On Hardy type integral inequalities with the gamma function

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R E S E A R C H

Open Access

On Hardy-type integral inequalities with

the gamma function

Jianquan Liao

*

_{and Bicheng Yang}

*_{Correspondence: lmath@163.com}

Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 51003, P.R. China

Abstract

By means of real analysis and weight functions, we obtain a few equivalent conditions of two kinds of Hardy-type integral inequalities with the non-homogeneous kernel and parameters. The constant factors related to the gamma function are proved to be the best possible. We also consider the operator expressions and some cases of homogeneous kernel.

MSC: 26D15

Keywords: Hardy-type integral inequality; weight function; equivalent form; operator; norm

1 Introduction

If  <_∞f₍_x₎_dx_<_∞_{and  <}∞

 g(y)dy<∞, then we have the following Hilbert’s

inte-gral inequality (cf.[]):

∞



∞



f(x)g(y)

x+y dx dy<π

∞



f(x)dx

∞



g(y)dy 



, ()

where the constant factorπis the best possible. In , by introducing one pair of con-jugate exponents (p,q), Hardy [] gave an extension of () as follows: Forp> , _p+_q= , f(x),g(y)≥,  <_∞fp₍_x₎_dx_<_∞_{and  <}∞

 gq(y)dy<∞, we have

∞



∞



f(x)g(y) x+y dx dy<

π sin(π/p)

∞



fp(x)dx 

p ∞



gq(y)dy 

q

, ()

where the constant factor π

sin(π/p)is the best possible. Inequalities () and () are important

in analysis and its applications (cf.[, ]).

In , Hardy et al. gave an extension of () as follows: If k(x,y) is a non-negative

homogeneous function of degree –,kp=

∞

 k(u, )u –

p _du∈_R

+= (,∞), then

∞



∞



k(x,y)f(x)g(y)dx dy<kp

∞



fp(x)dx 

p ∞



gq(y)dy 

q

, ()

where the constant factorkp is the best possible (cf.[], Theorem ). Additionally,

a Hilbert-type integral inequality with the non-homogeneous kernel is proved as follows:

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Ifh(u) > ,φ(σ) =_∞h(u)uσ–_du_∈_R +, then

∞



∞



h(xy)f(x)g(y)dx dy

<φ

 p

∞



xp–fp(x)dx 

p ∞



gq(y)dy 

q

, ()

where the constant factorφ(_p) is still the best possible (cf.[], Theorem ).

In , by introducing an independent parameter λ> , Yang gave a best extension of () with the kernel 

(x+y)λ (cf.[, ]). In , by introducing another pair conjugate exponents (r,s), Yang [] gave an extension of () as follows: Ifλ> ,r> ,_r+_s= ,f(x), g(y)≥,  <_∞xp(–λ

r)–_fp₍_x₎_dx_<∞_{and  <}∞

 y q(–λ

s)–_gq₍_y₎_dy_<∞_{, then}

∞



∞



f(x)g(y) xλ₊_yλ dx dy

< π

λsin(π/r)

∞



xp(–λr)–_fp₍_x₎_dx



p ∞



yq(–λs)–_gq₍_y₎_dy



q

, ()

where the constant factor π

λsin(π/r)is the best possible. Forλ= ,r=q,s=p, () reduces to

(); Forλ= ,r=p,s=q, () reduces to the dual form of () as follows:

∞



∞



f(x)g(y) x+y dx dy

< π sin(π/p)

∞



xp–fp(x)dx 

p ∞



yq–gq(y)dy 

q

. ()

Forp=q= , both () and () reduce to ().

In , in [] one also gave an extension of () and () with the kernel 

(x+y)λ. Krnićet al. [–] provided some extensions and particular cases of (), () and () with parameters. In , Yang gave an extension of () and () as follows (cf.[, ]): Ifλ+λ=λ∈R=

(–∞,∞),kλ(x,y) is a non-negative homogeneous function of degree –λ, satisfying

kλ(ux,uy) =u–λkλ(x,y) (u,x,y> ),

and

k(λ) =

∞



kλ(u, )uλ–du∈R+,

then

∞



∞



kλ(x,y)f(x)g(y)dx dy

<k(λ)

∞



xp(–λ)–fp(x)dx 

p ∞



yq(–λ)–_gq₍_y₎_dy



q

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where the constant factork(λ) is the best possible. Forλ= ,λ=_q,λ=_p, () reduces

to (). Additionally, an extension of () was given as follows:

∞



∞



h(xy)f(x)g(y)dx dy

<φ(σ)

∞



xp(–σ)–fp(x)dx 

p ∞



yq(–σ)–gq(y)dy 

q

, ()

where the constant factorφ(σ) is the best possible (cf.[]). Forσ=_p, () reduces to (). Some equivalent inequalities of () and () are considered by []. In , Yang [] studied the equivalency between () and (). In , Hong [] studied an equivalent condition between () with a few parameters.

Remark (cf.[]) Ifh(xy) = , forxy> ,φ(σ) =_h(u)uσ–_du₌_φ

(σ)∈R+, then ()

re-duces to the following Hardy-type integral inequality with the non-homogeneous kernel:

∞



g(y)



y



h(xy)f(x)dx

dy

<φ(σ)

∞



p ∞



q

; ()

ifh(xy) = , forxy< ,φ(σ) =_∞h(u)uσ–_du₌_φ

(σ)∈R+, then () reduces to the

follow-ing another kind of Hardy-type integral inequality with the non-homogeneous kernel:

∞



g(y)

∞



y

h(xy)f(x)dx

dy

<φ(σ)

∞



p ∞



q

. ()

In this paper, by real analysis and the weight functions, we obtain a few equivalent con-ditions of two kinds of Hardy-type integral inequalities with the non-homogeneous kernel and parameters as(min₍_max{xy_{,_xy})_,α_}|₎lnλ+xyα|β. The constant factors related to the gamma function are proved to be the best possible. We also consider the operator expressions and some cases of homogeneous kernel.

2 Two lemmas

Forβ> –,σ+μ=λ∈R, we set

h(u) :=(min{u, })

α_|_ln_u_|β

(max{u, })λ+α (u> ).

Then, forσ> –α, settingv= –(σ+α)lnu, we ﬁnd

k(σ) :=





(min{u, })α_|_ln_u_|β

(max{u, })λ+α u σ–_du

= 



uσ+α–(–lnu)βdu=  (σ+α)β+

_∞



vβe–vdv= (β+ )

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Forμ> –α, we ﬁnd

k(σ) :=

∞



(max{u, })λ+α u σ–_du

= 



vμ+α–(–lnv)βdv= (β+ )

(μ+α)β+ =k(μ)∈R+, ()

where(η) :=_∞vη–_e–v_dv_(η_{> ) is the gamma function (}_cf._[]).

Lemma  If p> , _p+_q = ,σ∈R,β> –,σ> –α,there exists a constant M,such that,

for any non-negative measurable functions f(x)and g(y)in(,∞),the following inequality:

∞



g(y)



y



(min{xy, })α_|_ln_xy_|β

(max{xy, })λ+α f(x)dx

dy

≤M

∞



xp(–σ)–_fp₍_x₎_dx



p ∞



yq(–σ)–_gq₍_y₎_dy



q

()

holds true,then we haveσ=σ,and then M≥k(σ).

Proof Ifσ>σ, then, forn≥σ–σ (n∈N), we set the following functions:

fn(x) :=

⎧ ⎨ ⎩

xσ+pn–_, _{ <}_x≤_,

, x> ,

gn(y) :=

⎧ ⎨ ⎩

,  <y< ,

yσ–qn–_, _y≥_,

and ﬁnd

J:=

∞



xp(–σ)–f_np(x)dx 

p ∞



yq(–σ)–g_nq(y)dy 

q

=





xn–_dx



p ∞



y–n–_dy



q

=n.

Settingu=xy, we obtain

I:=

∞



gn(y)



y



(max{xy, })λ+α fn(x)dx

dy

=

∞



y



(min{xy, })α_(–_ln_xy₎β

(max{xy, })λ+α x σ+_pn–

dx

yσ–qn–_dy

=

∞



y(σ–σ)–_n–_dy





(min{u, })α_(–_ln_u₎β

(max{u, })λ+α u σ+_pn–

du.

Then by (), we have

∞



y(σ–σ)–n–_dy





(max{u, })λ+α u

σ+pn–_du

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Since (σ –σ) – _n ≥ , it follows that

_∞

 y

(σ–σ)–_n–_dy ₌_∞_{. By (), in view of}



y



(min{u,})α_(–_ln_u₎β

(max{u,})λ+α u

σ+_pn–

du> , we ﬁnd∞<∞, which is a contradiction. Ifσ<σ, then, forn≥σ–σ (n∈N), we set the following functions:

˜

fn(x) :=

⎧ ⎨ ⎩

,  <x< ,

xσ–pn–_, _x≥_, ˜

gn(y) :=

⎧ ⎨ ⎩

yσ+qn–_, _{ <}_y≤_,

, y> ,

and ﬁnd

˜

J:=

∞



xp(–σ)–_f˜p n(x)dx



p ∞



yq(–σ)–_g_˜q n(y)dy



q

=

∞



x–n–_dx



p 



yn–_dy



q

=n.

˜

I:=

_∞

 ˜

fn(x)



x



(max{xy, })λ+α g˜n(y)dy

dx = ∞   x 

(min{xy, })α_(–_ln_xy₎β

(max{xy, })λ+α y σ+_qn–

dy

xσ–pn–_dx

=

_∞



x(σ–σ)–n–_dx





(max{u, })λ+α u σ+_qn–

du.

Then by the Fubini theorem (cf.[]) and (), we have

∞



x(σ–σ)–n–_dx





(max{u, })λ+α u σ+_qn–

du

=˜I=

_∞

 ˜

gn(y)



y



(max{xy, })λ+α f˜n(x)dx

dy

≤M˜J=Mn<∞. ()

Since (σ–σ) –_n≥, it follows that

∞

 x(

σ–σ)–_n–_dx₌_∞_{. By (), in view of}





(max{u, })λ+α u σ+_qn–

du> 

we ﬁnd∞<∞, which is a contradiction. Hence, we conclude thatσ=σ.

Forσ=σ, we reduce () as follows:

M≥





(max{u, })λ+α u σ+_qn–

du. ()

Since

(max{u, })λ+α u σ+_qn–

∞

(6)

is non-negative and increasing in (, ], by Levi theorem (cf.[]), we ﬁnd

M≥ lim n→∞





(max{u, })λ+α u σ+_qn–

du

= 



lim

n→∞

(max{u, })λ+α u σ+_qn–

du=k(σ).

The lemma is proved.

Lemma  If p> ,  p +



q = ,σ∈R,β> –,μ=λ–σ> –α,there exists a constant M,

such that,for any non-negative measurable functions f(x)and g(y)in(,∞),the following inequality:

∞



g(y)

∞



y

dy

≤M

∞



p ∞



yq(–σ)–gq(y)dy 

q

()

holds true,then we haveσ=σ,and then M≥k(μ).

Proof If σ<σ, then, for n≥ σ–σ (n∈N), we set two functionsf˜n(x) and g˜n(y) as in

Lemma , and ﬁnd

˜

J=

∞



xp(–σ)–f˜_np(x)dx 

p ∞



yq(–σ)–g˜_nq(y)dy 

q

=n.

˜

I:=

∞

 ˜

gn(y)

∞



y

(max{xy, })λ+α f˜n(x)dx

dy

= 



∞



y

(min{xy, })α₍_ln_xy₎β

(max{xy, })λ+α x σ–_pn–

dx

yσ+qn–_dy

= 



y(σ–σ)+_n–_dy

∞



(min{u, })α₍_ln_u₎β

(max{u, })λ+α u σ–_pn–

du,

and then by (), we obtain





y(σ–σ)+n–_dy

∞



(max{u, })λ+α u σ–_pn–

du

=˜I≤MJ˜=Mn<∞. ()

Since (σ–σ) +_n≤, it follows that

∞

 y(

σ–σ)+n–_dy₌∞_{. By (), in view of}

_∞



σ–pn–_du_{> ,}

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Ifσ>σ, then, forn≥σ–σ (n∈N), we set two functionsfn(x) andgn(y) as in Lemma ,

and we ﬁnd

J=

∞



xp(–σ)–_fp n(x)dx



p ∞



yq(–σ)–_gq n(y)dy



q

=n.

I:=

∞



fn(x)

∞



x

(max{xy, })λ+α gn(y)dy

dx

= 



∞



x

(max{xy, })λ+α y σ–_qn–

dy

xσ+pn–_dx

= 



x(σ–σ)+n–_dx

∞



(max{u, })λ+α u σ–_qn–

du,

and then by Fubini theorem (cf.[]) and (), we have





x(σ–σ)+n–_dx

∞



σ–qn–_du

=I=

∞



gn(y)

∞



y

(max{xy, })λ+α fn(x)dx

dy

≤MJ=Mn. ()

Since (σ–σ) +_n≤, it follows that

 x

(σ–σ)+_n–_dx₌_∞_{. By (), in view of}

∞



(max{u, })λ+α u σ–_qn–

du> ,

we have∞<∞, which is a contradiction. Hence, we conclude thatσ=σ.

Forσ=σ, we reduce () as follows:

M≥

∞



σ–qn–_du_. _()

Since

(max{u, })λ+α u σ–_qn–

∞

n=

is non-negative and increasing in [,∞), still by the Levi theorem (cf.[]), we have

M≥ lim n→∞

_∞



σ–qn–_du

=

_∞



lim

n→∞

σ–qn–_du₌_k

(μ).

(8)

3 Main results and corollaries

Theorem  If p> , _p+_q = ,σ∈R,β> –,σ> –α,then the following conditions are

equivalent:

(i) There exists a constantM,such that,for anyf(x)≥,satisfying

 <

∞



xp(–σ)–fp(x)dx<∞,

we have the following Hardy-type integral inequality of the ﬁrst kind with the

non-homogeneous kernel:

J:=

∞



ypσ–



y



p

dy 

p

<M

∞



p

. ()

(ii) There exists a constantM,such that,for anyf(x),g(y)≥,satisfying

 <_∞xp(–σ)–_fp₍_x₎_dx_<_∞_,_and_{ <}∞  yq(–

σ)–_gq₍_y₎_dy_<_∞_,_{we have the}

following inequality:

I:=

∞



g(y)



y



dy

<M

∞



p ∞



q

. ()

(iii) σ=σ.

If condition(iii)holds true,then M≥k(σ)and the constant factor

M=k(σ) =

(β+ ) (σ+α)β+

in()and()is the best possible.

Proof (i)⇒(ii). By Hölder’s inequality (cf.[]), we have

I=

∞



yσ–p



y



yp–σ_g₍_y₎_dy

≤J

∞



q

. ()

Then by (), we have ().

(ii)⇒(iii). By Lemma , we haveσ=σ.

(iii)⇒(i). Settingu=xy, we obtain the following weight function:

ω(σ,y) :=yσ



y



(max{xy, })λ+α x σ–_dx

= 



σ–_du₌_k

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By Hölder’s inequality with weight and (), fory∈(,∞), we have



y



p

=



y



(max{xy, })λ+α

y(σ–)/p

x(σ–)/qf(x)

x(σ–)/q

y(σ–)/p

dx

p

≤



y



(max{xy, })λ+α

yσ–_fp₍_x₎

x(σ–)p/q dx

×



y



(max{xy, })λ+α

xσ–

y(σ–)q/pdx

p–

=

ω(σ,y)

yq(σ–)+

p– 

y



(max{xy, })λ+α

yσ–

x(σ–)p/qf p₍_x₎_dx

=k(σ)

p–

y–pσ+



y



(max{xy, })λ+α

yσ–

x(σ–)p/qf

p₍_x₎_dx_. _()

If () obtains the form of equality for ay∈(,∞), then (cf.[]), there exist constants AandB, such that they are not all zero, and

A y

σ–

x(σ–)p/qf

p₍_x_{) =}_B xσ–

y(σ–)q/p a.e. in R+.

We suppose thatA=  (otherwiseB=A= ). It follows that

xp(–σ)–_fp₍_x_{) =}_yq(σ–) B

Ax a.e. in R+,

which contradicts the fact that  <_∞xp(–σ)–_fp₍_x₎_dx_<_∞_{. Hence, () assumes the form}

of strict inequality. Hence, forσ=σ, by () and by the Fubini theorem (cf.[]), we obtain

J<k(σ)



q

∞





y



(max{xy, })λ+α

yσ–_fp₍_x₎

x(σ–)p/q dx

dy



p

=k(σ)



q

∞





x



(max{xy, })λ+α

yσ–_dy

x(σ–)(p–)

fp(x)dx



p

=k(σ)



q

∞



ω(σ,x)xp(–σ)–fp(x)dx



p

=k(σ)

∞



p

.

SettingM≥k(σ), () follows.

Therefore, conditions (i), (ii) and (iii) are equivalent.

When condition (iii) follows, if there exists a constant factorM≥k(σ), such that ()

is valid, then by Lemma , we haveM≥k(σ). Hence, the constant factorM=k(σ) in

() is the best possible. The constant factorM=k(σ) in () is still the best possible.

Otherwise, by () (forσ=σ), we can conclude that the constant factorM=k(σ) in

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Settingy= _Y,G(Y) =Yλ–_g₍

Y),μ=λ–σin Theorem , then replacingY byyand

G(Y) byg(y), we have

Corollary  If p> ,_p+_q= ,μ∈R,β> –,σ=λ–μ> –α,then the following conditions

are equivalent:

(i) There exists a constantM,such that,for anyf(x)≥,satisfying

 <

_∞



we have the following Hardy-type inequality of the ﬁrst kind with the homogeneous

kernel:

∞



ypμ–

y



(min{x,y})α_|_ln₍_x_/_y₎_|β

(max{x,y})λ+α f(x)dx

p

dy 

p

<M

∞



xp(–σ)–_fp₍_x₎_dx



p

. ()

(ii) There exists a constantM,such that,for anyf(x),g(y)≥,satisfying

 <_∞xp(–σ)–_fp₍_x₎_dx_<_∞_,_and_{ <}∞  yq(–

μ)–_gq₍_y₎_dy_<_∞_,_{we have the}

∞



g(y)

y



dy

<M

∞



p ∞



yq(–μ)–gq(y)dy 

q

. ()

(iii) μ=μ.

If condition(iii)holds true,then we have M≥k(σ),and the constant M=k(σ)in()

and()is the best possible.

Remark  On the other hand, settingy=_Y,G(Y) =Yλ–_g₍

Y),σ=λ–μin Corollary ,

then replacingYbyyandG(Y) byg(y), we have Theorem . Hence, Theorem  and Corol-lary  are equivalent.

Similarly, we obtain the following weight function:

ω(σ,y) :=yσ

∞



y

(max{xy, })λ+α x σ–_dx

=

∞



σ–_du₌_k

(μ) (y> ),

and then in view of Lemma  and in the same way, we have

Theorem  If p> ,_p+_q= ,σ∈R,β> –,μ=λ–σ> –α,then the following conditions

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(i) There exists a constantM,such that,for anyf(x)≥,satisfying

 <

∞



we have the following Hardy-type inequality of the second kind with the

non-homogeneous kernel:

∞



ypσ–

∞



y

p

dy 

p

<M

∞



xp(–σ)–_fp₍_x₎_dx



p

. ()

(ii) There exists a constantM,such that,for anyf(x),g(y)≥,satisfying

 <_∞xp(–σ)–_fp₍_x₎_dx_<_∞_,_and_{ <}∞  yq(–

σ)–_gq₍_y₎_dy_<_∞_,_{we have the}

∞



g(y)

∞



y

dy

<M

∞



p ∞



q

. ()

(iii) σ=σ.

If condition(iii)holds true,then we have M≥k(μ)and the constant factor

M=k(μ) =

(β+ ) (μ+α)β+

in()and()is the best possible.

Settingy= _Y,G(Y) =Yλ–_g₍

Y),μ=λ–σ in Theorem , then replacingY byyand

G(Y) byg(y), we have

Corollary  If p> ,_p+_q= ,μ∈R,β> –,μ=λ–σ> –α,then the following conditions

are equivalent:

(i) There exists a constantM,such that,for anyf(x)≥,satisfying

 <

∞



xp(–σ)–_fp₍_x₎_dx_<_∞_,

we have the following Hardy-type inequality of the second kind with the

homogeneous kernel:

∞



ypμ–

∞

y

p

dy 

p

<M

∞



p

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(ii) There exists a constantM,such that,for anyf(x),g(y)≥,satisfying

 <_∞xp(–σ)–_fp₍_x₎_dx_<_∞_,_and_{ <}∞  yq(–

μ)–_gq₍_y₎_dy_<_∞_,_{we have the}

∞



g(y)

∞

y

dy

<M

∞



xp(–σ)–_fp₍_x₎_dx



p ∞



yq(–μ)–_gq₍_y₎_dy



q

. ()

(iii) μ=μ.

If condition(iii)holds true,then we have M≥k(μ),and the constant M=k(μ)in()

and()is the best possible.

Remark  Theorem  and Corollary  are still equivalent.

4 Operator expressions

Forp> , _p +_q= ,σ,λ> ,μ=λ–σ, we set the following functions:ϕ(x) :=xp(–σ)–_,

ψ(y) :=yq(–σ)–_, _φ(_y_{) :=}_yq(–μ)–_{, wherefrom,}_ψ–p₍_y_{) =}_ypσ–_,_φ–p₍_y_{) =}_ypμ– ₍_x_,_y_∈_R +).

Deﬁne the following real normed linear spaces:

Lp,ϕ(R+) :=

f : f p,ϕ:=

∞



ϕ(x)f(x)pdx 

p

<∞

,

wherefrom

Lq,ψ(R+) :=

g: g q,ψ:=

∞



ψ(y)g(y)qdy 

q

<∞

,

Lq,φ(R+) :=

g: g q,φ:=

∞



φ(y)g(y)qdy 

q

<∞

,

Lp,ψ–p(R+) =

h: h p,ψ–p=

∞



ψ–p(y)h(y)pdy 

p

<∞

,

L_q_,φ–p(R₊) =

h: h _p_,φ–p=

∞



φ–p(y)h(y)pdy 

p

<∞

.

(a) In view of Theorem  (σ=σ), forf∈Lp,ϕ(R+), setting

h(y) :=



y



(max{xy, })λ+α f(x)dx (y∈R+),

by (), we have

h _p_,ψ–p=

∞



ψ–p(y)hp_(y)dy 

p

<M f p,ϕ<∞. ()

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exists a unique representationT_()f=h∈Lp,ψ–p(R+), satisfying for anyy∈R+,T()f(y) =

h(y).

In view of (), it follows that T_()f p,ψ–p= h p,ψ–p≤M f p,ϕ, and then the

opera-torT_()is bounded satisfying

T_()= sup

f(=θ)∈Lp,ϕ(R+)

T_()f p,ψ–p

f p,ϕ

≤M.

If we deﬁne the formal inner product ofT_()f andgas follows:

T_()f,g:=

∞





y



g(y)dy,

then we can rewrite Theorem  (forσ=σ) as follows.

Theorem  If p> ,_p+_q= ,β> –,σ> –α,then the following conditions are equivalent:

(i) There exists a constantM,such that,for anyf(x)≥,f ∈Lp,ϕ(R+), f p,ϕ> ,we

have the following inequality:

T_()f_p_,_ψ–p<M f p,ϕ. ()

(ii) There exists a constantM,such that,for anyf(x),g(y)≥,f∈Lp,ϕ(R+),

g∈Lq,ψ(R+), f p,ϕ, g q,ψ> ,we have the following inequality:

T_()f,g<M f p,ϕ g q,ψ. ()

We still have T_() =k(σ)≤M.

(b) In view of Corollary  (μ=μ), forf ∈Lp,ϕ(R+), setting

h(y) :=

y



(max{x,y})λ+α f(x)dx (y∈R+),

by (), we have

h p,φ–p=

∞



φ–p(y)hp(y)dy



p

<M f p,ϕ<∞. ()

Definition  Define a Hardy-type integral operator of the first kind with the homoge-neous kernelT_():Lp,ϕ(R+)→Lp,φ–p(R₊) as follows: For anyf ∈L_p_,ϕ(R+), there exists a

unique representationT()f =h∈Lp,φ–p(R₊), satisfying for anyy∈R₊,T_()f(y) =h_(y).

In view of (), it follows that T_()f _p_,φ–p= h _p_,_φ–p≤M_ f _p_,ϕ, and then the operator

T_()is bounded satisfying

T_()= sup

f(=θ)∈Lp,ϕ(R+)

T_()f _p_,φ–p

f p,ϕ

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If we deﬁne the formal inner product ofT_()f andgas follows:

T_()f,g:=

∞



y



g(y)dy,

then we can rewrite Corollary  (forμ=μ) as follows.

Corollary  If p> ,_p +_q = ,β> –,σ =λ–μ> –α,then the following conditions are equivalent:

(i) There exists a constantM,such that,for anyf(x)≥,f ∈Lp,ϕ(R+), f p,ϕ> ,we

T_()f_p_,_φ–p<M f p,ϕ. ()

(ii) There exists a constantM,such that,for anyf(x),g(y)≥,f ∈Lp,ϕ(R+),g∈

Lq,φ(R+), f p,ϕ, g q,φ> ,we have the following inequality:

T_()f,g<M f p,ϕ g q,φ. ()

We still have T_() =k(σ)≤M.

Remark  Theorem  and Corollary  are equivalent.

(c) In view of Theorem  (σ=σ), forf ∈Lp,ϕ(R+), setting

H(y) :=

_∞



y

(max{xy, })λ+α f(x)dx (y∈R+),

by (), we have

H p,ψ–p=

∞



ψ–p(y)H_p(y)dy 

p

<M f p,ϕ<∞. ()

Deﬁnition  Deﬁne a Hardy-type integral operator of the second kind with the non-homogeneous kernelT_():Lp,ϕ(R+)→Lp,ψ–p(R+) as follows: For anyf ∈Lp,ϕ(R+), there

exists a unique representationT_()f =H∈Lp,ψ–p(R+), satisfying for anyy∈R+,T()f(y) =

H(y).

In view of (), it follows that T_()f _p_,ψ–p= H _p_,_ψ–p≤M f _p_,ϕ, and then the

oper-atorT_()is bounded satisfying

T_()= sup

f(=θ)∈Lp,ϕ(R+)

T_()f p,ψ–p

f p,ϕ ≤

M.

If we deﬁne the formal inner product ofT_()f andgas follows:

T_()f,g:=

∞



∞



y

g(y)dy,

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Theorem  If p> , _p +_q = ,β> –,μ=λ–σ> –α,then the following conditions are equivalent:

(i) There exists a constantM,such that,for anyf(x)≥,f∈Lp,ϕ(R+), f p,ϕ> ,we

T_()f_p_,_ψ–p<M f p,ϕ. ()

(ii) There exists a constantM,such that,for anyf(x),g(y)≥,f∈Lp,ϕ(R+),

g∈Lq,ψ(R+), f p,ϕ, g q,ψ> ,we have the following inequality:

T_()f,g<M f p,ϕ g q,ψ. ()

We still have T_() =k(μ)≤M.

(d) In view of Corollary  (μ=μ), forf ∈Lp,ϕ(R+), setting

H(y) :=

∞

y

(max{x,y})λ+α f(x)dx (y∈R+),

by (), we have

H p,φ–p=

∞



φ–p(y)Hp(y)dy



p

<M f p,ϕ<∞. ()

Deﬁnition  Deﬁne a Hardy-type integral operator of the second kind with the homo-geneous kernelT_():Lp,ϕ(R+)→Lp,φ–p(R₊) as follows: For anyf ∈L_p_,ϕ(R+), there exists a

unique representationT_()f =H∈Lp,φ–p(R+), satisfying for anyy∈R+,T()f(y) =H(y).

In view of (), it follows that T_()f _p_,φ–p= H _p_,_φ–p≤M f _p_,ϕ, and then the

opera-torT()is bounded satisfying

T_()= sup

f(=θ)∈Lp,ϕ(R+)

T_()f p,φ–p

f p,ϕ ≤

M.

If we deﬁne the formal inner product ofT_()f andgas follows:

T_()f,g:=

∞



∞

y

g(y)dy,

then we can rewrite Corollary  (forμ=μ) as follows.

Corollary  If p> , _p+_q = ,β> –,μ=λ–σ > –α,then the following conditions are equivalent:

(i) There exists a constantM,such that,for anyf(x)≥,f ∈Lp,ϕ(R+), f p,ϕ> ,we

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(ii) There exists a constantM,such that,for anyf(x),g(y)≥,f ∈Lp,ϕ(R+),

g∈Lq,φ(R+), f p,ϕ, g q,φ> ,we have the following inequality:

T_()f,g<M f p,ϕ g q,φ. ()

We still have T_() =k(μ)≤M.

Remark  Theorem  and Corollary  are equivalent.

5 Conclusions

In this paper, by means of real analysis and weight functions a few equivalent conditions of two kinds of Hardy-type integral inequalities with the non-homogeneous kernel and parameters are obtained by Theorem , . The constant factors related to the gamma function are proved to be the best possible. We also consider the operator expressions in Theorem , . The dependent cases of homogeneous kernel are assumed by Corollary -. The method of weight functions is very important, it is the key to help us proving the main inequalities with the best possible constant factor. The lemmas provide an extensive account of this type of inequalities.

Acknowledgements

This work is supported by the National Natural Science Foundation (No. 11401113), and Appropriative Researching Fund for Professors and Doctors, Guangdong University of Education (No. 2015ARF25). We are grateful for their help.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

BY carried out the mathematical studies, participated in the sequence alignment and drafted the manuscript. JL participated in the design of the study and performed the numerical analysis. All authors read and approved the ﬁnal manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 29 March 2017 Accepted: 15 May 2017 References

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