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
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.
+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
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[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.
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tion:26D15
A
1674-5647(2013)01-0014-09
*Received date:Jan.26,2010.
Communications in Mathematical Research2013年1期