A multiple Hilbert type integral inequality with a non homogeneous kernel

R E S E A R C H

Open Access

A multiple Hilbert-type integral inequality

with a non-homogeneous kernel

Qiliang Huang

*

_{and Bicheng Yang}

*_{Correspondence:}

[email protected] Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, P.R. China

Abstract

By using the way of weight functions and the technic of real analysis, a multiple Hilbert-type integral inequality with a non-homogeneous kernel is given. The operator expression with the norm, the reverses and some examples with the particular kernels are also considered.

MSC: 47A07; 26D15

Keywords: multiple Hilbert-type integral inequality; kernel; weight function; norm; operator

1 Introduction

Ifp> , _p+_q = ,f(≥)∈Lp_{(,∞),g(≥)∈L}q_{(,∞),fp,gq> , then we have the} following equivalent inequalities (cf.[]):

∞

 ∞



f(x)g(y)

x+y dx dy<

π

sin(π/p)fpgq, ()

∞

 ∞

 f(x)

x+ydx p

dy 

p

< π

sin(π/p)fp, ()

where the constant factor π

sin(π/p)is the best possible. () is the well-known Hardy-Hilbert

integral inequality. Define the Hardy-Hilbert integral operatorT:Lp_{(,∞)→L}p_{(,∞) as} follows: forf ∈Lp(,∞),Tf(y) :=_∞_x+_yf(x)dx(y∈(,∞)).

Then in view of (), it followsTfp< π

sin(π/p)fpandT ≤

π

sin(π/p). Since the constant

is the best possible, we findT=_sin(π_π/p).

Inequalities () and () and the operator are important in analysis and its applications (cf.[, ]). In , [] considered the property of the Hardy-Hilbert integral operator and gave an improvement of () (forp=q= ). In , by introducing another pair of conjugate exponents (r,s) (r> , _r+_s= ) and an independent parameterλ> , [] gave the best extensions of () as follows:

∞

 ∞



f(x)g(y)

xλ_+yλ dx dy< π

λsin(π/r)fp,φgq,ψ, ()

where φ(x) =xp(–λ_r)–_{, ψ(x) =x}q(–λ_s)–_{, fp} ,φ={

∞

 φ(x)fp(x)dx} 

p _{> ,g}q

,ψ > . In , [] gave the following inequality with the best constantB(λ_,λ_) (λ> ;B(u,v) is

the beta function):

∞

 ∞



f(x)g(y)

( +xy)λdx dy<B

λ ,

λ 

∞



x–λf(x)dx ∞



x–λg(x)dx 



. ()

In , [] gave an extension of () inR_{with the kernel} 

|+xy|λ( <λ< ); [] gave another

extension of () to the general kernelkλ(,xy) (λ> ) with a pair of conjugate exponents (p,q) and obtained the following multiple Hilbert-type integral inequality. Suppose that

n∈N\{},pi> ,

n

i=pi= ,λ> ,kλ(x, . . . ,xn)≥ is a measurable function of –λ-degree

inRn

+, and for any (r, . . . ,rn) (ri> ), satisfies

n

i=ri=  and

kλ=

Rn+–

kλ(u, . . . ,un–, )

n–

j= u

λ rj–

j du· · ·dun–> .

Ifφi(x) =x pi(–_riλ)–

,fi(≥)∈Lpφii(,∞),fpi,φi>  (i= , . . . ,n), then we have the following

inequality:

Rn+

kλ(x, . . . ,xn) n

i=

fi(xi)dx· · ·dxn<kλ n

i=

fip_i,φi, ()

where the constant factorkλis the best possible. Forn= ,kλ(x,y) =_xλ_+yλ in (), we

ob-tain (). Inequality () is an extension of the results in [–] and []. In recent years, [] and [] considered some Hilbert-type operators relating ()-(); [] also considered a multiple Hilbert-type integral operator with the homogeneous kernel of –n+ -degree and the relating particular case of () (forλ=n– , 

ri = 

n–( – 

pi)).

In this paper, by using the way of weight functions and the technic of real analysis, a multiple Hilbert-type integral inequality with a non-homogeneous kernel is given. The operator expression with the norm, the reverses and some examples with the particular kernels are considered.

2 Some lemmas

Lemma  If n∈N\{},λi∈R(i= , . . . ,n),

n i=



pi = ,then we have

A:= n

i=

x(λi–)(–pi)

i

n

j=(j=i) xλ_jj–

 pi

= . ()

Proof We find

A:= n

i=

x(λi–)(–pi)+–λi

i

n

j= xλ_jj–

 pi

= n

i=

x(–λi)pi

i n

j= xλ_jj–

 pi

= n

i= x–λi

i

_n

j= xλ_jj–

n i=_pi

,

Definition  Ifn∈N,Rn

+:={(x, . . . ,xn)|xi>  (i= , . . . ,n)},λ∈R,kλ(x, . . . ,xn) is a mea-surable function in Rn

+ such that for any u>  and (x, . . . ,xn)∈Rn+, kλ(ux, . . . ,uxn) = u–λ_kλ(x

, . . . ,xn), then we callkλ(x, . . . ,xn) the homogeneous function of –λ-degree inRn+.

Lemma  Suppose n∈N\{},λi∈R(i= , . . . ,n),λn=

n–

i= λi=λ_,kλ(x, . . . ,xn)≥is a

homogeneous function of–λ-degree.If

H(i) :=

Rn+–

kλ(u, . . . ,ui–, ,ui+, . . . ,un)

× n

j=(j=i)

uλ_jj–du· · ·dui–dui+· · ·dun (i= , . . . ,n)

satisfying kλ:=H(n)∈R,then each H(i) =H(n) =kλand for any i= , . . . ,n,

ωi(xi) :=xλi

i

Rn+–

kλ(xxn, . . . ,xn–xn, )

× n

j=(j=i)

xλ_jj–dx· · ·dxi–dxi+· · ·dxn=kλ (∈R). ()

Proof Settinguj=unvj(j=i,n) in the integralH(i), we find

H(i) =

Rn+– kλ

v, . . . ,vi–,u–n,vi+, . . . ,vn–,  n–

j=(j=i) vλ_jj–

×u––λi

n dv· · ·dvi–dvi+· · ·dvn–dun.

Settingvi=u–n in the above integral, we obtainH(i) =H(n). Settingxn=x–n in (), since λ–λn=λn, we find

ωi(xi) =x λi

i

Rn+– kλ

x, . . . ,xn–,x–n

x–λn–

n

× n–

j=(j=i)

xλ_jj–dx· · ·dxi–dxi+· · ·dxn

=xλi

i

Rn₊– kλ

x, . . . ,xn–,xn

x_nλn+

× n–

j=(j=i)

xλ_jj–dx· · ·dxi–dxi+· · ·dxn–

x_n–dx_n

=xλi

i

Rn+– kλ

x, . . . ,xn–,xn

x_nλn–

× n–

j=(j=i)

xλ_jj–dx· · ·dxi–dxi+· · ·dxn–dxn.

Setting uj=xj/xi (j=i,n) and un=xn/xi in the above integral, we findωi(xi) =H(i) =

Lemma  With the assumptions given in Lemma,then

k(λ, . . . ,λn–) :=

Rn+–

kλ(u, . . . ,un–, )

n–

j=

uλ_jj–du· · ·dun–

is finite in a neighborhood of (λ, . . . ,λn–) if any only if k(λ, . . . ,λn–)is continuous at

(λ, . . . ,λn–).

Proof The sufficiency property is obvious. We prove the necessary property of the con-dition by mathematical induction in the following. For n= , there existsI:={λ|λ=

λ+δ,|δ| ≤δ,δ> }such that for anyλ∈I,k(λ)∈R. Since forλ=λ+δ∈I(δ= ),

k(λ+δ) = 



kλ(u, )uλ_+δ–du+

∞



kλ(u, )uλ_+δ–du,

kλ(u, )u

λ+δ–

 ≤kλ(u, )u

λ–δ–

 du, u∈(, ];

kλ(u, )uλ_+δ–≤kλ(u, )uλ_+δ–du, u∈(,∞),

andk(λ–δ) +k(λ+δ) <∞, then by the Lebesgue control convergence theorem (cf.

[]), it followsk(λ+δ) =k(λ) +o() (δ→). Assuming that forn(≥),k(λ, . . . ,λn–)

is continuous at (λ, . . . ,λn–), then forn+ , by the result ofn= , sincek(λ+δ, . . . ,λn+δn) is finite in a neighborhood of (λ, . . . ,λn), we find

lim

δn→k(λ+δ, . . . ,λn+δn)

= lim δn→

∞



Rn+–

kλ(u, . . . ,un, ) n–

j=

uλ_jj+δj–du· · ·dun–

uλn+δn–

n dun

=

∞



Rn+–

kλ(u, . . . ,un, ) n–

j=

uλ_jj+δj–du· · ·dun–

uλn–

n dun

=

Rn+– ∞



kλ(u, . . . ,un, )uλn–

n dun

n–

j=

uλ_jj+δj–du· · ·dun–,

then by the assumption forn, it follows

lim δn→

k(λ+δ, . . . ,λn+δn) =k(λ, . . . ,λn) +o() (δi→,i= , . . . ,n– ).

By mathematical induction, we prove that forn∈N\ {},k(λ, . . . ,λn–) is continuous at

(λ, . . . ,λn–).

Lemma  With the assumptions given in Lemma, if there existsδ > such that for

max≤i≤n–{|δi|}<δ, k(λ +δ, . . . ,λn–+δn–)∈ R, pi ∈ R\{, } (i= , . . . ,n),  < ε< min≤i≤n{|pi|}δ,then we have

Iε:=ε

∞



· · ·

∞

 

 xλn+

ε pn–

n kλ(xxn, . . . ,xn–xn, )dxn

× n–

j= x

λj–_pjε–

j dx· · ·dxn–=kλ+o()

Proof Settingx_n=x–

n in (), we find

Iε:=ε

_∞ 

· · ·

_∞ 

∞



x_n–λn–pnε–_k

λ

x x_n, . . . ,

x x_n, 

dx_n

× n–

j= x

λj–_pjε–

j dx· · ·dxn–=kλ+o().

Settinguj=xj/xn(j= , . . . ,n– ) in the above integral, sinceλ–λn=λn, we find (replacing

x_nbyxn)

Iε=ε

∞

 x––_n ε

∞

x–n

· · ·

∞

x–n

kλ(u, . . . ,un–, )

n–

j= u

λj–_pjε–

j du· · ·dun–

dxn. ()

SettingDj:={(u, . . . ,un–)|uj∈(,x–n),uk∈(,∞) (k=j)}and

Aj(xn) :=

· · ·

Dj

kλ(u, . . . ,un–, )

n–

j= u

λj–_pjε–

j du· · ·dun–,

by (), it follows

Iε≥

Rn+–

kλ(u, . . . ,un–, )

n–

j= u

λj–_pjε–

j du· · ·dun––ε

n–

j= ∞



x–_nAj(xn)dxn. ()

Without loss of generality, we estimate the case ofj=n, e.t.

∞

 x –

n An–(xn)dxn=O(). In fact, settingα> , such that|_pnε_– +α|<δ, since –uαn–×

lnun–→ (un–→+), there existsM>  such that –uαn–lnun–≤M(un–∈(, ]), and

then by the Fubini theorem, it follows

≤

∞



x–_n An–(xn)dxn

=

∞

 x–_n

Rn+– x–n



kλ(u, . . . ,un–, )

n–

j= u

λj–_pjε–

j dun–du· · ·dun–

dxn

=





Rn+–

kλ(u, . . . ,un–, )

n–

j= u

λj–_pjε–

j

u–_n–



x–_n dxn

du· · ·dun–

=





Rn+–

kλ(u, . . . ,un–, )

n–

j= u

λj–_pjε–

j (–lnun–)du· · ·dun–

≤M 



Rn+–

kλ(u, . . . ,un–, )

n–

j= u

λj–_pjε–

j u

λn––(_pnε_–+α)–

n– du· · ·dun–

≤M

Rn+–

kλ(u, . . . ,un–, )

n–

j= u

λj–_pjε–

j u

λn––(_pnε_–+α)–

n– du· · ·dun–

=M·k

λ–

ε

p, . . . ,λn––

ε

pn–

,λn––

ε

pn–

+α

Hence by (), we have

Iε≥

Rn+–

kλ(u, . . . ,un–, )

n–

j= u

λj–_pjε–

j du· · ·dun––o(). ()

Since by Lemma  we find

Iε≤ε

∞

 x––ε

n

∞

 · · · ∞



kλ(u, . . . ,un–, )

n–

j= u

λj–_pjε–

j du· · ·dun–

dxn

=

∞



· · ·

∞



kλ(u, . . . ,un–, )

n–

j= u

λj–_pjε–

j du· · ·dun–

=k

λ–

ε

p, . . . ,λn––

ε

pn–

=kλ+o(),

then combining with (), we have ().

Lemma  Suppose that n∈N\{}, pi∈R\{, } (i= , . . . ,n),

n i=pi = ,



qn =  – 

pn,

(λ, . . . ,λn)∈Rn, λn=

n–

i= λi= λ_, kλ(x, . . . ,xn) (≥)is a measurable function of –λ

-degree inRn₊such that

kλ=

Rn+–

kλ(u, . . . ,un–, )

n–

j=

uλ_jj–du· · ·dun–∈R.

If fi≥are measurable functions inR+(i= , . . . ,n– ),kλ(x, . . . ,xn) :=kλ(xxn, . . . ,xn–xn, ),then()for pi>  (i= , . . . ,n),we have

J:=

∞

 x

λqn  –

n

Rn+–

kλ(x, . . . ,xn) n–

i=

fi(xi)dx· · ·dxn– qn

dxn

 qn

≤kλ n–

i= ∞



xpi(–λi)–_fpi_(x)dx 

pi

; ()

()for <p< ,pi<  (i= , . . . ,n),we have the reverse of().

Proof () Forpi>  (i= , . . . ,n), by the Hölder inequality (cf.[]) and (), it follows

Rn+–

kλ(x, . . . ,xn) n–

i=

fi(xi)dx· · ·dxn– qn

=

Rn+–

kλ(x, . . . ,xn) n–

i=

x(λi–)(–pi)

i

n

j=(j=i) xλ_jj–

 pi

fi(xi)

×

x(λn–)(–pn)

n

n–

j= xλ_jj–

 pn

≤

Rn+–

kλ(x, . . . ,xn) n–

i=

x(λi–)(–pi)

i

n

j=(j=i) xλ_jj–

qn pi

fqn

i (xi)dx· · ·dxn–

×

Rn+–

kλ(x, . . . ,xn)x(λn–)(–pn)

n

n–

j=

xλ_jj–dx· · ·dxn– qn–

= (kλ)qn–_x–qnλn

n

Rn+–

kλ(x, . . . ,xn)

× n–

i=

x(λi–)(–pi)

i

n

j=(j=i) xλ_jj–

qn pi

fqn

i (xi)dx· · ·dxn–, ()

J≤(kλ)pn ∞



Rn+–

kλ(x, . . . ,xn)

× n–

i=

x(λi–)(–pi)

i

n

j=(j=i) xλ_jj–

qn pi

fqn

i (xi)dx· · ·dxn–dxn

 qn

= (kλ)pn

Rn+– ∞



kλ(x, . . . ,xn)xλn–

n dxn

× n–

i=

x(λi–)(–pi)

i

n–

j=(j=i) xλ_jj–

qn pi

fqn

i (xi)dx· · ·dxn– 

qn

.

Forn≥, by the Hölder inequality again, it follows

J≤(kλ)pn _n–

i=

Rn+– ∞



kλ(x, . . . ,xn)xλn–

n dxn

× x(λi–)(–pi)

i

n–

j=(j=i) xλ_jj–fpi

i (xi)dx· · ·dxn– qn

pi  qn

= (kλ)pn

n–

i= ∞



Rn+–

kλ(x, . . . ,xn)

×xλi

i n

j=(j=i)

xλ_jj–dx· · ·dxi–dxi+· · ·dxn

xpi(–λi)–

i f

pi

i (xi)dxi

 pi

= (kλ)pn

n–

i= ∞



ωi(xi)xpi(–λi)–

i f

pi

i (xi)dxi

 pi

. ()

Then by (), we have (). (Note: forn= , we do not use the Hölder inequality again in the above.) () For  <p< ,pi<  (i= , . . . ,n), by the reverse Hölder inequality and in

the same way, we obtain the reverses of ().

3 Main results and applications

With the assumptions given in Lemma , settingφi(x) :=xpi(–λi)–(x∈(,∞);i= , . . . ,n), then we findφ/(–pn)

spaces:

Lpi

φi(,∞) :=

f;fp_i,φ i=

∞



φi(x)f(x) pi

dx 

pi

<∞

(i= , . . . ,n),

n–

i= Lpi

φi(,∞) :=

(f, . . . ,fn–);fi∈Lpφii(,∞),i= , . . . ,n– 

,

and a multiple Hilbert-type integral operatorT:n_i=–Lpi

φi(,∞)→L

qn

φn/(–pn)

as follows: for

f = (f, . . . ,fn–)∈ n–

i= L

pi

φi(,∞),

(Tf)(xn) :=

Rn+–

kλ(x, . . . ,xn) n–

i=

fi(xi)dx· · ·dxn–, xn∈(,∞). ()

Then by (), it followsTf ∈Lqn

φn/(–pn)

,Tis bounded,Tf

qn,φ/(–n pn)≤kλ n–

i= fipi,φiand

T ≤kλ, where

T:= sup f∈n–

i=Lpiφ_i(,∞)(fi=θ,i=,...,n–)

Tf_q n,φ/(–n pn) n–

i= fipi,φi

. ()

Define the formal inner product ofT(f, . . . ,fn–) andfnas

T(f, . . . ,fn–),fn

:=

Rn+

kλ(x, . . . ,xn) n

i=

fi(xi)dx· · ·dxn. ()

Theorem  With the assumptions given in Lemma,suppose that for any(λ, . . . ,λn)∈Rn, it satisfiesλn=

n–

i= λi=λ_,and

 <kλ=

Rn+–

kλ(u, . . . ,un–, )

n–

j=

uλ_jj–du· · ·dun–<∞. ()

If fi(≥)∈Lφpii(,∞),fpi,φi>  (i= , . . . ,n),then(i)for pi>  (i= , . . . ,n),we haveT= kλand the following equivalent inequalities:

T(f, . . . ,fn–)_q

n,φ/(–n pn)<kλ

n–

i=

fipi,φi, ()

T(f, . . . ,fn–),fn

<kλ n

i=

fip_i,φi, ()

where the constant factor kλis the best possible; (ii)for <p< ,pi<  (i= , . . . ,n),using

Proof (i) For allpi> , if () takes the form of equality, then forn≥ in (), there exist

CiandCk(i=k) such that they are not all zero and

Cix(iλi–)(–pi) n–

j=(j=i)

xλ_jj–f_jpj(xj)

=Ckx(

λk–)(–pk)

k

n–

j=(j=k)

xλ_jj–f_jpj(xj) a.e. inRn+,

e.t. Cixpi(– λi)

i f

pi

i (xi) = Ckx pk(–λk)

k f

pk

k (xk) = C a.e. in Rn+. Assuming that Ci > , then

xpi(–λi)–

i f

pi

i (xi) =C/(Cixi), which contradictsfpi,φ_i > . (Note: forn= , we consider

() forfpi

k (xk) =  in the above.) Hence we have (). By the Hölder inequality, it follows

(Tf,fn) =

∞



xλn–  qn

n

Rn+–

kλ(x, . . . ,xn) n–

i=

fi(xi)dx· · ·dxn–

x

 qn–λn

n fn(xn)

dxn

≤T(f, . . . ,fn–)_q

n,φn/(–pn)fnpn,φn, ()

and then by (), we have (). Assuming that () is valid, setting

fn(xn) :=xqnnλn–

Rn+–

kλ(x, . . . ,xn) n–

i=

fi(xi)dx· · ·dxn– qn–

,

thenJ={_∞xpn(–λn)–

n fnpn(xn)dxn}



qn_{. By (), it followsJ<}∞_{. IfJ= , then () is naturally}

valid. Assuming that  <J<∞, by (), it follows

∞



xpn(–λn)–

n fnpn(xn)dxn=Jqn= (Tf,fn) <kλ n

i=

fipi,φi,

∞



xpn(–λn)–

n fnpn(xn)dxn

 qn

=J<kλ n–

i=

fipi,φi,

and then () is valid, which is equivalent to ().

Forε>  small enough, settingfi(x) as:fi(x) = ,x∈(, );fi(x) =x

λi–_piε–_{,x∈[,∞)}

(i= , . . . ,n– ),fn(x) =xλn+_pnε–

,x∈(, );fn(x) = ,x∈[,∞), if there existsk≤kλsuch that () is still valid as we replacekλbyk, then in particular, by Lemma , we have

kλ+o() =Iε=ε

T(f, . . . ,fn–),fn

<εk

n

i=

fip_i,φi=k

andkλ≤k(ε→+). Hencek=kλis the best value of (). We confirm that the constant factorkλin () is the best possible, otherwise we can get a contradiction by () that the constant factor in () is not the best possible. ThereforeT=kλ.

(ii) For  <p< ,pi<  (i= , . . . ,n), by using the reverse Hölder inequality and in the same way, we have the equivalent reverses of () and () with the same best constant

Example  Forλ> ,λi=_rλ_i (i= , . . . ,n),rn= ,

n

i=ri= ,kλ(x, . . . ,xn) =  (n_i=xi)λ, by

mathematical induction, we can show

kλ=

Rn+–

 (n_i=–ui+ )λ

n–

j= u

λ rj–

j du· · ·dun–=

 (λ)

n i= λ ri . ()

In fact, forn= , we obtain

kλ=

R+

 (u+ )λu

λ r_–  du=

 (λ)

λ r λ r .

Assuming that forn(≥) () is valid, then forn+ , it follows

kλ=

Rn+

 (n_i=ui+ )λ

n

j= u

λ rj–

j du· · ·dun

=

Rn+–

n j= u λ rj– j R+ 

[u+ (n_i=ui+ )]λ

u λ r–  du

du· · ·dun

=

Rn+–

 (n_i=ui+ )λ

n j= u λ rj– j R+  (v+ )λv

λ r–  dv

du· · ·dun

=( λ r)(λ–

λ r)

(λ)

Rn+–



(n_i=ui+ ) λ(–r_)

n

j= u

λ rj–

j du· · ·dun

=( λ r)(λ–

λ r)

(λ)

 (λ– λ

r)

n+ i= λ ri = 

(λ) n+ i= λ ri .

Then by mathematical induction, () is valid forn∈N\{}.

Example  Forλ> ,λi=_rλ_i (i= , . . . ,n),rn= ,

n

i=ri = ,kλ(x, . . . ,xn) =  (max≤i≤n{xi})λ,

by mathematical induction, we can show

kλ=

Rn+–



(max_{≤i≤n–}{ui}+ )λ n–

j= u

λ rj–

j du· · ·dun–= n

i=ri

λn– . ()

In fact, forn= , we obtain

kλ=

R+ u

λ r_– 

(max{u, })λdu=



 u

λ r–  du+

∞

 u

–λ r–  du=

 λrr.

Assuming that forn(≥), () is valid, then forn+ , it follows

kλ=

Rn+–

n j= u λ rj– j ∞   (max_≤i≤n{ui, })λ

u λ r_–  du

du· · ·dun

=

Rn+–

n j= u λ rj– j

max{u,...,un,} 



(max≤i≤n{ui, })λu

+

∞

max{u,...,un,}



uλ

 u

λ r–  du

du· · ·dun

= r

 

λ(r– )

Rn+–



(max≤i≤n{ui, })λ(–

 r)

n

j= u

λ rj–

j du· · ·dun

= r

 

λ(r– )

r r– 

n–

Rn+–

 (max≤i≤n{vi, })λ

n j= v λ rj r_ r––

j dv· · ·dvn

= r

 

λ(r– )

r r– 

n–

 λn–

n+

i= r– 

r ri=

 λn

n+

i= ri.

Then by mathematical induction, () is valid forn∈N\{}.

Example  For λ > , λi = –_rλ_i (i = , . . . ,n), rn = ,

n

i=ri = , kλ(x, . . . ,xn) =

(min≤i≤n{xi})λ, by mathematical induction, we can show

k–λ=

Rn+–

min{u, . . . ,un–, } λn–

j= u

–λ rj–

j du· · ·dun–= n

i=ri

λn– . ()

In fact, forn= , we obtain

k–λ=



 u

λ r_–  du+

∞

 u

–λ r_–  du=

 λrr.

Assuming that forn(≥), () is valid, then forn+ , it follows

k–λ=

Rn+–

n j= u –λ rj– j ∞ 

min{u, . . . ,un, }

λ

u –λ r–  du

du· · ·dun

=

Rn+–

n j= u –λ rj– j

min{u,...,un,} 

uλ

u –λ r–  du

+

∞

min{u,...,un,}

min{u, . . . ,un, }λu –λ

r–  du

du· · ·dun

= r

 

λ(r– )

Rn+–

min{u, . . . ,un, }

λ(–_r )

n

j= u

–λ(– _r ) (– _r

)rj –

j du· · ·dun

= r

 

λ(r– )

 [λ( –_r

)] n– n+ i=  –  r

ri=  λn

n+

i= ri.

Then, by mathematical induction, () is valid forn∈N\{}.

Remarks (i) In particular, forn=  in (), we have

∞

 ∞



kλ(,xy)f(x)g(y)dx dy

<kλ

∞



xp(–λ)–_fp_(x)dx 

p ∞ 

xq(–λ)–_gq_(x)dx 

q

wherekλ=

_∞

 kλ(u, )u λ

–_{du>  (λ}∈_{R) is the best possible. Inequality () is an}

exten-sion of () and (..) in [].

(ii) In Examples  and , by Theorem , since for any (λ, . . . ,λn)∈Rn(λn=

n

i=λi=λ_), we obtain  <kλ<∞, then we haveT=kλ and the equivalent inequalities () and () with the particular kernels and some equivalent reverses. In Example , still using Theorem , we find  <T=k–λ<∞and the relating particular inequalities.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

QH carried out the study, and wrote the manuscript. BY participated in the design of the study, and reformed the manuscript. All authors read and approved the final manuscript.

Acknowledgements

This work is supported by Guangdong Modern Information Service industry Develop Particularly item 2011 (No. 13090).

Received: 3 October 2012 Accepted: 11 February 2013 Published: 28 February 2013 References

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doi:10.1186/1029-242X-2013-73

Cite this article as:Huang and Yang:A multiple Hilbert-type integral inequality with a non-homogeneous kernel.