A Hilbert’s Inequality with a Best Constant Factor

Journal of Inequalities and Applications Volume 2009, Article ID 820176,8pages doi:10.1155/2009/820176

Research Article

A Hilbert’s Inequality with a Best Constant Factor

Zheng Zeng

and Zi-tian Xie

1Department of Mathematics, Shaoguan University, Shaoguan, Guangdong 512005, China

2Department of Mathematics, Zhaoqing University, Zhaoqing, Guangdong 526061, China

Correspondence should be addressed to Zi-tian Xie,[email protected] Received 6 February 2009; Revised 3 May 2009; Accepted 23 July 2009 Recommended by Yong Zhou

We give a new Hilbert’s inequality with a best constant factor and some parameters.

Copyrightq2009 Z. Zeng and Z.-t. Xie. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Ifp >1, 1/p1/q1,an, bn >0 such that∞ >_∞

n1a^pn >0 and∞>_∞

n1b^qn > 0, then the well-known Hardy-Hilbert’s inequality and its equivalent form are given by

∞ n1

∞ m1

ambn

mn < π sin

π/p _∞

n1

a^pn

_1/p∞

n1

bn^q

_1/q

, 1.1

∞ n1

_∞

m1

am

mn

p

<

π sin

π/p p_∞

n1

a^pn

, 1.2

where the constant factors are all the best possible 1. It attracted some attention in the recent years. Actually, inequalities1.1and 1.2 have many generalizations and variants.

Equation1.1has been strengthened by Yang and othersincluding integral inequalities 2–11.

In 2006, Yang gave an extension of2as follows.

Ifp >1, 1/p1/q1,r >1,1/r1/s1, t∈0,1,2−min{r, s}tmin{r, s} ≥λ >

2−min{r, s}t,such that∞ > _∞

n1np1−t2t−λ/r−1a^pn > 0,∞ > _∞

n1nq1−t2t−λ/s−1b^qn >0, then

∞ n1

∞ m1

ambn

mn^λ

< B

r−2tλ

r ,s−2tλ s

_∞

n1

np1−t2t−λ/r−1a^pn

_1/p∞

n1

nq1−t2t−λ/s−1b^qn

_1/q . 1.3

Bu, vis the Beta function.

In 2007 Xie gave a new Hilbert-type Inequality3as follows.

Ifp >1,1/p1/q1, a, b, c >0,2/3≥μ >0, and the right of the following inequalities converges to some positive numbers, then

∞ m1

∞ n1

ambn

n^μa²m^μn^μb²m^μn^μa²m^μ

< π

μabbcca _∞

n1

n^{1−3μ/2p−1}a^pn

_1/p∞

n1

n^{1−3μ/2q−1}b^qn

_1/q .

1.4

The main objective of this paper is to build a new Hilbert’s inequality with a best constant factor and some parameters.

In the following, we always suppose that

11/p1/q1, p >1,a≥0,−1< α <1,

2both functionsuxandvxare differentiable and strict increasing inn0−1,∞ andm0−1,∞,respectively,

3ux/u^αx, vx/v^αx are strictly increasing in n0 − 1,∞ and m0 − 1,∞, respectively.{unvm/u²_n2aunvmv_m²u^α_nv_m^α}is strict decreasing onnandm,

4un un, un0 u0, un0−1 vm0−1 0, u∞ ∞, v∞ ∞, un un, vm vm, vm0 v0, vm v_m.

2. Some Lemmas

Lemma 2.1. Define the weight coefficients as follows:

W p, m

: ^∞

nn0

1

u²n2aunvmvm²

·v^αp−1m

u^αn

· u_n

vm ^p−1, 2.1

ω p, m

: _∞

no−1

1

u²x 2auxvmv²m

·vm^αp−1

u^αx · ux

vm^p−1dx, 2.2

W q, n

: ^∞

mm0

1

u²n2aunvmvm²

·u^αq−1n

v^αm

· v_m

un^q−1, 2.3

ω

q, n :

_∞

m0−1

1 u²n2aunv

y v²

y·u^αq−1n

v^α

y · v y

un^q−1dy, 2.4

then

W p, m

< ω p, m

Kv^{pα−2α−1}m

vm^p−1 , W q, n

<ω q, n

Ku^{qα−2α−1}n

un^q−1 , 2.5

where

K _∞

0

dσ 12aσσ²σ^α

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

π 2√

a²−1 sinαπ

a√

a²−1^α− 1 a√

a²−1^α

, ifα /0, a >1,

|απ|/sin|απ|, ifα /0, a1,

πcscθcscαπsinαθ, ifα /0, acosθ,0< θ < π,

√ 1

a²−1ln a√

a²−1

, ifα0, a >1,

θcscθ, ifα0, acosθ,0< θ < π

2 ,

1, ifα0, a1,

2.6

Proof. Let fz 1/1 2az z²z^α 1/z − z1z − z2z^α then K 2πi/1 − e^−2απiResf, z1 Resf, z2ifa >1 thenz1−a−√

a²−1, z2 −a√ a²−1

K 2πi

1−e^−2απi

⎡

⎢⎣

−a−√

a²−1_−α

−2√

a²−1

−a√

a²−1_−α 2√

a²−1

⎤

⎥⎦

π 2√

a²−1 sinαπ

⎡

⎢⎣

a

a²−1_α

− 1

a√

a²−1_α

⎤

⎥⎦,

2.7

ifacosθ0< θ < π/2, thenz1−e^iθ, z2−e^−iθ

K 2πi

1−e^−2απi

1

−2isinθ−e^iθα 1 2isinθ−e^−iθα

πcscθcscαπsinαθ. 2.8

On the other hand, Wp, m < ωp, m. Setting ux vmσ, then ωp, m Kvm^{pα−2α−1}/ v_m^p−1.Similarly,Wq, n <ωq, n Ku^{qα−2α−1}n /u_n^q−1.

Lemma 2.2. For 0< ε <min{p, p1−α}one has _∞

0

dσ

12aσσ²σ^αε/p Ko1 ε−→0. 2.9

Proof.

_∞

0

1

12aσσ²σ^αε/pdσ−K

≤

₁

0

σ^−α

1−σ^−ε/p 12aσσ² dσ

_∞

1

σ^−α

1−σ^−ε/p 12aσσ² dσ

≤

₁

0

σ^−α

1−σ^−ε/p dσ

_∞

1

σ^−2−α

1−σ^−ε/p dσ

1

1−α− 1 1−α−ε/p

1

1α− 1

1αε/p

−→0 for ε−→0.

2.10

The lemma is proved.

Lemma 2.3. Settingwn un(orvmandw0 n0(orm0, resp.), thenk > 0.{τ_w /τ_w^k}is strictly decreasing, then

N ww0

τ_w τw^k

_N

w0

τx

τ^kxdxA. 2.11

ThereA∈0, τ_w0/τ_w^k₀,for anyN).

Proof. We have _N

w0

τx τ^kxdx <

N ww0

τ_w τw^k

τ_w0 τw^k0

^N

ww01

τ_w τw^k

< τ_w0 τw^k0

_N

w0

τx

τ^kxdx. 2.12

Easily,Ahad up bounded whenN → ∞.

3. Main Results

Theorem 3.1. If an > 0, bn > 0, 0 < _∞

n1v^{pα−2α−1}m /v_m ^p−1a^pn < ∞, 0 < _∞

nn0u^{qα−2α−1}n / u_n^q−1b^qn<∞, then

∞ nn0

∞ mm0

ambn

u²n2aunvmv²m

< K _∞

mm0

v^{pα−2α−1}m

v_m ^p−1a^pm

1/p_∞

nn0

u^{qα−2α−1}n

u_n^q−1b^qn

1/q

, 3.1

∞ nn0

u^{pαp−2α−1}n u_{n ∞}

mm0

am

u²n2aunvmv²m

p

< K^p ∞ mm0

v^{pα−2α−1}m

vm^p−1a^pm. 3.2

Kis defined byLemma 2.1.

Proof. By H ¨older’s inequality12and2.5,

J: ^∞

nn0

∞ mm0

ambn

u²n2aunvmvm²

^∞

nn0

∞ mm0

1

u²n2aunvmvm²

·v^α/qm

u^α/pn

· u_n^1/p

vm ^1/qam·u^α/pn

v^α/qm

·v_m ^1/q un^1/pbn

≤ _∞

mm0

Wp, ma^pm

_1/p∞

nn0

Wq, nb^qn

_1/q

< K _∞

mm0

vm^{pα−2α−1}

vm^p−1a^pm

1/p_∞

nn0

u^{qα−2α−1}n

un^q−1bn^q

1/q

,

3.3

settingbnu^{pα−2αp−1}n u_n∞

mm0am/u²_n2aunvmv²_m^p−1>0.By3.1we have

∞ nn0

u^{qα−2α−1}n

un^q−1b^qn ^∞

nn0

u^{pα−2αp−1}n u_{n ∞}

mm0

am

u²n2aunvmvm² p

J≤K _∞

mm0

v^{pα−2α−1}m

v_m^p−1a^pm

1/p_∞

nn0

u^{qα−2α−1}n

u_n^q−1b^qn

1/q

.

3.4

By 0<_∞

nn0u^{qα−2α−1}n /u_n^q−1b^qn<∞and3.4taking the form of strict inequality, we have 3.1. By H ¨older’s inequality12, we have

J ^∞

nn0

u^{−α2α/q1/q}n u_n^−11/q∞

mm0

am

u²n2aunvmv²m

u^{α−2α/q−1/q}n bn

u_n^1−1/q

≤ _∞

nn0

u^{pα−2αp−1}n u_{n ∞}

mm0

am

u²n2aunvmv²m

p1/p_∞

nn0

u^{qα−2α−1}n

un^q−1 b^qn

1/q

.

3.5

as 0<{_∞

nn0u^{qα−2α−1}n /u_n^q−1bn^q}^1/q<∞. By3.2,3.5taking the form of strict inequality, we have3.1.

Theorem 3.2. Ifα0, then both constant factors,KandK^pof3.1and3.2, are the best possible.

Proof. We only prove thatKis the best possible. If the constant factorKin3.1is not the best possible, then there exists a positiveHwithH < K, such that

J < H _∞

mm0

v_m⁻¹ vm^p−1a^pm

1/p_∞

nn0

u⁻¹_n un^q−1b^qn

1/q

. 3.6

For 0< ε < min{p, q}, settingamvm^−ε/pv_m ,bnu^−ε/qn u_n, then

_∞

mm0

v_m⁻¹ vm^p−1a^pm

1/p_∞

nn0

u⁻¹_n un^q−1bn^q

1/q

_∞

mm0

v_m v^1εm

1/p_∞

nn0

u_n u^1εn

1/q

. 3.7

On the other handux σvyandvy τ, ∞

mm0

∞ nn0

u^−ε/pn u_nvm^−ε/qv_m u²n2aunvmv²m

>

_∞

m0

_∞

n0

u^−ε/pxuxdx u²x 2auxv

y v²

y vy^−ε/qv y

dy

_∞

m0

_∞

u0/vy

σ^−ε/pdσ

σ²2aσ1 vy^−1−εv y

dy

_∞

v0

_∞

0

σ^−ε/pdσ

σ²2aσ1 τ^−1−εdτ

− _∞

v0

_u0/τ

0

σ^−ε/pdσ

σ²2aσ1 τ^−1−εdτ

≥Ko1 _∞

v0

τ^−1−εdτ− _∞

v0

τ⁻¹ _u0/τ

0

σ^−ε/pdσ dτ

Ko1 _∞

v0

τ^−1−εdτ−u^1−ε/p₀ v₀^−1ε/p 1−ε/p² Ko1

_∞

v0

τ^−1−εdτ−O1.

3.8

By3.6,3.7,3.8, andLemma 2.3, we have

Ko1−_∞O1

v0τ^−1−εdτ < H _∞

mm0

v_m/v_m^1ε _∞

v0τ^−1−εdτ

1/p_∞

nn0

u_n/u^1ε_{n ∞}

v0τ^−1−εdτ 1/q

, 3.9

Ko1−_∞O1

v0τ^−1−εdτ < H

1_∞O1

v0τ^−1−εdτ 1/p

1_∞O1

v0τ^−1−εdτ 1/q

. 3.10

We haveK≤H,ε → 0. This contracts the fact thatH < K.

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