Key words and phrases: Multiple Hardy-Hilbert integral inequality, theΓ-function, Best constant factor

(1)

http://jipam.vu.edu.au/

Volume 7, Issue 4, Article 139, 2006

A MULTIPLE HARDY-HILBERT INTEGRAL INEQUALITY WITH THE BEST CONSTANT FACTOR

HONG YONG

DEPARTMENT OFMATHEMATICS

GUANGDONGUNIVERSITY OFBUSINESSSTUDY

GUANGZHOU510320 PEOPLE’SREPUBLIC OFCHINA.

hongyong59@sohu.com

Received 14 September, 2005; accepted 14 December, 2005 Communicated by B. Yang

ABSTRACT. In this paper, by introducing the normkxkα(x∈Rⁿ), we give a multiple Hardy- Hilbert’s integral inequality with a best constant factor and two parametersα,λ.

Key words and phrases: Multiple Hardy-Hilbert integral inequality, theΓ-function, Best constant factor.

2000 Mathematics Subject Classification. 26D15.

1. INTRODUCTION

Ifp >1, ¹_p + ¹_q = 1, f ≥0,g ≥0,0<R∞

0 f^p(x)dx < +∞, 0<R∞

0 g^q(x)dx < +∞, then the well known Hardy-Hilbert integral inequality is given by (see [3]):

(1.1)

Z ∞ 0

f(x)g(x)

x+y dxdy < π sin

π p

Z ∞

0

f^p(x)dx

¹_pZ ∞ 0

g^q(x)dx ¹_q

,

where the constant factor ^π

sin(^π_p) is the best possible. Its equivalent form is:

(1.2)

Z ∞ 0

f(x) x+ydx

p

dy <



 π sin

π p





p

Z ∞ 0

f^p(x)dx,

where the constant factor

π sin(^π_p)

p

is also the best possible.

Hardy-Hilbert’s inequality is valuable in harmonic analysis, real analysis and operator theory. In recent years, many valuable results (see [4] – [10]) have been obtained in the form of

ISSN (electronic): 1443-5756

273-05

(2)

generalizations and improvements of Hardy-Hilbert’s inequality. In 1999, Kuang [5] gave a generalization with a parameterλof (1.1) as follows:

(1.3)

Z ∞ 0

f(x)g(y)

x^λ+y^λ dxdy < hλ(p) Z ∞

0

x^1−λf^p(x)dx

¹_p Z ∞ 0

x^1−λg^q(x)dx ¹_q

,

wheremaxn

1 p,¹_qo

< λ ≤ 1, h_λ(p) = πh

λsin^p¹

π pλ

sin¹^q

π qλ

i−1

. Because of the constant factorhλ(p)being not the best possible, Yang [8] gave a new generalization of (1.1) in 2002 as follows:

(1.4) Z ∞

0

Z ∞ 0

f(x)g(y) x^λ+y^λ dxdy

< π λsin

π p

Z ∞

0

x^{(1−λ)(p−1)}f^p(x)dx

¹_pZ ∞ 0

x^{(1−λ)(q−1)}g^q(x)dx ¹_q

,

its equivalent form is:

(1.5)

Z ∞ 0

y^λ−1 Z ∞

0

f(x) x^λ+y^λdx

p

dy <



 π λsin

π p





p

Z ∞ 0

x^{(1−λ)(p−1)}f^p(x)dx,

where the constant factors ^π

λsin(^π_p) in (1.4) and

π λsin(^π_p)

p

in (1.5) are all the best possible.

At present, because of the requirement of higher-dimensional harmonic analysis and higher- dimensional operator theory, multiple Hardy-Hilbert integral inequalities have been studied.

Hong [11] obtained: If a > 0, Pn i=1

1

pi = 1, p_i > 1, f_i ≥ 0, r_i = _p¹

i

Qn

j=1p_j, λ >

1 a

n−1− _r¹

i

,i= 1,2, . . . , n, then

(1.6) Z ∞

α

· · · Z ∞

α

1 [Pn

i=1(x_i−α_i)^a]^λ

n

Y

i=1

f_i(x)dx₁· · ·dx_n ≤ Γⁿ⁻² _α¹ αⁿ⁻¹Γ(λ)

×

n

Y

i=1

Γ

1 a

1− 1

r_i

Γ

λ− 1 a

n−1− 1 r_i

Z ∞ α

(t−α)^n−1−αλf_i^pⁱdt _pi¹

.

Afterwards, Bicheng Yang and Kuang Jichang etc. obtained some multiple Hardy-Hilbert integral inequalities (see [9, 6]).

In this paper, by introducing the Γ-function, we generalize (1.3) and (1.4) into multiple Hardy-Hilbert integral inequalities with the best constant factors.

2. SOME LEMMAS

First of all, we introduce the signs as:

Rⁿ+ ={x= (x₁, . . . , x_n) :x₁, . . . , x_n>0},

kxk_α = (x^α₁ +· · ·+x^α_n)^α¹, α >0.

(3)

Lemma 2.1 (see [1]). Ifp_i >0,a_i >0,α_i >0,i= 1,2, . . . , n,Ψ(u)is a measurable function, then

(2.1) Z

. . . Z

x1,...,xn>0;

_x

1 a1

α1

+···+(^xn_an)^αn^≤1 Ψ

x₁ a₁

α1

+· · ·+ x_n

a_n αn

×x^p₁¹⁻¹· · ·x^p_nⁿ⁻¹dx₁· · ·dx_n

=

a^p₁¹· · ·a^p_nⁿΓ

p1

α1

· · ·Γ

pn

αn

α₁· · ·α_nΓ

p1

α1 +· · ·+ _α^pⁿ

n

Z 1

0

Ψ(u)u

p1

α1+···+^pn_αn−1

du.

whereΓ(·)is theΓ-function.

Lemma 2.2. Ifp > 1, ¹_p+¹_q = 1,n∈Z+,α >0,λ >0, setting the weight functionω_α,λ(x, p, q) as:

ω_α,λ(x, p, q) = Z

Rⁿ+

1 kxk^λ_α+kyk^λ_α



 kxk

1 q

α

kyk

1

αp





(n−λ)p

kxk_α kyk_α

^λ_q dy,

then

(2.2) ω_α,λ(x, p, q) =kxk^{(n−λ)(p−1)}_α πΓⁿ _α¹ sin

π p

λαⁿ⁻¹Γ _αⁿ.

Proof. By Lemma 2.1, we have

ω_α,λ(x, p, q) =kxk(n−λ)(p−1)+^λ_q α

Z

Rⁿ+

1

kxk^λ_α+kyk^λ_αkyk⁻(^n−λ+^λq)

α dy

=kxk(n−λ)(p−1)+^λ

α q lim

r→+∞

Z

· · · Z

y1,...,yn>0;y^α₁+···+y^α_n<r^α

× h

r ^y_r¹α

+· · ·+ ^y_rⁿα_α¹i^−(n−λ+^λ_q⁾

kxk^λ_α+h

r ^y_r¹α

+· · ·+ ^y_rⁿα_α¹i^λy₁¹⁻¹· · ·y_n¹⁻¹dy₁· · ·dy_n

=kxk(n−λ)(p−1)+^λ_q

α lim

r→+∞

rⁿΓⁿ _α¹ αⁿΓ ⁿ_α

Z 1 0

ru^α¹−(^n−λ+^λq)

kxk^λ_α+

ru^α¹λuⁿ^α⁻¹du

=kxk(n−λ)(p−1)+^λ_q α

Γⁿ _α¹

αⁿ⁻¹Γ ⁿ_α lim

r→+∞

Z r 0

1

kxk^λ_α+u^λu^λ−^λ^q⁻¹du

=kxk(n−λ)(p−1)+^λ_q α

Γⁿ _α¹ αⁿ⁻¹Γ ⁿ_α

Z ∞ 0

1

kxk^λ_α+u^λu^λ−^λ^q⁻¹du

=kxk^{(n−λ)(p−1)}_α Γⁿ _α¹ λαⁿ⁻¹Γ ⁿ_α

Z ∞ 0

1

1 +uu¹^p⁻¹du

=kxk^{(n−λ)(p−1)}_α Γⁿ _α¹ λαⁿ⁻¹Γ ⁿ_αΓ

1 p

Γ

1− 1

p

(4)

=kxk^{(n−λ)(p−1)}_α πΓⁿ _α¹ sin

π p

λαⁿ⁻¹Γ _αⁿ,

hence (2.2) is valid.

Lemma 2.3. Ifp >1, ¹_p+¹_q = 1,n ∈Z⁺,α >0,λ >0,0< ε < λ(q−1), settingω˜_α,λ(x, q, ε) as:

˜

ω_α,λ(x, q, ε) = Z

Rⁿ+

1

kxk^λ_α+kyk^λ_αkyk⁻

(n−λ)(q−1)+n+ε

α q dy,

then we have

(2.3) ω˜_α,λ(x, q, ε) =kxk⁻

λ q−^ε_q α

Γⁿ _α¹ λαⁿ⁻¹Γ ⁿ_αΓ

1 p− ε

λq

Γ 1

q + ε λq

.

Proof. Lemma 2.3 can be proved in the same manner as Lemma 2.2.

3. MAINRESULTS

Theorem 3.1. Ifp >1, ¹_p +¹_q = 1,n∈Z⁺,α >0,λ >0,f ≥0,g ≥0, and

(3.1) 0<

Z

Rⁿ+

kxk^{(n−λ)(p−1)}_α f^p(x)dx <∞, 0<

Z

Rⁿ+

kxk^{(n−λ)(q−1)}_α g^q(x)dx <∞,

then

(3.2) Z

Rⁿ₊

Z

Rⁿ₊

f(x)g(y)

kxk^λ_α+kyk^λ_αdxdy < πΓⁿ _α¹ sin

π p

λαⁿ⁻¹Γ ⁿ_α

× Z

Rⁿ+

kxk^{(n−λ)(p−1)}_α f^p(x)dx

!¹_p Z

Rⁿ+

kxk^{(n−λ)(q−1)}_α g^q(x)dx

!¹_q

;

(3.3)

Z

Rⁿ+

kyk^λ−n_α Z

Rⁿ+

f(x)

kxk^λ_α+kyk^λ_αdx

!p

dy

<





πΓⁿ _α¹ sin

π p





p

Z

Rⁿ₊

kxk^{(n−λ)(p−1)}_α f^p(x)dx,

where the constant factors ^πΓ

n(α¹)

sin(^πp)^λαⁿ⁻¹^Γ(ⁿα) and

πΓⁿ(α¹)

sin(^πp)^λαⁿ⁻¹^Γ(ⁿα) p

are all the best possible.

(5)

Proof. By Hölder’s inequality, we have

A:=

Z

Rⁿ+

Z

Rⁿ+

f(x)g(y)

kxk^λ_α+kyk^λ_αdxdy

= Z

Rⁿ+

Z

Rⁿ+

f(x) (kxk^λ_α+kyk^λ_α)^p¹



 kxk

1 q

α

kyk

1

αp





n−λ

kxk_α kyk_α

_pq^λ

× g(y)

(kxk^λ_α+kyk^λ_α)¹^q



 kyk

1 p

α

kxk

1

αq





n−λ

kyk_α kxk_α

_pq^λ dxdy

≤





 Z

Rⁿ₊

Z

Rⁿ₊

f^p(x) kxk^λ_α+kyk^λ_α



 kxk

1

αq

kyk

1 p

α





(n−λ)p

kxk_α kyk_α

^λ_q dxdy







1 p

×





 Z

Rⁿ₊

Z

Rⁿ₊

g^q(x) kxk^λ_α+kyk^λ_α



 kyk

1

αp

kxk

1 q

α





(n−λ)q

kyk_α kxk_α

^λ_p dxdy







1 q

=





 Z

Rⁿ₊

f^p(x)





 Z

Rⁿ₊



 kxk

1

αq

kyk

1 p

α





(n−λ)p

kxk_α kyk_α

^λ_q dy





dx







1 p

×





 Z

Rⁿ₊

g^q(y)





 Z

Rⁿ₊



 kyk

1

αp

kxk

1 q

α





(n−λ)q

kyk_α kxk_α

^λ_p dx





dy







1 q

= Z

Rⁿ+

f^p(x)ω_α,λ(x, p, q)dx

!¹_p Z

Rⁿ+

g^q(y)ω_α,λ(y, q, p)dy

!¹_q ,

according to the condition of taking equality in Hölder’s inequality, if this inequality takes the form of an equality, then there exist constantsC₁ andC₂, such that they are not all zero, and

C1f^p(x) kxk^λ_α+kyk^λ_α



 kxk

1 q

α

kyk

1 p

α





(n−λ)p

kxkα

kyk_α ^λ_q

= C₂g^q(y) kxk^λ_α+kyk^λ_α



 kyk

1 p

α

kxk

1 q

α





(n−λ)q

kyk_α kxkα

^λ_p

, a.e. inRⁿ+×Rⁿ+, it follows that

C₁kxk(n−λ)(p−1)+n

α f^p(x) = C₂kyk(n−λ)(q−1)+n

α g^q(y) =C (constant), a.e. in Rⁿ+×Rⁿ+, which contradicts (3.1), hence we have

A <

Z

Rⁿ₊

f^p(x)ω_α,λ(x, p, q)dx

!¹_p Z

Rⁿ₊

g^q(y)ω_α,λ(y, q, p)dy

!¹_q .

(6)

By Lemma 2.2 andsin

π p

= sin(^π_q), we have

A <





πΓⁿ _α¹ sin

π p

λαⁿ⁻¹Γ _αⁿ





1 p

Z

Rⁿ₊

!¹_p

×

"

πΓⁿ _α¹ sin(^π_q)λαⁿ⁻¹Γ ⁿ_α

#¹_q Z

Rⁿ+

kyk^{(n−λ)(q−1)}_α g^q(y)dy

!¹_q

= πΓⁿ _α¹ sin

π p

λαⁿ⁻¹Γ ⁿ_α Z

Rⁿ+

!¹_p Z

Rⁿ+

kxk^{(n−λ)(q−1)}_α g^q(x)dx

!¹_q .

Hence (3.2) is valid.

For0< a < b <∞, setting

ga,b(y) =







kyk^λ−n_α R

Rⁿ+

f(x)

kxk^λ_α+kyk^λ_αdxp−1

, a <kyk_α < b,

0, 0<kyk_α ≤a or kyk_α ≥b,

˜

g(y) = kyk^λ−n_α Z

Rⁿ+

f(x)

!p−1

, y∈Rⁿ+,

by (3.1), for sufficiently smalla >0and sufficiently largeb > 0, we have

0<

Z

a<kykα<b

kyk^{(n−λ)(q−1)}_α g^q_a,b(y)dy <∞.

Hence by (3.2), we have Z

a<kyk_α<b

kyk^{(n−λ)(q−1)}_α g˜^q(y)dy

= Z

a<kykα<b

kyk^λ−n_α Z

Rⁿ₊

f(x) kxk^λ_α+kyk^λ_αdx

!p

dy

= Z

a<kykα<b

kyk^λ−n_α Z

Rⁿ+

!p−1

Z

Rⁿ+

f(x)

! dy

= Z

Rⁿ₊

Z

Rⁿ₊

f(x)g_a,b(y) kxk^λ_α+kyk^λ_αdxdy

(7)

< πΓⁿ _α¹ sin

π p

Rⁿ₊

!¹_p

× Z

Rⁿ+

kyk^{(n−λ)(q−1)}_α g^q_a,b(y)dy

!¹_q

= πΓⁿ _α¹ sin

π p

Rⁿ₊

!¹_p

× Z

a<kykα<b

kyk^{(n−λ)(q−1)}_α ˜g^q(y)dy ¹_q

,

it follows that

Z

a<kykα<b

kyk^{(n−λ)(q−1)}_α g˜^q(y)dy <





πΓⁿ _α¹ sin

π p





p

Z

Rⁿ+

kxk^{(n−λ)(p−1)}_α f^p(x)dx.

Fora→0⁺,b→+∞, by (3.1), we obtain

0<

Z

Rⁿ+

≤





πΓⁿ _α¹ sin

π p

λαⁿ⁻¹Γ _αⁿ





p

Z

Rⁿ+

kxk^{(n−λ)(p−1)}_α f^p(x)dx <∞,

hence by (3.2), we have

Z

Rⁿ+

kyk^λ−n_α Z

Rⁿ+

f(x)

!p

dy

= Z

Rⁿ+

Z

Rⁿ+

f(x)˜g(y)

< πΓⁿ _α¹ sin

π p

Rⁿ+

!¹_p Z

Rⁿ+

!¹_q

= πΓⁿ _α¹ sin

π p

Rⁿ₊

!¹_p

×

"

Z

Rⁿ₊

kyk^λ−n_α Z

Rⁿ₊

!p

dy

#¹_q ,

(8)

it follows that Z

Rⁿ₊

kyk^λ−n_α Z

Rⁿ₊

f(x)

!p

dy

<





πΓⁿ _α¹ sin

π p





p

Z

Rⁿ+

kxk^{(n−λ)(p−1)}_α f^p(x)dx.

Hence (3.3) is valid.

Remark 3.2. By (3.3), we can also obtain (3.2), hence (3.3) and (3.2) are equivalent.

If the constant factor C_n,α(λ, p) := ^πΓ

n(α¹)

sin(^πp)^λαⁿ⁻¹^Γ(ⁿα) in (3.2) is not the best possible, then there exists a positive constantK < C_n,α(λ, p), such that

(3.4) Z

Rⁿ+

Z

Rⁿ+

f(x)g(y)

< K Z

Rⁿ+

!_p¹ Z

Rⁿ+

kyk^{(n−λ)(q−1)}_α g^q(y)dy

!¹_q .

In particular, setting f(x) = kxk⁻

(n−λ)(p−1)+n+ε p

α , g(y) =kyk⁻

(n−λ)(q−1)+n+ε q

α , 0< ε < λ(q−1), (3.4) is still true. By the properties of limit, there exists a sufficiently smalla >0, such that

Z

kxkα>a

Z

Rⁿ+

1

kxk^λ_α+kyk^λ_αkxk⁻

(n−λ)(p−1)+n+ε p

α kyk⁻

(n−λ)(q−1)+n+ε q

α dxdy

< K Z

kxkα>a

kxk^{(n−λ)(p−1)}_α kxk−(n−λ)(p−1)−n−ε

α dx

¹_p

× Z

kyk_α>a

kyk^{(n−λ)(q−1)}_α kyk−(n−λ)(q−1)−n−ε

α dy

¹_q

=K Z

kxkα>a

kxk^−n−ε_α dx.

On the other hand, by Lemma 2.3, we have Z

kxkα>a

Z

Rⁿ+

1

kxk^λ_α+kyk^λ_αkxk⁻

(n−λ)(p−1)+n+ε p

α kyk⁻

(n−λ)(q−1)+n+ε q

α dxdy

= Z

kxkα>a

kxk⁻ⁿ⁺

λ q−_p^ε α

Z

Rⁿ+

1

kxk^λ_α+kyk^λ_αkyk⁻

(n−λ)(q−1)+n+ε q

α dydx

= Z

kxkα>a

kxk⁻ⁿ⁺

λ q−^ε

α pω˜_α,λ(x, q, ε)dx

= Γⁿ _α¹ λαⁿ⁻¹Γ ⁿ_αΓ

1 p− ε

λq

Γ 1

q + ε λq

Z

kxkα>a

kxk^−n−ε_α dx,

hence we obtain

Γⁿ _α¹ λαⁿ⁻¹Γ ⁿ_αΓ

1 p − ε

λq

Γ 1

q + ε λq

< K.

(9)

Forε→0⁺, we have

C_n,α(λ, p) = πΓⁿ _α¹ sin

π p

λαⁿ⁻¹Γ ⁿ_α ≤K,

this contradicts the fact that K < Cn,α(λ, p).Hence the constant factor in (3.2) is the best possible.

Since (3.3) and (3.2) are equivalent, the constant factor in (3.3) is also the best possible.

Corollary 3.3. Ifp >1, ¹_p +¹_q = 1,n∈Z⁺,α >0,f ≥0,g ≥0, and

(3.5) 0<

Z

Rⁿ+

f^p(x)dx <∞, 0<

Z

Rⁿ+

g^q(x)dx <∞,

then

(3.6)

Z

Rⁿ₊

Z

Rⁿ₊

f(x)g(y)

kxkⁿ_α+kykⁿ_αdxdy

< πΓⁿ _α¹ sin

π p

nαⁿ⁻¹Γ ⁿ_α Z

Rⁿ+

f^p(x)dx

!¹_p Z

Rⁿ+

g^q(x)dx

!¹_q

;

(3.7)

Z

Rⁿ+

Z

Rⁿ+

f(x)

kxkⁿ_α+kykⁿ_αdx

!p

dy <





πΓⁿ ¹_α sin

π p

nαⁿ⁻¹Γ _αⁿ





p

Z

Rⁿ+

f^p(x)dx,

where the constant factors in (3.6) and (3.7) are all the best possible.

Proof. By takingλ =nin Theorem 3.1, (3.6) and (3.7) can be obtained.

Corollary 3.4. Ifp >1, ¹_p +¹_q = 1,n∈Z⁺,f ≥0,g ≥0, and

(3.8) 0<

Z

Rⁿ₊

f^p(x)dx <∞, 0<

Z

Rⁿ₊

g^q(x)dx <∞,

then

(3.9)

Z

Rⁿ₊

Z

Rⁿ₊

f(x)g(y) (Pn

i=1x_i)ⁿ+ (Pn

i=1y_i)ⁿdxdy

< π n! sin

π p

Z

Rⁿ+

f^p(x)dx

!¹_p Z

Rⁿ+

g^q(x)dx

!¹_q

;

(3.10) Z

Rⁿ+

Z

Rⁿ+

f(x) (Pn

i=1x_i)ⁿ+ (Pn

i=1y_i)ⁿdx

!p

dy <



 π n! sin

π p





p

Z

Rⁿ+

f^p(x)dx,

where the constant factors in (3.9) and (3.10) are all the best possible.

Proof. By takingα = 1in Corollary 3.3, (3.9) and (3.10) can be obtained.

(10)

REFERENCES

[1] G.M. FICHTINGOLOZ, A Course in Differential and Integral Calculus, Renmin Jiaoyu Publishers, Beijing, 1957.

[2] MINGZHE GAOANDTAN LI, Some improvements on Hilbert’s integral inequality, J. Math. Anal.

Appl., 229 (1999), 682–689.

[3] G.H. HARDY, J.E. LITTLEWOODANDG. POLYA, Inequalities, Cambridge Univ. Press, London, 1952.

[4] A.E. INGHAM, A note on Hilbert’s inequality, J. London Math. Soc., 11 (1936), 237–240.

[5] JICHANG KUANG, On new extensions of Hilbert’s integtal inequality, Math. Anal. Appl., 235 (1999), 608–614.

[6] JICHANG KUANG, Applied Inequalities, Shandong Science and Technology Press, Jinan, 2004.

[7] B.G. PACHPATTE. On some new inequalities similar to Hilbert’s inequality, J. Math. Anal. Appl., 226 (1998), 166–179.

[8] BICHENG YANG, On a generalization of Hardy-Hilbert’s inequality, Chin. Ann. of Math., 23 (2002), 247–254.

[9] BICHENG YANG, On a multiple Hardy-Hilbert’s integral inequality, Chin. Ann. of Math., 24(A):6(2003), 743–750.

[10] BICHENG YANG ANDTh.M. RASSIAS, On the way of weight coefficient and research for the Hilbert-type inequalities, Math. Ineq. Appl., 6(4) (2003), 625–658.

[11] HONG YONG, All-sided generalization about Hardy-Hilbert’s integral inequalities, Acta Math.

Sinica, 44 (2001), 619–626.