On a more accurate half discrete Hardy Hilbert type inequality related to the kernel of exponential function

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R E S E A R C H

Open Access

On a more accurate half-discrete

Hardy-Hilbert-type inequality related to the

kernel of exponential function

Jianquan Liao

*

_{and Bicheng Yang}

*_{Correspondence: [email protected]}

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

Abstract

By applying the weight functions, the technique of real analysis and

Hermite-Hadamard’s inequality, a half-discrete Hardy-Hilbert-type inequality related to the kernel of exponential function with the best possible constant factor expressed by the gamma function is given. The more accurate equivalent forms, the operator expressions with the norm, the reverses, and some particular cases are considered.

MSC: 26D15

Keywords: Hardy-Hilbert-type inequality; weight function; equivalent form; reverse; operator

1 Introduction

Suppose thatp> ,_p+_q= ,f(x),g(y)≥,f ∈Lp_(R+),_g_∈_Lq_(R+),_f p= (

∞

 fp(x)dx)



p_>

,gq> , and we have the following Hardy-Hilbert integral inequality (cf.[]):

∞



∞



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

π

sin(π/p)fpgq, ()

where the constant factor _sin₍π_π_/_p₎ is the best possible. Ifam,bn≥,a={am}∞m=∈lp,b=

{bn}∞n=∈lq,ap= (

∞

m=a p m)



p_{> ,}_b

q> , then we have the following discrete analogy

of () with the same best possible constant_sin₍π_π_/_p₎(cf.[]):

∞

m=

∞

n= ambn m+n<

π

sin(π/p)apbq. ()

Inequalities () and () are important in analysis and its applications (cf.[–]). Ifμi,υj>  (i,j∈N={, , . . .}),

Um:= m

i=

μi, Vn:= n

j=

νj(m,n∈N), ()

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then we have the following Hardy-Hilbert-type inequality (cf.[], Theorem , replacing

μ/mqamandυn/pbnbyamandbn):

∞

m=

∞

n= ambn Um+Vn

< π

sin(π

p)

_∞

m= apm

μpm–



p∞

n= bqn

νnq–



q

. ()

Forμi=υj=  (i,j∈N), inequality () reduces to ().

Note The authors of [] did not prove that () is valid with the best possible constant factor.

In , by introducing an independent parameterλ∈(, ], Yang [] gave an extension of () with the kernel ₍_x₊_y₎λ forp=q= . Following [], Yang [] gave some extensions of () and () as follows:

Ifλ,λ∈R,λ+λ=λ,kλ(x,y) is a non-negative homogeneous function of degree –λ,

withk(λ) =_∞kλ(t, )tλ–dt∈R+,φ(x) =xp(–λ)–,ψ(x) =xq(–λ)–,f(x),g(y)≥,

f ∈Lp,φ(R+) =

f;fp,φ:=

∞



φ(x)f(x)pdx



p

<∞

,

g∈Lq,ψ(R+),fp,φ,gq,ψ> , then we have

∞



∞



kλ(x,y)f(x)g(y)dx dy<k(λ)fp,φgq,ψ, ()

where the constant factork(λ) is the best possible. Moreover, if kλ(x,y) keeps a ﬁnite

value andkλ(x,y)xλ–(kλ(x,y)yλ–) is decreasing with respect tox>  (y> ), then, for

am,bn≥,

a∈lp,φ=

a;ap,φ:= _∞

n=

φ(n)|an|p



p

<∞

,

b={bn}∞n=∈lq,ψ,ap,φ,bq,ψ> , we have

∞

m=

∞

n=

kλ(m,n)ambn<k(λ)ap,φbq,ψ, ()

where the constant factork(λ) is still the best possible.

In , by adding some conditions, Yang [] gave an extension of () as follows:

∞

m=

∞

n=

ambn (Um+Vn)λ

<B(λ,λ)

_∞

m=

Up(–λ)– m apm

μpm–



p∞

n=

Vq(–λ)– n bqn

νnq–



q

, ()

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Some other results including multidimensional Hilbert-type inequalities are provided by [–].

About the topic of half-discrete Hilbert-type inequalities with the non-homogeneous kernels, Hardyet al.provided a few results in Theorem  of []. But they did not prove that the constant factors are the best possible. However, Yang [] gave a result with the kernel _(+_nx ₎λ by introducing a variable and proved that the constant factor is the best possible. In  Yang [] gave the following half-discrete Hardy-Hilbert inequality with the best possible constant factorB(λ,λ):

∞

 f(x)

_∞

n= an (x+n)λ

dx<B(λ,λ)fp,φaq,ψ, ()

whereλ> ,  <λ≤,λ+λ=λ. Zhonget al.([, , ]) investigated several half-discrete Hilbert-type inequalities with particular kernels. Applying weight functions, a half-discrete Hilbert-type inequality with a general homogeneous kernel of degree –λ∈R

and a best constant factork(λ) are obtained as follows:

∞

 f(x)

∞

n=

kλ(x,n)andx<k(λ)fp,φaq,ψ, ()

which is an extension of () (cf.[]). At the same time, a half-discrete Hilbert-type in-equality with a general non-homogeneous kernel and a best constant factor are given by Yang []. In -, Yanget al.published three books [, ] and [] concerned with building the theory of half-discrete Hilbert-type inequalities.

In this paper, by applying weight functions, the technique of real analysis, and Hermite-Hadamard’s inequality, a half-discrete Hardy-Hilbert-type inequality related to the ker-nel of exponential function with a best possible constant factor expressed by the gamma function is given, which is similar to () and an extension of () in the following particular kernel:

k(x,n) = 

eα(n_x)γ (α> ,  <γ≤).

Furthermore, the more accurate equivalent forms, the operator expressions with the norm, the reverses, and some particular cases are considered.

2 An example and some lemmas

In the following, we agree thatνn> , ≤τn≤νn (n∈N),Vn=

n

i=νi,μ(t) is a positive continuous function in R+= (,∞),

U() := ; U(x) :=

x



μ(t)dt<∞ x∈(,∞),

ν(t) :=νn,t∈(n–_,n+_] (n∈N), and

V 



:= ; V(y) :=

y

 

ν(t)dt y∈  ,∞

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p= , , _p +_q = ,δ∈ {–, },f(x),an≥ (x∈R+,n∈N),fp,δ= (

_∞

 δ(x)fp(x)dx)



p_,

aq,= ( ∞

n=(n)b q n)



q_{, where}

δ(x) :=U

p(–δσ)–_(x)

μp–_(x) (x∈R+), (n) :=

(Vn–τn)q(–σ)–

νnq–

(n∈N).

Example  Forα> ,  <γ,σ≤, we seth(t) = 

eαtγ (t∈R+).

(i) Settingu=αtγ_{, we ﬁnd}

k(σ) :=

_∞

 tσ– eαtγ dt=



γ ασ/γ _∞



e–uuσγ–_du₌(σ/γ)

γ ασ/γ ∈R+, ()

where

(y) :=

∞



e–vvy–dv (y> )

is called the gamma function (cf.[]).

(ii) We obtain, fort> ,α> ,  <γ ≤,h(t) = 

eαtγ > ,h(t) = –αγtγ– 

eαtγ <  and

h(t) = –αγ(γ – )tγ–  eαtγ +

αγtγ–  eαtγ > .

(iii) Ifg(u) > ,g(u) < ,g(u) > , then we ﬁnd that, fory∈(n–_,n+_),g(V(y)) > , d

dyg(V(y)) =g(V(y))νn< , and d

dyg

V(y)=gV(y)ν_n>  (n∈N);

For g(u) > , g_(u) < , g_(u) > , g(u) > , g_(u)≤, g_(u)≥ (u> ), we obtain g(u)g(u) > , (g(u)g(u))=g(u)g(u) +g(u)g(u) < , and

g(u)g(u)=g_(u)g(u) + g_(u)g_(u) +g(u)g_(u) >  (u> ). (iv) Forα> ,  <γ,σ≤,c> , we haveh(cV(y))Vσ–_{(y) > ,} d

dy(h(cV(y))V

σ–_{(y)) < ,}

and

d dy

hcV(y)Vσ–(y)>  y∈ n–  ,n+

 

,n∈N

.

Then by Hermite-Hadamard’s inequality (cf.[]), we have

hcV(n)Vσ–_{(n) <}

n+_

n–_

hcV(y)Vσ–_(y)_dy _(n_∈_N_). _()

Lemma  If g(t) (> )is a strictly decreasing continuous function in(

,∞),which is strictly convex satisfying∞

 g(t)dt∈R+,then we have

∞



g(t)dt<

∞

n= g(n) <

∞

 

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Proof By Hermite-Hadamard’s inequality and the decreasing property, we have

n+

n

g(t)dt<

n+

n

g(n)dt=g(n) <

n+_

n–_

g(t)dt (n∈N), ()

and, forn∈N, it follows that

n+



g(t)dt< n

n= g(n) <

n

n=

n+

n–

g(t)dt=

n+

 

g(t)dt,

∞

n+

g(t)dt≤

∞

n=n+

g(n)≤

∞

n+

g(t)dt<∞.

Hence, choosing plus for the above two inequalities, we have ().

Lemma  Ifα> ,  <γ,σ≤,deﬁne the following weight coeﬃcients:

ωδ(σ,x) :=

∞

n=

 eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

Uδσ_(x)_ν

n (Vn–τn)–σ

, x∈R+, ()

δ(σ,n) :=

∞



 eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

(Vn–τn)σμ(x)

U–δσ_(x) dx, n∈N. ()

Then we have the following inequalities:

ωδ(σ,x) <k(σ) (x∈R+), ()

δ(σ,n)≤k(σ) (n∈N), ()

where k(σ)is indicated by().

Proof SinceVn–τn≥

n+_

 

ν(t)dt–νn

 =

n 

 ν(t)dt=V(n), and, fort∈(n–

 ,n+

 ),νn= V(t), by () (forc=Uδ_{(x)) and (), we have}

 eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

Uδσ_(x)

(Vn–τn)–σ

≤ 

eαUδγ₍_x₎_Vγ₍_n₎ Uδσ_(x)

V–σ_(n)

<

n+_

n–_  eαUδγ₍_x₎_Vγ₍_t₎

Uδσ_(x)

V–σ_(t)dt (n∈N),

ωδ(σ,x) <

∞

n=

νn

n+_

n–_  eαUδγ₍_x₎_Vγ₍_t₎

Uδσ_(x)

V–σ_(t)dt

=

∞

n=

n+

n–

 eαUδγ₍_x₎_Vγ₍_t₎

Uδσ_(x)V_(t)

V–σ_(t) dt

=

∞

 

Uδσ_(x)V_(t)

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Settingu=Uδ_(x)V_{(t), by (), we ﬁnd}

ωδ(σ,x) <

Uδ₍_x₎_V₍_∞₎

Uδ₍_x₎_V₍ )

 eαuγ

Uδσ_(x)U–δ_(x)

(uU–δ_(x))–σ du

≤ ∞

  eαuγu

σ–_du₌_k(_σ_).

Hence, () follows.

Settingu= (Vn–τn)Uδ(x) in (), we ﬁnddu=δ(Vn–τn)Uδ–(x)μ(x)dxand

δ(σ,n) =



δ

(Vn–τn)Uδ(∞)

(Vn–τn)Uδ()

 eαuγu

σ–_du.

Ifδ= , then

(σ,n) =

(Vn–τn)U(∞)



 eαuγu

σ–_du_≤

∞

  eαuγu

σ–_du;

ifδ= –, then

–(σ,n) = –

(Vn–τn)U–(∞)

∞

 eαuγu

σ–_du_≤

_∞

  eαuγu

σ–_du.

Hence, by (), we have ().

Remark  (i) We do not need the condition ofσ≤ in obtaining (). (ii) IfU(∞) =∞,

then we have

δ(σ,n) =k(σ) (n∈N). ()

For example, we setμ(t) =_(+_t₎a(t> ; ≤a≤), then forx≥, we ﬁnd

U(x) =

x

 dt ( +t)a =

(+x)–a–

–a , ≤a< ,

ln( +x), a=  <∞,

U() = , andU(∞) =_∞_(+dt_t₎a=∞.

Lemma  Ifα> ,  <γ,σ≤,there exists n∈N,such that{νn}∞n=nis decreasing and

V(∞) =∞,then: (i)for x∈R+,we have

k(σ) –θδ(σ,x)<ωδ(σ,x), ()

where

θδ(σ,x) :=

 k(σ)

Uδ₍_x₎_V₍_n

+)



uσ– eαuγ du=O

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(ii)for any b> ,we have

∞

n=

νn (Vn–τn)+b

= b



ν_b +bO()

. ()

Proof SinceVn–τn≤Vn≤Vn+–νn_+ =V(n+ ), andνn≥V(t) (t∈(n,n+ );n≥n), by (), we ﬁnd

ωδ(σ,x)≥

∞

n=n

 eαUδγ₍_x₎_Vγ₍_n_+)

Uδσ_(x)_ν

n+ V–σ_(n_{+ )}

=

∞

n=n+

 eαUδγ₍_x₎_Vγ₍_n₎

Uδσ_(x)_ν

n V–σ_(n)

>

∞

n=n+

n+

n

Uδσ_(x)V_(t)

V–σ_(t) dt

=

∞

n+

Uδσ_(x)V_(t)

V–σ_(t) dt.

Settingu=Uδ_(x)V_{(t), in view of}_V₍_∞_{) =}_∞_{, by (), we ﬁnd}

ωδ(σ,x) >

∞

Uδ₍_x₎_V₍_n

+)

uσ–

eαuγ du=k(σ) –

Uδ(x)V(n+)



uσ–

eαuγ du=k(σ)

 –θδ(σ,x).

We ﬁnd

 <θδ(σ,x)≤  k(σ)

Uδ₍_x₎_V₍_n

+)



uσ–du=(U

δ_(x)V_(n_{+ ))}σ

σk(σ) (x∈R+),

and then () follows. Forb> , we ﬁnd

∞

n=

νn (Vn–τn)+b ≤

∞

n=

νn V+b_(n)=

ν V+b_()+

∞

n=

νn V+b_(n)

< +b

ν_b +

∞

n=

n+_

n–_

V(x)dx V+b_(x) =

+b

ν_b + _∞

 

V(x)dx V+b_(x)

= +b

ν_b + ν–b



b =

 b



ν_b +b

+b

ν_b , ∞ n= νn (Vn–τn)+b ≥

∞

n=n

νn (Vn–τn)+b ≥

∞

n=n

νn+ V+b_(n_{+ )}

=

∞

n=n+

νn V+b_(n)>

∞

n=n+

n+

n

V(x)dx V+b_(x) =

∞

n+

V(x)dx V+b_(x)

= 

bVb_(n_{+ )}=  b



νb 

+bV –b_(n

+ ) –ν–b b

.

SinceV–b(n+)–ν–b

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Note For example,νn=_na(n∈N; ≤a≤) satisﬁes the conditions of{νn}∞n=in Lemma  (forn= ).

3 Main results and operator expressions

Theorem  Ifα> ,  <γ,σ ≤,then for p> ,  <fp,δ,aq,<∞,we have the

following equivalent inequalities:

I:= ∞ n= ∞ 

anf(x)

eαUδγ₍_x₎₍_V_n_–_τ_n₎γ dx<

(σ/γ)

γ ασ/γ fp,δaq,, ()

J:=

∞

n=

νn (Vn–τn)–pσ

∞



f(x)dx eαUδγ₍_x₎₍_V

n–τn)γ p

<(σ/γ)

γ ασ/γ fp,δ, ()

J:=

∞



μ(x) U–qδσ_(x)

_∞

n=

an eαUδγ₍_x₎₍_V

n–τn)γ q

dx



q

<(σ/γ)

γ ασ/γ aq,. ()

Proof By Hölder’s inequality with weight (cf.[]), we have

∞



f(x) eαUδγ₍_x₎₍_V_n_–_τ_n₎γ dx

p

=

∞



 eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

U–qδσ_(x)f_(x)

(Vn–τn) –σ

p _μq_(x)

(Vn–τn) –σ

p _μq_(x)

U–qδσ_(x)

dx

p

≤ ∞



(Vn–τn)γ eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

U

p(–δσ)

q _(x)fp_(x)

(Vn–τn)–σμ

p q_(x) dx × ∞ 

(Vn–τn)γ eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

(Vn–τn)(–σ)(p–)μ(x)

U–δσ_(x) dx

p–

= (δ(σ,n)) p–

(Vn–τn)pσ–νn

∞



 eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

U(–δσ)(p–)_(x)_ν nfp(x) (Vn–τn)–σμp–(x)

dx. ()

In view of () and the Lebesgue term by term integration theorem (cf.[]), we ﬁnd

J≤

k(σ)q _∞ n= ∞   eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

U(–δσ)(p–)_(x)_ν n (Vn–τn)–σμp–(x)

fp(x)dx



p

=k(σ)q

∞  ∞ n=  eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

fp(x)dx



p

=k(σ)



q

∞



ωδ(σ,x)U

p(–δσ)–_(x)

μp–_(x) f p_(x)dx



p

. ()

Then by (), we have ().

By Hölder’s inequality (cf.[]), we have

I = ∞ n= ν  p n

(Vn–τn) 

p–σ ∞



(Vn–τn) 

p–σ_a

n

νn/p

(9)

Then by (), we have (). On the other hand, assuming that () is valid, we set

an:=

∞





eαUδγ₍_x₎₍_V_n_–_τ_n₎γf(x)dx

p–

, n∈N.

Then we ﬁndJ_p=aq_q_,. IfJ= , then () is trivially valid; ifJ=∞, then () remains impossible. Suppose that  <J<∞. By (), we have

aq_q_,=J_p=I<k(σ)fp,δaq,,

aq_q–_,=J<k(σ)fp,δ,

and then () follows, which is equivalent to ().

Still by Hölder’s inequality with weight (cf.[]), we have

_∞

n=

an eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

q

=

_∞

n=

 eαUδγ₍_x₎₍_V_n_–_τ_n₎γ ·

U–qδσ_(x)_ν



p

n

(Vn–τn) –σ

p ·

(Vn–τn) –σ

p _a

n

U–qδσ_(x)_ν/p

n

q

≤ _∞

n=

 eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

U(–δσ)(p–)_(x)_ν n (Vn–τn)–σ

q–

×

∞

n=

 eαUδγ₍_x₎₍_V

n–τn)γ

(Vn–τn)

q(–σ)

p

U–δσ_(x)_νq–

n aq_n

= (ωδ(σ,x)) q–

Uqδσ–_(x)_μ_(x)

∞

n=

 eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

(Vn–τn)(–σ)(q–)μ(x)

U–δσ_(x)_νq–

n

aq_n. ()

Then by () and the Lebesgue term by term integration theorem (cf.[]), it follows that

J<

k(σ)p

∞



∞

n=

 eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

(Vn–τn)(–σ)(q–)μ(x) U–δσ_(x)_νq–

n

aq_ndx



q

=k(σ)p _∞

n=

∞



 eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

(Vn–τn)(–σ)(q–)μ(x) U–δσ_(x)_νq–

n

aq_ndx



q

=k(σ)p _∞

n=

δ(σ,n)

νnq– aq_n



q

. ()

Then by (), we have ().

By Hölder’s inequality (cf.[]), we have

I =

∞



Uq–δσ_(x)

μq_(x)

f(x)

μq_(x)

Uq–δσ_(x)

∞

n=



eαUδγ₍_x₎₍_V_n_–_τ_n₎γan

dx

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Then by (), we have (). On the other hand, assuming that () is valid, we set

f(x) := μ(x) U–qδσ_(x)

_∞

n=



q–

, x∈R+.

Then we ﬁndJ_q=fp_p_,δ. IfJ= , then () is trivially valid; ifJ=∞, then () keeps impossible. Suppose that  <J<∞. By (), we have

fp_p_,

δ=J

q

 =I<k(σ)fp,δaq,, f

p–

p,δ=J<k(σ)aq,,

and then () follows, which is equivalent to ().

Therefore, (), (), and () are equivalent.

Theorem  As regards the assumptions of Theorem ,if there exists n∈N, such that

{νn}∞n=n is decreasing and U(∞) =V(∞) =∞,then the constant factor k(σ) =

(σ/γ)

γ ασ/γ in (), (),and()is the best possible.

Proof Forε∈(,qσ), we setσ=σ–ε_q(∈(, )), andf=f(x),x∈R+,a={an}∞n=,

f(x) =

Uδ(σ+ε)–_(x)_μ_(x), _{ <}_xδ_≤_,

, xδ_{> ,} ()

an= (Vn–τn)σ–νn= (Vn–τn)σ–

ε

q–_ν

n, n∈N. ()

Then forδ=±, sinceU(∞) =∞, we ﬁnd

{x>;<xδ_≤__}

μ(x) U–δε_(x)dx=



εU δε

(). ()

By (), (), and (), we obtain

fp,δaq,=

{x>;<xδ_≤__}

μ(x)dx U–δε_(x)



p∞

n=

νn (Vn–τn)+ε



q

=

εU

δε

p_()  νε



+εO()



q

, ()

I:=

∞



∞

n=

n–τn)γan f(x)dx

=

{x>;<xδ_≤__}

∞

n=

 eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

(Vn–τn)σ–νnμ(x) U–δ(σ+ε)_(x) dx

=

{x>;<xδ_≤__}

ωδ(σ,x) μ(x) U–δε_(x)dx

≥k(σ)

{x>;<xδ_≤__}

 –θδ(σ,x) μ(x)

U–δε_(x)dx

=k(σ)

{x>;<xδ_≤__}

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=k(σ)

{x>;<xδ_≤__}

μ(x)dx U–δε_(x)–

{x>;<xδ_≤__}

O μ(x)

U–δ(σ+εp)_(x)

dx

=

εk σ– ε

q

Uδε() –εO().

If there exists a positive constant K ≤k(σ), such that () is valid when replacing k(σ) to K, then in particular, by Lebesgue term by term integration theorem, we have

εI<εKfp,δaq,, namely,

k σ–ε q

Uδε() –εO()<K·Uδεp_() 

ν_ε+εO() 

q

.

It follows thatk(σ)≤K(ε→+_{). Hence,}_K₌_k(_σ_{) is the best possible constant factor of} ().

The constant factork(σ) in () (()) is still the best possible. Otherwise, we would reach a contradiction by () (()) that the constant factor in () is not the best

possi-ble.

Forp> , we ﬁnd–p_{(n) =} νn

(Vn–τn)–pσ (n∈N),

–q

δ (x) = μ(x)

U–qδσ₍_x₎(x∈R+), and we deﬁne the following real normed spaces:

Lp,δ(R+) =

f;f=f(x),x∈R+,fp,δ<∞

,

lq,=

a;a={an}∞n=,aq,<∞

,

L_q_,–q

δ (R+) =

h;h=h(x),x∈R+,h_q_,–q

δ <∞

,

lp,–p=

c;c={cn}∞n=,cp,–p<∞

.

Assuming thatf ∈Lp,δ(R+), setting

c={cn}∞n=, cn:=

∞





eαUδγ₍_x₎₍_V_n_–_τ_n₎γf(x)dx, n∈N,

we can rewrite () ascp,–p<k(σ)fp,δ<∞, namely,c∈lp,–p.

Deﬁnition  Deﬁne a half-discrete Hardy-Hilbert-type operatorT:Lp,δ(R+)→lp,–p

as follows: For any f ∈Lp,δ(R+), there exists a unique representationTf =c∈lp,–p.

Deﬁne the formal inner product ofTf anda={an}∞n=∈lq,as follows:

(Tf,a) :=

∞

n=

∞





an. ()

Then we can rewrite () and () as follows:

(Tf,a) <k(σ)fp,δaq,, ()

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Deﬁne the norm of operatorTas follows:

T:= sup f(=θ)∈Lp,_δ(R+)

Tfp,–p fp,δ

.

Then by (), it follows thatT ≤k(σ). Since, by Theorem , the constant factor in () is the best possible, we have

T=k(σ) =(σ/γ)

γ ασ/γ . ()

Assuming thata={an}n∞=∈lq,, setting

h(x) :=

∞

n=



eαUδγ₍_x₎₍_V_n_–_τ_n₎γan, x∈R+,

we can rewrite () ash_q_,–q

δ <k(

σ)aq,<∞, namely,h∈L_q_,–_δq(R+).

Deﬁnition  Deﬁne a half-discrete Hardy-Hilbert-type operatorT:lq,→L_q_,–_δq(R+)

as follows: For any a={an}∞n= ∈lq,, there exists a unique representation Ta=h∈

L_q_,–q

δ (R+). Deﬁne the formal inner product ofTaandf ∈Lp,δ(R+) as follows:

(Ta,f) :=

∞



_∞

n=



f(x)dx. ()

Then we can rewrite () and () as follows:

(Ta,f) <k(σ)fp,δaq,, ()

Ta_q_,–q

δ

<k(σ)a_q_,. ()

Deﬁne the norm of operatorTas follows:

T:= sup

a(=θ)∈lq,

Ta_q_,–q

δ

aq,

.

Then by (), we ﬁndT ≤k(σ). Since, by Theorem , the constant factor in () is the best possible, we have

T=k(σ) =(σ/γ)

γ ασ/γ =T. ()

4 Some equivalent reverses In the following, we also set

δ(x) :=

 –θδ(σ,x)

Up(–δσ)–_(x)

μp–_(x) (x∈R+).

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Theorem  As regards the assumptions of Theorem,for p< ,  <fp,δ,aq,<∞,

we have the following equivalent inequalities with the best possible constant factor k(σ) =

(σ/γ)

γ ασ/γ :

I=

∞

n=

∞



anf(x)

eαUδγ₍_x₎₍_V_n_–_τ_n₎γ dx>

(σ/γ)

γ ασ/γ fp,δaq,, ()

J=

∞

n=

∞



p

>(σ/γ)

γ ασ/γ fp,δ, ()

J=

∞



μ(x) U–qδσ_(x)

_∞

n=

q dx



q

>(σ/γ)

γ ασ/γ aq,. ()

Proof By the reverse Hölder inequality with weight (cf.[]), sincep< , in the similar way to obtaining () and (), we have

∞





p

≤ (δ(σ,n))p–

_∞



 eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

fp(x)dx.

Then by () and the Lebesgue term by term integration theorem, it follows that

J≥

k(σ)q _∞

n=

∞



n–τn)γ

fp_(x)_dx



p

=k(σ)



q

∞



ωδ(σ,x)

Up(–δσ)–_(x)



p

.

Then by (), we have ().

By the reverse Hölder inequality (cf.[]), we have

I =

∞

n=

ν



p

n

(Vn–τn) 

p–σ ∞



(Vn–τn) 

p–σ_a

n

νn/p

≥Jaq,. ()

Then by (), we have (). On the other hand, assuming that () is valid, we setanas in Theorem . Then we ﬁndJ_p=aq_q_,. IfJ=∞, then () is trivially valid; ifJ= , then () keeps impossible. Suppose that  <J<∞. By (), it follows that

aq_q_,=J_p=I>k(σ)fp,δaq,, a

q–

q,=J>k(σ)fp,δ,

(14)

Still by the reverse Hölder’s inequality with weight (cf.[]), since  <q< , in the similar way to obtaining () and (), we have

_∞

n=

n–τn)γan q

≥ (ωδ(σ,x))q–

Uqδσ–_(x)_μ_(x)

∞

n=

 eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

(Vn–τn)(–σ)(q–)μ(x) U–δσ_(x)_νq–

n

aq_n.

Then by () and the Lebesgue term by term integration theorem, it follows that

J>k(σ)



p

_∞



∞

n=

 eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

(Vn–τn)(–σ)(q–)μ(x) U–δσ_(x)_νq–

n

aq_ndx



q

=k(σ)



p _∞

n=

δ(σ,n)(Vn–τn) q(–σ)–

νnq– aq_n



q

.

Then by (), we have ().

I =

∞



Uq–δσ_(x)

μq_(x)

f(x)

μq_(x)

Uq–δσ_(x)

∞

n=

dx

≥ fp,δJ. ()

Then by (), we have (). On the other hand, assuming that () is valid, we setf(x) as in Theorem . Then we ﬁndJ_q=fp_p_,δ. IfJ=∞, then () is trivially valid; ifJ= , then () remains impossible. Suppose that  <J<∞. By (), it follows that

fp_p_,δ=J

q

 =I>k(σ)fp,δaq,, f

p–

p,δ=J>k(σ)aq,,

and then () follows, which is equivalent to ().

Therefore, inequalities (), (), and () are equivalent. Forε∈(,qσ), we setσ=σ–ε

q (∈(, )) andf=f(x),x∈R+,a={an}∞n=,

f(x) =

Uδ(σ+ε)–_(x)_μ_(x), _{ <}_xδ_≤_,

, xδ_{> ,}

an= (Vn–τn)σ–νn= (Vn–τn)σ–

ε

q–_ν

n, n∈N.

By (), (), and (), we obtain

fp,δaq,=



εU

δε

p_() 

ν_ε+εO() 

q

,

I=

∞

n=

∞



anf(x) eαUδγ₍_x₎₍_V

n–τn)γ dx=

{x>;<xδ_≤__}

ωδ(σ,x) μ(x) U–δε_(x)dx

≤k(σ)

{x>;<xδ_≤__}

μ(x) U–δε_(x)dx=



εk σ– ε

q

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If there exists a positive constantK≥k(σ), such that () is valid when replacingk(σ) toK, then in particular, we haveεI>εKfp,δaq,, namely,

k σ–ε q

Uδε_{() >}_K_·_Uδεp_()  νε



+εO()



q

.

It follows thatk(σ)≥K(ε→+_{). Hence,}_K₌_k(_σ_{) is the best possible constant factor of} ().

The constant factork(σ) in () (()) is still the best possible. Otherwise, we would reach a contradiction by () (()) that the constant factor in () is not the best

possi-ble.

Theorem  As regards the assumptions of Theorem,if <p< ,  <fp,δ,aq,<∞,

then we have the following equivalent inequalities with the best possible constant factor k(σ) = _{γ α}(σσ//γγ):

I=

∞

n=

∞



anf(x)

eαUδγ₍_x₎₍_V_n_–_τ_n₎γ dx>

(σ/γ)

γ ασ/γ fp,δaq,, ()

J=

∞

n=

∞



f(x)dx eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

p

>(σ/γ)

γ ασ/γ fp,δ, ()

J:=

∞



( –θδ(σ,x))–q_μ_(x) U–qδσ_(x)

_∞

n=

q dx



q

>(σ/γ)

γ ασ/γ aq,. ()

Proof By the reverse Hölder inequality with weight (cf.[]), since  <p< , in a similar way to obtaining () and (), we have

∞



p

≥ (δ(σ,n))p–

∞



n–τn)γ

fp_(x)_dx.

In view of () and the Lebesgue term by term integration theorem, we ﬁnd

J≥k(σ)



q _∞

n=

∞



 eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

fp(x)dx



p

=k(σ)



q

∞



ωδ(σ,x)U

p(–δσ)–_(x)



p

.

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I =

∞

n=

νn/p

(Vn–τn) 

p–σ ∞



f(x) eαUδγ₍_x₎₍_V

n–τn)γ dx

(Vn–τn) 

p–σ_a

n

νn/p

≥Jaq,. ()

Then by (), we have (). On the other hand, assuming that () is valid, we setanas in Theorem . Then we ﬁndJ_p=aq_q_,. IfJ=∞, then () is trivially valid; ifJ= , then () remains impossible. Suppose that  <J<∞. By (), it follows that

aq_q_,=J_p=I>k(σ)fp,δaq,, a

q–

q,=J>k(σ)fp,δ,

and then () follows, which is equivalent to ().

Still by the reverse Hölder inequality with weight (cf.[]), sinceq< , we have

_∞

n=

q

≤ (ωδ(σ,x))q–

Uqδσ–_(x)_μ_(x)

∞

n=

 eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

(Vn–τn)(–σ)(q–)μ(x) U–δσ_(x)_νq–

n

aq_n.

Then by () and the Lebesgue term by term integration theorem, it follows that

J>k(σ)p

∞



∞

n=

 eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

(Vn–τn)(–σ)(q–)μ(x) U–δσ_(x)_νq–

n

aq_ndx



q

=k(σ)



p _∞

n=

δ(σ,n)

νnq–

aq_n



q

.

Then by (), we have ().

I =

∞



 –θδ(σ,x)



pU



q–δσ_(x)

μq_(x)

f(x)

×

( –θδ(σ,x))–p_μq_(x)

Uq–δσ_(x)

∞

n=

dx

≥ f_p_,δJ. ()

Then by (), we have (). On the other hand, assuming that () is valid, we setf(x) as in Theorem . Then we ﬁndJq=fp_p_,

δ. IfJ=∞, then () is trivially valid; ifJ= , then () keeps impossible. Suppose that  <J<∞. By (), it follows that

fp p,δ=J

q₌_I_>_k(_σ₎_f

p,δaq,, f

p–

p,δ=J>k(σ)aq,,

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Therefore, inequalities (), (), and () are equivalent. Forε∈(,pσ), we setσ=σ+ε_p andf =f(x),x∈R+,a={an}∞n=,

f(x) =

Uδσ–_(x)_μ_(x), _{ <}_xδ_≤_,

, xδ_{> ,}

an= (Vn–τn)σ–ε–νn= (Vn–τn)

σ–ε_q–

νn, n∈N.

By (), (), and (), we obtain

fp,δaq,

=

{x>;<xδ_≤__}

 –OU(x)δσμ(x)dx U–δε_(x)



p∞

n=



q

=

ε

Uδε() –εO()



p 

ν_ε+εO() 

q

,

I=

∞

n=

∞



n–τn)γan f(x)dx

=

∞

n=

{x>;<xδ_≤__}  eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

(Vn–τn)σμ(x) U–δσ_(x) dx

≤

∞

n=

∞



 eαUδγ₍_x₎₍_V_n_–_τ_n₎γ

(Vn–τn)σμ(x) U–δσ_(x) dx

νn (Vn–β)+ε

=

∞

n=

δ(σ,n) νn (Vn–τn)+ε

=k(σ)

∞

n=

=

εk σ+ ε

p 

ν_ε+εO()

.

If there exists a positive constantK≥k(σ), such that () is valid when replacingk(σ) toK, then, in particular, we haveεI>εKf_p_,δaq,, namely,

k σ+ε p



ν_ε +εO()

>KUδε() –εO()



p 

ν_ε+εO() 

q

.

It follows thatk(σ)≥K(ε→+). Hence,K=k(σ) is the best possible constant factor of ().

The constant factork(σ) in () (()) is still the best possible. Otherwise, we would reach the contradiction by () (()) that the constant factor in () is not the best

possi-ble.

5 Some corollaries and a remark

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Corollary  As regards the assumptions of Theorem, (i)for p> ,  <fp,,aq,<∞,

we have the following equivalent inequalities:

∞

n=

∞



anf(x) eαUδγ₍_x₎₍_V

n–τn)γ dx< (σ/γ)

γ ασ/γ fp,aq,, ()

∞

n=

∞



f(x) eαUγ₍_x₎₍_V

n–τn)γ dx p

<(σ/γ)

γ ασ/γ fp,, ()

∞



μ(x) U–qσ_(x)

_∞

n=

an eαUγ₍_x₎₍_V_n_–_τ_n₎γ

q dx



q

<(σ/γ)

γ ασ/γ aq,; ()

(ii)for p< ,  <fp,,aq,<∞,we have the following equivalent inequalities:

∞

n=

_∞



anf(x)

eαUγ₍_x₎₍_V_n_–_τ_n₎γ dx>

(σ/γ)

γ ασ/γ fp,aq,, ()

∞

n=

∞



f(x) eαUγ₍_x₎₍_V_n_–_τ_n₎γ dx

p

>(σ/γ)

γ ασ/γ fp,, ()

∞



μ(x) U–qσ_(x)

_∞

n=

an eαUγ₍_x₎₍_V_n_–_τ_n₎γ

q dx



q

>(σ/γ)

γ ασ/γ aq,; ()

(iii)for <p< ,  <fp,,aq,<∞,we have the following equivalent inequalities:

∞

n=

∞



anf(x)

eαUγ₍_x₎₍_V_n_–_τ_n₎γ dx>

(σ/γ)

γ ασ/γ fp,aq,, ()

∞

n=

∞



f(x) eαUγ₍_x₎₍_V_n_–_τ_n₎γ dx

p

>(σ/γ)

γ ασ/γ fp,, ()

_∞



( –θ(σ,x))–qμ(x) U–qσ_(x)

_∞

n=

an eαUγ₍_x₎₍_V

n–τn)γ q

dx



q

>(σ/γ)

γ ασ/γ aq,. ()

The above inequalities are with the best possible constant factor _{γ α}(σσ//γγ).

Forδ= – in Theorems -, we have the following inequalities with the homogeneous kernel of degree :

Corollary  As regards the assumptions of Theorem, (i)for p> ,  <fp,–,aq,< ∞,we have the following equivalent inequalities:

∞

n=

∞



anf(x)

eα(VnU–(xτ)n)γ

dx<(σ/γ)

γ ασ/γ fp,–aq,, ()

∞

n=

∞



f(x)

eα(VnU–(xτ)n)γ

dx

p

<(σ/γ)