HONG YONG

(Department of Mathematics,Guangdong University of Business Studies,Guangzhou,510320)

An Extended Multiple Hardy-Hilbert's Integral Inequality*

HONG YONG

(Department of Mathematics,Guangdong University of Business Studies,Guangzhou,510320)

Communicated by Ji You-qing

In this paper,by introducing the norm‖x‖α(x∈Rn),a multiple Hardy-Hilbert's integral inequality with the best constant factor and it's equivalent form are given.

multiple Hardy-Hilbert's integral inequality,weight function,best constant factor,β-function,Γ-function

1 Introduction

If and

then the well known Hardy-Hilbert's integral inequality is(see[1])

Its equivalent form is

where the constant factors in(1.1)and(1.2)are optimal.

Hardy-Hilbert's inequality is important in harmonic analysis,real analysis and operator theory.In recent years,many valuable results(see[2–5])have been obtained in generalization and improvement of Hardy-Hilbert's inequality.In 1999,Kuang[6]gave a generalization with a parameter λ of(1.1)as follows:

where λ＞2-min{p,q},and the constant factoris optimal.

At present,for multiple Hardy-Hilbert's integral inequality,many new results have be obtained(see[8–10]).In this paper,by the method of weight function,a higher-dimensional generalization of(1.4)is obtained,and its equivalent form is researched.For the sake of convenience,we introduce the following symbols:

Lemma 1.1[11]If pi＞ 0,ai＞ 0,αi＞0,i=1,2,···,n,and Ψ(u)is a measurable function,then

+the weight function

then

Proof.By(1.5)one has

and hence(1.6)is valid.

+0＜ε＜n(q-2)+λ,then∫

Proof.By a method similar to the proof of Lemma 1.2,Lemma 1.3 can be proved.

2 Main Result

+g ≥0,and

then

where

and the constant factors hα,λ(n,p,q)in(2.2)and(n,p,q)in(2.3)are optimal.

In particular,

(1)for λ=n,one has

where the constant factors

and

are optimal;

(2)for α=1,one has

where the constant factors

and

are optimal;

(3)for p=q=2,one has

where the constant factors

and

are optimal.

Proof. By the H¨older's inequality,one has

According to the condition of taking equality in the H¨older's inequality,if this inequality takes the form of an equality,then there exist constants C1and C2with+/=0 such that

It follows that

Without loss of generality,suppose that C1/=0.Then

which contradicts(2.1).Hence

Further,by(1.6),one has that(2.2)is valid.

For 0＜a＜b＜∞,set

By(2.1),for sufficiently small a＞0 and sufficiently large b＞0,one has

Hence,by(2.2),one has

which implies that

For a→0+,b→+∞,we get

Hence,by(2.1),one has

By(2.2),one has

Hence(2.3)can be obtained.

By(2.3),one can also obtain(2.2).Hence(2.2)and(2.3)are equivalent.

If the constant factor hα,λ(n,p,q)in(2.2)is not optimal,then there exists a positive constant K＜hα,λ(n,p,q)such that

In particular,setting

by(2.4)and the properties of limit,we see that there exists a sufficiently small a＞0 such that

On the other hand,by(1.7),one has

Hence,

For ε→0+,one has

which contradicts the fact that K＜h(n,p,q).Hence the constant factor h(n,p,q)inα,λα,λ(2.2)is optimal.

Since(2.2)and(2.3)are equivalent,the constant factor(n,p,q)in(2.3)is also optimal.

The proof of Theorem 2.1 is completed.

References

[1]Hardy G H,Littewood J E,Polya G.Inequalities.London:Gambridge Univ.Press,1952.

[2]Zhao C J,Debnath L.Some new type Hilbert integral inequalities.J.Math.Anal.Appl.,2001, 262:411–418.

[3]Hu K.On Hilbert's inequality and it's applications.Adv.in Math.(China),1993,22:160–163.

[4]Pachpatte B G.On some new inequalities similar to Hilbert's inequality.J.Math.Anal.Appl., 1998,226:166–179.

[5]Gao M Z,Tan L.Some improvements on Hilbert's integral inequality.J.Math.Anal.Appl., 1999,229:682–689.

[6]Kuang J C.On new extensions of Hilbert's integtal inequality.J.Math.Anal.Appl.,1999, 235:608–614.

[7]Yang B C.On a generalization of Hardy-Hilbert's inequality.Chinese Ann.Math.Ser.A,2002, 23(5):247–254.

[8]Hong Y.All-sided generalization about Hardy-Hilbert integral inequalities.Acta Math.Sinica, 2001,44(4):619–626.

[9]Yang B C.On a multiple Hardy-Hilbert's integral inequality.Chinese Ann.Math.Ser.A,2003, 24(6):743–750.

[10]Yang B C,Themistocles M.On the way of weight coefficient and research for the Hilbert-type inequalities.Math.Inequal.Appl.,2003,6:625–658.

[11]Fichtingoloz G M.A Course in Dif f erential and Integral Calculus.Beijing:Renmin Jiaoyu Publisgers,1957.

tion:26D15

A

1674-5647(2013)01-0014-09

*Received date:Jan.26,2010.