Define the functiong(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 haveθ(n,∞)> ²ⁿ⁺¹_n+1g(n). Sinceg(1)>0.0787> ₁₃¹ , and forx∈[1,∞),

g⁰(x) = 1

3 (2x+ 1)² − 1

12 (x+ 1)² − 1

3 (2x+ 1)³ = 4x²+ 2x−1

12 (x+ 1)²(2x+ 1)³ >0, then forn ≥ 1, we haveθ(n,∞)> _(n+1)²ⁿ⁺¹g(1) > _13(n+1)²ⁿ⁺¹ .Hence by (2.7), inequality (2.10) is valid forn ≥1. Sinceln 2−γ = 0.1159⁺> ₁₃¹, then by (1.5), we find

(2.11) W(0, r)< π

sin ^π_r − ln 2−γ (2×0 + 1)^2−1r

< π

sin ^π_r − 1

13 (0 + 1) (2×0 + 1)^1−1r .

It follows that (2.10) is valid forr >1,andn∈N₀. This proves the lemma.

3. THEOREM AND REMARKS

Theorem 3.1. If p > 1,¹_p + ¹_q = 1, an, bn ≥ 0, 0 < P∞

n=0a^p_n < ∞, and0 < P∞

n=0b^q_n < ∞, then

(3.1)

∞

X

m=0

∞

X

n=0

a_mb_n m+n+ 1 <







∞

X

n=0



 π sin

π p

− 1

13 (n+ 1) (2n+ 1)^1p



a^p_n







1 p

×







∞

X

n=0



 π sin

π p

− 1

13 (n+ 1) (2n+ 1)^1q



b^q_n







1 q

,

(3.2)

∞

X

m=0

∞

X

n=0

a_n m+n+ 1

!p

<



 π sin

π p





p−1 ∞

X

n=0



 π sin

π p

− 1

13 (n+ 1) (2n+ 1)^p1



a^p_n.

Proof. By Hölder’s inequality, we have

∞

X

m=0

∞

X

n=0

a_mb_n m+n+ 1 =

∞

X

m=0

∞

X

n=0

"

a_m (m+n+ 1)^1p

m+ ¹₂ n+¹₂

pq¹ # "

b_n (m+n+ 1)^1q

n+¹₂ m+ ¹₂

pq¹ #

≤ ( _∞

X

m=0

∞

X

n=0

"

1 (m+n+ 1)

m+¹₂ n+ ¹₂

¹q# a^p_m

)¹p

× ( _∞

X

m=0

∞

X

n=0

"

1 (m+n+ 1)

n+ ¹₂ m+¹₂

1 p#

b^q_n )¹q

= ( _∞

X

m=0

∞

X

n=0

"

1 (m+n+ 1)

m+ ¹₂ n+¹₂

1 q#

a^p_m )

1 p

× ( _∞

X

n=0

∞

X

m=0

"

1 (m+n+ 1)

n+ ¹₂ m+¹₂

1 p#

b^q_n )

1 q

= ( _∞

X

m=0

W(m, q)a^p_m

)_p¹ ( _∞ X

n=0

W(n, p)b^q_n )¹_q

. Sincesin(π/q) = sin(π/p), by (2.10) forr=p, q, we have (3.1).

By (2.10), we haveW(n, p)< π/sin(π/p). Then by Hölder’s inequality, we obtain

∞

X

n=0

a_n

m+n+ 1 =

∞

X

n=0

"

a_n (m+n+ 1)^p1

n+ ¹₂ m+¹₂

pq¹ # "

1 (m+n+ 1)^1q

m+ ¹₂ n+¹₂

pq¹ #

≤ ( _∞

X

n=0

"

1 (m+n+ 1)

n+ ¹₂ m+¹₂

¹q# a^p_n

)

1

p ( _∞

X

n=0

1 (m+n+ 1)

m+ ¹₂ n+¹₂

¹p)

1 q

= ( _∞

X

n=0

"

1 (m+n+ 1)

n+¹₂ m+ ¹₂

¹q# a^p_n

)¹p

{W(m, p)}^1q

<

( _∞ X

n=0

"

1 (m+n+ 1)

n+¹₂ m+ ¹₂

1 q#

a^p_n )

1

p



 π sin

π p







1 q

. Then we find

∞

X

m=0

∞

X

n=0

a_n m+n+ 1

!p

<

∞

X

m=0

∞

X

n=0

"

1 (m+n+ 1)

n+¹₂ m+¹₂

¹_q# a^p_n



 π sin

π p





p q

=



 π sin

π p





p−1 ∞

X

n=0

∞

X

m=0

"

1 (m+n+ 1)

n+¹₂ m+ ¹₂

¹q# a^p_n

=



 π sin

π p





p−1 ∞

X

n=0

W(n, q)a^p_n.

Hence by (2.10) forr=q, we have (3.2). The theorem is proved.

Remark 3.1. Inequality (3.1) is a definite improvement over (1.1) forλ = 1.

Remark 3.2. Since forn ≥2,13 (2−γ)<2− _n+1¹ , then

(3.3) π

sin ^π_r − ln 2−γ (2n+ 1)^2−1r

> π

sin ^π_r − 1

13 (n+ 1) (2n+ 1)^1−1r

(r >1, n≥2). 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 (3.4)

∞

X

m=0

∞

X

n=0

a_n m+n+ 1

!p

<



 π sin

π p





p ∞

X

n=0

a^p_n.

This is an equivalent form of the more accurate Hardy-Hilbert’s inequality (see [3, Chapter 9]).

REFERENCES

[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 AND G. PÓLYA, Inequalities, Cambridge University Press, 1934.

[4] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´C 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.

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Sci., 21(2) (1998), 403–408.

[7] B. YANG ANDM. GAO, On a best value of Hardy-Hilbert’s inequality, Adv. Math., 26(2) (1997), 156–164.