A Hilbert type operator with a symmetric homogeneous kernel of two parameters and its applications

R E S E A R C H

Open Access

A Hilbert-type operator with a symmetric

homogeneous kernel of two parameters and

its applications

Keomkyo Seo

*_{Correspondence:} [email protected] Department of Mathematics, Sookmyung Women’s University, Hyochangwongil 52, Yongsan-ku, Seoul, 140-742, Korea

Abstract

We introduce a general homogeneous kernel whose degree is given by two parameters to establish the equivalent inequalities with the norm of a new Hilbert-type operator. As applications, we provide new extended Hilbert-type inequalities with the best possible constant factors.

MSC: 47A07; 26D15

Keywords: Hilbert-type operator; Hilbert-type inequality; beta function; kernel; norm

1 Introduction

Let {an} and {bm} be two sequences of nonnegative real numbers. The well-known Hilbert’s inequality says that ifp> , _p +_q = ,  <∞_m=apm<∞and  <

_∞ n=b

q n<∞,

then

∞

m= ∞

n=

ambn m+n<

π

sin(π_p)

_∞

m=

ap_m

/p_∞

n=

bq_n /q

, ()

where the constant factor π

sin(πp) is the best possible []. This inequality has been

gener-alized in numerous ways with introducing suitable parameters and weight coefficients. (For example, see [–] and the references therein.) In particular, by introducing a Hilbert-type linear operator with a symmetric homogeneous kernel, one can obtain var-ious Hilbert-type inequalities with the best constant factors. For this purpose, letk(x,y) be a nonnegative symmetric function defined on (,∞)×(,∞),i.e.,k(x,y) =k(y,x). For

p>  and_p +_q= , letr (r=p,q) be two normed spaces. IfT is a bounded self-adjoint semi-positive definite operator defined by

(Ta)(n) := ∞

m=

k(m,n)am, n∈N

fora={am}∞_m=∈p_{, or similarly,}

(Tb)(m) := ∞

n=

k(m,n)bn, m∈N

forb={bn}∞n=∈q. The operatorT is called theHilbert-type operatorand the function

k(x,y) is called thesymmetric kernelofT. In view of this point, Hilbert’s inequality () can be expressed by

(Ta,b)≤ π

sin(π

p)

apbq,

where the kernel k(x,y) = _x+_y and the formal inner product (Ta,b) between Taandb

is given by (Ta,b) :=∞_n=(Ta)(n)bn. Motivated by this observation, Yang [] defined a

Hilbert-type linear operatorT:r→r (r=p,q) with the kernelk(x,y) = (xy)

λ– 

(x+y)λ of

de-gree –. As a consequence, he was able to prove that if p> , _p + _q = , am,bn ≥,

 – min{_p,_q}<λ<  + min{_p,_q}, then the following two inequalities are equivalent: ∞

n= ∞

m=

(mn)λ–a_mb_n mλ_+nλ <



λB

q(λ+ ) – 

qλ ,

p(λ+ ) –  pλ

apbq,

_∞

n=

_∞

m=

(mn)λ– a_m mλ_+nλ

p  p

<

λB

q(λ+ ) – 

qλ ,

p(λ+ ) –  pλ

ap,

whereB(u,v) denotes the beta function defined by

B(u,v) :=

∞



tu–

( +t)u+vdt=B(u,v) (u,v> ).

Moreover, the constant factor _λB(q(λ_+)–_qλ ,p(λ_+)–_pλ ) is the best possible. In , Jin and Debnath [] generalized the Hilbert-type linear operator whose kernel is symmetric and homogeneous of degree –. In fact, they obtained several extended Hilbert-type inequal-ities by using the kernelk(x,y) = 

(xλ+yλ)λ (λ> ). For instance, they proved that ifp> ,



p+



q = ,α,β> ,  <λ≤min{ q

α,

p

β}, then the following two inequalities are equivalent:

∞

n= ∞

m=

ambn

(mα_+nβ₎λ<

B(λ_p,λ_q)

αq_βp _∞

m=

m(p–)(–αλ)|am|p 

p∞

n=

n(q–)(–βλ)|bn|q 

q

,

_∞

n=

nβλ– _∞

m=

am

(mα_+nβ₎λ

p  p

<B( λ

p,

λ

q)

αq_βp _∞

m=

m(p–)(–αλ)|am|p 

p

,

where the constant factor B(

λ

p,λq)

α



q_βp is the best possible. See [–] for other Hilbert-type

operators and the corresponding extended Hilbert-type inequalities with the best factors. Most of the previous results were, however, obtained by using the Hilbert-type opera-tor with the symmetric homogeneous kernel of –λ-order, which depends on a parameter

given by two parameters (Definition .). We establish the equivalent inequalities with the norm of a new Hilbert-type operator (Theorem .). As applications, we provide new extended Hilbert-type inequalities with the best possible constant factors (Corollary . and Cases -).

2 Hilbert-type operator with a symmetric homogeneous kernel whose degree is given by two parameters

For completeness, we begin with the following definitions and notations.

Definition . Letp> ,n∈Z,w(n)≥ (n≥n,n∈Z). Define the normed spacepw,n

by

p_w,n:=

a={an}∞_n=n:ap,w:=

_∞

n=n

w(n)|an|p /p

<∞ .

Definition . Letλ,λ,λ>  satisfying thatλ=λ+λ. Denote byFn(r) (n∈Z) the

set of all real-valuedC-functionsφ(x) satisfying the following conditions: () φ(x)is strictly increasing in(n– ,∞)withφ((n– )+) = ,φ(∞) =∞. () Forα> , φ(x)

φ(x)α+–λi is decreasing in(n– ,∞).

Letp> ,_p +_q= ,λ=λ+λ,λ,λ,λ> . Forφ(x)∈Fm(r) andψ(y)∈Fn(s),r,s> ,

we define the following weight functions:

w(m) :=φ(m)

p(α+–λ)–

φ(m)p– , w(n) :=

ψ(n)q(α+–λ)– ψ(n)q– ,

w(n) := ψ _(n)

ψ(n)p(α–λ)+, w(m) :=

φ(m)

φ(m)q(α–λ)+.

Definition . Letλ,λ,λ>  satisfying thatλ=λ+λ. Forα>  andx,y> ,Kα,λ(x,y) is a continuous real-valued function on (,∞)×(,∞) satisfying the following properties:

() Kα,λ(x,y)is a symmetric homogeneous function of degreeα–λ, that is,

Kα,λ(x,y) =Kα,λ(y,x),

Kα,λ(tx,ty) =tα–λKα,λ(x,y) for anyt> .

() Kα,λ(x,y)is decreasing with respect toxandy, respectively.

() For sufficiently smallε≥, the following integral

Kα,λ(λi,ε) := ∞



Kα,λ(,t)t–+λi–α–εdt

exists fori= , . Moreover, assume thatKα,λ(λi, ) :=Kα(λi) > and

Kα,λ(λi,ε) =Kα(λi) +o()asε→+.

() Givenp> ,φ(x)∈Fm(r), andψ(y)∈Fn(s)(r,s> ),

∞

n=n ψ(n)

ψ(n)+ε

φ(m_)

ψ(n)



Kα,λ(,t)t–+

λi–α–εp_dt=O()

Lemma . Letλ,λ,λ> satisfying thatλ=λ+λ.For anyα> ,we have

Kα(λ) =Kα(λ).

Proof Since

Kα(λ) =Kα,λ(λ, ) =

_∞



Kα,λ(,t)t–+λ–αdt,

lettingt=_s gives

Kα(λ) =

_∞



Kα,λ(,s)s–+λ–αds=Kα(λ).

In view of Lemma ., we may assume that

Kα(λ) :=Kα(λ) =Kα(λ).

Lemma . Let p> ,_p+_q= ,λ+λ=λ,λ,λ> ,α> .Forφ(x)∈Fm(r)andψ(y)∈ Fn(s),r,s> ,define the weight coefficients W(m)and W(n)by

W(m) := ∞

n=n Kα,λ

φ(m),ψ(n)φ(m) λ–α

ψ(n)α+–λψ

_(n),

W(n) := ∞

m=m Kα,λ

φ(m),ψ(n) ψ(n) λ–α

φ(m)α+–λφ

_(m).

Then

W(m) <Kα(λ) and W(n) <Kα(λ)

for any m≥m,n≥n(m,n∈Z).

Proof We have

W(m) = ∞

n=n Kα,λ

,ψ(n)

φ(m)

φ(m)α–λ ψ(n)α+–λψ

_(n)

<

∞

n– Kα,λ

,ψ(x)

φ(m)

ψ(x)

ψ(x)α+–λφ(m)

α–λ_dx.

Settingt=_φψ₍(_mx)₎, we get

W(m) <

_∞



Kα,λ(,t)t–+λ–αdt=Kα(λ).

Similarly, one can obtainW(n) <Kα(λ).

Lemma . Let p> , _p +_q = ,λ+λ=λ,λ,λ> .For am,bn≥ (m,n∈Z),let

a={am}∞_m=m∈pw,m and b={bn}∞n=n∈ q

(r,s> ),we have

∞

m=m Kα,λ

φ(m),ψ(n)am

p,w

≤Kα(λ)ap,w and

∞

n=n Kα,λ

φ(m),ψ(n)bn

q,w

≤Kα(λ)bq,w,

and hence

_∞

m=m Kα,λ

φ(m),ψ(n)am

∞

n=n ∈p_w˜

,n and

_∞

n=n Kα,λ

φ(m),ψ(n)bn

∞

m=m ∈q_w˜

,m.

Proof Applying Hölder’s inequality, we observe

∞

m=m Kα,λ

φ(m),ψ(n)am

= ∞

m=m

Kα,λ

φ(m),ψ(n)φ(m)

α+–λ q

ψ(n)α

+–λ p

ψ(n)p

φ(m)q am

ψ(n)α+–pλ

φ(m)α

+–λ q

φ(m)q

ψ(n)p

≤

_∞

m=m Kα,λ

φ(m),ψ(n)φ(m)

(α+–λ)(p–)

ψ(n)α+–λ

ψ(n)

φ(m)p–a

p m

 p

×

_∞

m=m Kα,λ

φ(m),ψ(n)ψ(n)

(α+–λ)(q–)

φ(m)α+–λ

φ(m)

ψ(n)q–

 q

.

By Definition ., we get

∞

m=m Kα,λ

φ(m),ψ(n)am

≤ ∞



Kα,λ(,t)t–+λ–αdt



q_ψ(n)q(α+–λ)– ψ(n)q–

 q

×

_∞

m=m Kα,λ

φ(m),ψ(n)φ(m)

(α+–λ)(p–)

ψ(n)α+–λ

ψ(n)

φ(m)p–a

p m

 p

.

Therefore, by using Lemma ., we get

∞

m=m Kα,λ

φ(m),ψ(n)am

p,w

=

_∞

n=n

ψ(n)

ψ(n)p(α–λ)+

_∞

m=m Kα,λ

φ(m),ψ(n)am p 

≤Kα(λ)

 q

_∞

n=n

∞

m=m Kα,λ

φ(m),ψ(n)φ(m)

(α+–λ)(p–)

ψ(n)α+–λ

ψ(n)

φ(m)p–a

p m

 p

=Kα(λ)

 q

_∞

m=m

W(m)φ(m)

p(α+–λ)–

φ(m)p– a

p m

 p

<Kα(λ)ap,w.

In the same manner, one can obtain

∞

n=n Kα,λ

φ(m),ψ(n)bn

q,w

≤Kα(λ)bq,w.

In view of Lemma ., we can define a Hilbert-type operatorT:pw,m→ p

w,nby

(Ta)(n) := ∞

m=m Kα,λ

φ(m),ψ(n)am, n≥n,n∈Z.

Similarly, defineT:qw,n→ q

w,mby

(Ta)(m) := ∞

n=n Kα,λ

φ(m),ψ(n)bn, m≥m,m∈Z.

It immediately follows from Lemma . that

Tp:= sup

ap,w=

Tap,w≤Kα(λ)

and

Tq:= sup

ap,w_=

Tbq,w≤Kα(λ).

Hence the operatorTis bounded. The formal inner product (Ta,b) ofTaandbis defined by

(Ta,b) := ∞

n=n

∞

m=m Kα,λ

φ(m),ψ(n)ambn.

Lemma . Let p> , _p+_q = .Leta={am}∞_m=mandb={bn}_n∞_=nwitham= φ

_(m)

φ(m)α+–λ+

ε

p andbn= ψ

_(n)

ψ(n)α+–λ+

ε

q for <ε<pλi,i= , .Then,asε→+,

Kα(λ)

 –o() ∞

n=n ψ(n)

ψ(n)+ε < (Ta,b) <Kα(λ)

 +o() ∞

n=n ψ(n)

Proof We have

(Ta,b) = ∞

n=n

∞

m=m Kα,λ

φ(m),ψ(n) φ _(m)

φ(_m)α+–λ+pε

ψ(n)

ψ(n)α+–λ+εq

< ∞

n=n ∞

m– Kα,λ

φ(x),ψ(n) φ _(x)

φ(x)α+–λ+εp

ψ(n)

ψ(n)α+–λ+εq dx.

Settingt=_ψφ(₍x_n)₎, we get

(Ta,b) < ∞

n=n

∞



Kα,λ(,t)t–+λ–α–

ε

p_dt

ψ(n)

ψ(n)+ε

=Kα(λ)

 +o() ∞

n=n ψ(n)

ψ(n)+ε.

Moreover,

(Ta,b) > ∞

n=n

∞

φ(m)

ψ(n)

Kα,λ(,t)t–+

λ–α–pε_dt

ψ(n)

ψ(n)+ε

= ∞

n=n ψ(n)

ψ(n)+ε

∞



Kα,λ(,t)t–+

λ–α–pε_dt– φ(m_)

ψ(n)



Kα,λ(,t)t–+

λ–α–εp_dt

.

Note that the definition ofKα,λ(x,y) implies that

∞



Kα,λ(,t)t–+λ–α–

ε

p_dt=K

α(λ) +o()

and

∞

n=n ψ(n)

ψ(n)+ε

φ(m_)

ψ(n)



Kα,λ(,t)t–+

λ–α–εp_dt=O().

Thus, using the fact that fora> ,

∞

n=n ψ(n)

ψ(n)+ε = 

ε

 +o() and ∞

n=n

ψ(n)

ψ(n)+a+εq =O()

asε→+, we obtain

(Ta,b) >Kα(λ)

 +o()

_∞

n=n ψ(n)

ψ(n)+ε–O()

=Kα(λ)

 +o() –O() ∞

n=n

ψ(n)

ψ(n)+ε

–∞

n=n ψ(n)

ψ(n)+ε

=Kα(λ)

 –o() ∞

n=n ψ(n)

ψ(n)+ε,

Theorem . Let p> , _p +_q= ,λ+λ=λ,λ,λ> .For am,bn≥ (m,n∈Z),let

a={am}∞_m=m∈pw,m and b={bn}∞n=n∈ q

w,n.Then,forφ(x)∈Fm(r)andψ(y)∈Fn(s)

(r,s> ),

Tp=Tq=Kα(λ).

Proof Suppose thatTp <Kα(λ). Consideram =φ(m)φ(m)–+λ–α–

ε

p _andb

n=φ(n)× ψ(n)–+λ–α–εq_{, wherem}≥_m,n≥_n,m,n∈Z_{,  <ε<pλi,i= , . A simple computation}

shows thata∈pw,mandb∈ q

w,nwithap,w>  andbq,w> . Then

Tap,w=

_∞

n=n

ψ(n)ψ(n)p(λ–α)–

_∞

m=m Kα,λ

φ(m),ψ(n)am p 

p

≤ Tpap,w.

Moreover, we have

(Ta,b) = ∞

n=n

∞

m=m Kα,λ

φ(m),ψ(n)ambn

= ∞

n=n

ψ(n)ψ(n)p(λ–α)–

_∞

m=m Kα,λ

φ(m),ψ(n)am p 

p bq,w

≤ Tpap,wbq,w

=Tp

_∞

m=m φ(m)

φ(m)+ε

 p∞

n=n ψ(n)

ψ(n)+ε

 q

. ()

On the other hand, from Lemma . it follows

Kα(λ)

 –o() ∞

n=n ψ(n)

ψ(n)+ε < (Ta,b). ()

Therefore, combining these inequalities () and (),

Kα(λ)

 –o()

_∞

n=n ψ(n)

ψ(n)+ε

 p

≤ Tp

_∞

m=m φ(m)

φ(m)+ε

 p

.

Since

∞

n=n ψ(n)

ψ(n)+ε = 

ε

 +o() and ∞

m=m φ(m)

φ(m)+ε= 

ε

 +o()

asε→+, we obtain thatKα(λ)≤ Tp, which is a contradiction. Thus we conclude that

Tp=Kα(λ). Applying the same argument, we haveTq=Kα(λ), which completes the

3 Two equivalent inequalities for the Hilbert-type operator

Equipped with the Hilbert-type operator defined as above, we have the following theorem.

Theorem . Let p> , _p +_q= ,λ+λ=λ,λ,λ> .For am,bn≥ (m,n∈Z),let

a={am}∞_m=m∈pw,m,b={bn}∞n=n∈ q

w,nap,w> ,bq,w> .Then,forφ(x)∈Fm(r) andψ(y)∈Fn(s) (r,s> ),we have the following equivalent inequalities:

(Ta,b) = ∞

n=n

∞

m=m Kα,λ

φ(m),ψ(n)ambn<Kα(λ)ap,wbq,w, ()

Tap,w<Kα(λ)ap,w. ()

Furthermore,the constant factor Kα(λ)is the best possible.

Proof It follows from Hölder’s inequality that

(Ta,b) = ∞

n=n

∞

m=m Kα,λ

φ(m),ψ(n)φ(m)

α+–λ q

ψ(n)

α+–λ p

ψ(n)p

φ(m)q am

×

ψ(n)

α+–λ_

p

φ(m)

α+–λ q

φ(m)q

ψ(n)p bn

≤

_∞

n=n

∞

m=m Kα,λ

φ(m),ψ(n)φ(m)

(α+–λ)(p–)

ψ(n)α+–λ

ψ(n)

φ(m)p–a

p m

 p

×

_∞

n=n

∞

m=m Kα,λ

φ(m),ψ(n)ψ(n)

(α+–λ)(q–)

φ(m)α+–λ

φ(m)

ψ(n)q–b

q n

 q

=

_∞

m=m

W(m)φ(m)

p(α+–λ)–

φ(m)p– a

p m

 p∞

n=n

W(n)ψ(n)

q(α+–λ)–

ψ(n)q– b

q n

 q

.

Applying Lemma ., we see that

(Ta,b) <Kα(λ)ap,wbq,w.

In order to prove that inequality () implies inequality (), we definebas follows:

bn:=

ψ(n)

ψ(n)p(α–λ)+

_∞

m=m Kα,λ

φ(m),ψ(n)

p–

forn≥n,n∈Z. Then we see thatb∈wq,n andbq,w>  as before. Thus using

in-equality () shows that

bq_q,w= ∞

n=n

ψ(n)

ψ(n)p(α–λ)+

_∞

m=m Kα,λ

φ(m),ψ(n)am p

= ∞

n=n

∞

m=m Kα,λ

which gives Tap,w =b q–

q,w <Kα(λ)ap,w. Hence inequality () implies

inequal-ity ().

Now suppose that inequality () holds for anya∈pw,m.

(Ta,b) = ∞

n=n

∞

m=m Kα,λ

φ(m),ψ(n)ambn

= ∞

n=n

ψ(n)p

ψ(n)α–λ+p

∞

m=m Kα,λ

φ(m),ψ(n)am

ψ(n)α–λ+p

ψ(n)p bn

≤

_∞

n=n

ψ(n)

ψ(n)p(α–λ)+

_∞

m=m Kα,λ

φ(m),ψ(n)am p 

p bq,w

<Kα(λ)ap,wbq,w,

which means that inequality () implies inequality (). Therefore inequality () is equiva-lent to inequality (). Furthermore, Theorem . implies that the constant factorKα(λ) in

inequalities () and () is the best possible, which completes the proof.

4 Applications to various Hilbert-type inequalities

In this section, we apply our previous theorems to obtain several Hilbert-type inequalities. Recall that the beta functionB(u,v) is defined by

B(u,v) :=

∞



tu–

( +t)u+vdt=B(u,v) (u,v> ).

Define the functionKα,λ(x,y) by

Kα,λ(x,y) := (xy)α (x+y)λ

forλ>α≥. ThenKα,λ(x,y) is a symmetric homogeneous function of degree α–λand is decreasing with respect toxandy, respectively. Moreover,

∞

n=n ψ(n)

ψ(n)+ε

φ(m_)

ψ(n)



Kα,λ(,t)t–+λ–α–

ε

p_dt=O().

To see this, for  <ε<pλ,

∞

n=n ψ(n)

ψ(n)+ε

φ(m)

ψ(n)



t–+λ–pε

( +t)λ dt≤ ∞

n=n ψ(n)

ψ(n)+ε

φ(m)

ψ(n)



t–+λ–εp_dt

= ∞

n=n ψ(n)

ψ(n)+ε 

λ–_pε

φ(m)

ψ(n)

λ–εp

=φ(m) λ–εp

λ–_pε ∞

n=n

ψ(n)

ψ(n)+λ+εq

Note that since

Kα,λ(λi,ε) := ∞



Kα,λ(,t)t–+λi–α–εdt

=

∞



t–+λi–ε

( +t)λdt,

we see that

Kα,λ(λi,ε)→

∞



tλi–

( +t)λdt=B(λ,λ) =Kα(λi) =Kα(λ)

asε→+. Therefore from Theorem . we observe the following.

Corollary . Let p> , _p + _q = , λ +λ =λ, λ,λ > , λ>α ≥.For am,bn≥

(m,n∈Z),let a={am}∞m=m∈ p

w,m,b={bn}∞n=n∈ q

w,n andap,w> ,bq,w > . Then,forφ(x)∈Fm(r)andψ(y)∈Fn(s) (r,s> ),we have the following equivalent in-equalities:

∞

n=n

∞

m=m

φ(m)α_ψ(n)α_a

mbn

(φ(m) +ψ(n))λ <B(λ,λ)ap,wbq,w,

_∞

n=n

ψ(n)ψ(n)p(λ–α)–

_∞

m=m

φ(m)α_ψ(n)α_a

m

(φ(m) +ψ(n))λ

p 

p

<B(λ,λ)ap,w.

Furthermore,the constant factor B(λ,λ)is the best possible.

As applications, we have the following.

Case. Letφ(x) =xβ_{andψ(x) =x}γ _{(β,γ> ) form}

=n= . For  <λi<α+min{β,  γ} and ≤α<λ, one has the following equivalent inequalities:

∞

n= ∞

m=

(mβ_nγ₎α

(mβ_+nγ₎λambn<

B(λ,λ)

βq_γp

ap,wbq,w,

_∞

n=

nγp(λ–α)–

_∞

m=

(mβ_nγ₎α (mβ_+nγ₎λam

p 

p

<B(λ,λ)

βq_γp ap,w,

wherew(m) =mp(–λβ+αβ)–_{andw(n) =n}q(–λγ+αγ)–_. (I) Forλ=λ_p andλ=λ_q with <λi<α+min{β,



γ}and≤α<λ, one has the

following equivalent inequalities:

∞

n= ∞

m=

(mβ_nγ₎α

(mβ_+nγ₎λambn<

B(λ_p,λ_q)

βq_γp

ap,wbq,w,

_∞

n=

nγ(λ–pα)–

_∞

m=

(mβ_nγ₎α (mβ_+nγ₎λam

p  p

<B( λ

p,

λ

q)

βq_γp

ap,w,

(II) Forλ=λ_q andλ=λ_p with <λi<α+min{β, 

γ}and≤α<λ, one has the

following equivalent inequalities:

∞

n= ∞

m=

(mβ_nγ₎α

(mβ_+nγ₎λambn<

B(λ

p,

λ

q)

βq_γp

ap,wbq,w,

_∞

n=

nγ λ(p–)–pαγ– _∞

m=

(mβ_nγ₎α (mβ_+nγ₎λam

p  p

<B( λ

p,

λ

q)

βq_γp

ap,w,

wherew(m) =mp––βλ+pαβandw(n) =nq––γ λ+qαγ.

(III) Letλ=p+λ_p–,λ=q+λ_q–,λ>max{ –p,  –q}, <β<_{p+λ––}p _pα, <γ<_{q+λ––}q _qα, ≤α<min{p+_pλ–,q+λ_q–}. Then one has the following equivalent inequalities:

∞

n= ∞

m=

(mβ_nγ₎α

(mβ_+nγ₎λambn<

B(p+λ_p–,q+λ_q–)

βq_γp

ap,wbq,w,

_∞

n=

nγ(p+λ–)–pαγ– _∞

m=

(mβ_nγ₎α (mβ_+nγ₎λam

p 

p

<B(

p+λ–

p , q+λ–

q )

βq_γp

ap,w,

wherew(m) =m(p–)(–β(q+λ–))+pαβ_{andw(n) =n}(q–)(–γ(p+λ–))+qαγ_.

(IV) Letλ=q+λ_q–,λ=p+λ_p–,λ>max{ –p,  –q}, <β<_q+λ––q qα, <γ <

p p+λ––pα, ≤α<min{p+_pλ–,q+λ_q–}. Then one has the following equivalent inequalities:

∞

n= ∞

m=

(mβ_nγ₎α

(mβ_+nγ₎λambn<

B(p+λ_p–,q+λ_q–)

βq_γp

ap,wbq,w,

_∞

n=

nγ(p–)(q+λ–)–pαγ–

_∞

m=

(mβ_nγ₎α (mβ_+nγ₎λam

p  p

<B(

p+λ–

p , q+λ–

q )

βq_γp

ap,w,

wherew(m) =mp––β(p+λ–)+pαβ _{andw(n) =n}q––γ(q+λ–)+qαγ_.

Case. ForA,B> , letφ(x) =A(lnx)β_{andψ(x) =B(lnx)}γ _{(β,γ > ),m}

=n= . For  <λi<α+min{β,



γ}and ≤α<λ, one has the following equivalent inequalities:

∞

n= ∞

m=

((lnm)β_(lnn)γ₎α

(A(lnm)β_+B(lnn)γ₎λambn<

B(λ,λ)

Aλ_Bλ_β  q_γp

ap,wbq,w,

_∞

n= 

n(lnn)

pγ(λ–α)–

_∞

m=

((lnm)β_(lnn)γ₎α (A(lnm)β_+B(lnn)γ₎λam

p  p

< B(λ,λ)

Aλ_Bλ_β  q_γp

ap,w,

wherew(m) =mp–_(lnm)p(–λβ+αβ)–_{andw(n) =n}q–_(lnn)q(–λγ+αγ)–_. (I) Forλ=λ_p andλ=λ_q with <λi<α+min{β,



γ}and≤α<λ, one has the

following equivalent inequalities:

∞

n= ∞

m=

((lnm)β_(lnn)γ₎α

(A(lnm)β_+B(lnn)γ₎λambn<

B(λ_p,λ_q)

Aλ_Bλ_β  q_γp

_∞

n= 

n(lnn)

γ–pαγ–

_∞

m=

((lnm)β_(lnn)γ₎α (A(lnm)β_+B(lnn)γ₎λam

p  p

< B( λ

p,

λ

q)

Aλ_Bλ_β  q_γp

ap,w,

wherew(m) =mp–(lnm)(p–)(–λβ)+pαβandw(n) =nq–(lnn)(q–)(–λγ)+qαγ. (II) Letλ=p+λ_p–,λ=q+λ_q–,λ>max{ –p,  –q}, <β<_p+λ––p pα, <γ<

q q+λ––qα, ≤α<min{p+_pλ–,q+λ_q–}. Then one has the following equivalent inequalities:

∞

n= ∞

m=

((lnm)β_(lnn)γ₎α

(A(lnm)β_+B(lnn)γ₎λambn<

B(p+λ_p–,q+λ_q–)

Aλ_Bλ_β  q_γp

ap,wbq,w,

_∞

n= 

n(lnn)

γ(p+λ–)–pαγ–

_∞

m=

((lnm)β_(lnn)γ₎α (A(lnm)β_+B(lnn)γ₎λam

p  p

<B(

p+λ–

p , q+λ–

q )

Aλ_Bλ_β  q_γp

ap,w,

wherew(m) =mp–(lnm)(p–)(–β(q+λ–))+pαβ and

w(n) =nq–_(lnn)(q–)(–γ(p+λ–))+qαγ_.

Case. ForA,B> , letφ(x) =A(lnx)β _{andψ(x) =Bx}γ _{(β,γ > ),m}

= ,n= . For  <λi<α+min{β,



γ}and ≤α<λ, one has the following equivalent inequalities:

∞

n= ∞

m=

((lnm)β_nγ₎α

(A(lnm)β_+Bnγ₎λambn<

B(λ,λ)

Aλ_Bλ_β  q_γp

ap,wbq,w,

_∞

n=

npγ(λ–α)– _∞

m=

((lnm)β_nγ₎α (A(lnm)β_+Bnγ₎λam

p  p

< B(λ,λ)

Aλ_Bλ_β  q_γp

ap,w,

wherew(m) =mp–_(lnm)p(–λβ+αβ)–_{andw(n) =n}q(–λγ+αγ)–_.

(I) Forλ=λ_p andλ=λ_q with <λi<α+min{β, 

γ}and≤α<λ, one has the

following equivalent inequalities:

∞

n= ∞

m=

((lnm)β_nγ₎α

(A(lnm)β_+Bnγ₎λambn<

B(λ_p,λ_q)

Aλ_Bλ_β  q_γp

ap,wbq,w,

_∞

n=

nγ(–pα)–

_∞

m=

((lnm)β_nγ₎α (A(lnm)β_+Bnγ₎λam

p  p

< B( λ

p,

λ

q)

Aλ_Bλ_β  q_γp

ap,w,

wherew(m) =mp–_(lnm)(p–)(–λβ)+pαβ_{andw(n) =n}(q–)(–λγ)+qαγ_.

(II) Letλ=p+λ_p–,λ=q+λ_q–,λ>max{ –p,  –q}, <β<_p+λ––p pα, <γ<

q q+λ––qα, ≤α<min{p+_pλ–,q+λ_q–}. Then one has the following equivalent inequalities:

∞

n= ∞

m=

((lnm)β_nγ₎α

(A(lnm)β_+Bnγ₎λambn<

B(p+λ_p–,q+λ_q–)

Aλ_Bλ_β  q_γp

_∞

n=

nγ(p+λ–)–pαγ–

_∞

m=

((lnm)β_nγ₎α (A(lnm)β_+Bnγ₎λam

p  p

<B(

p+λ–

p , q+λ–

q )

Aλ_Bλ_β  q_γp

ap,w,

wherew(m) =mp–(lnm)(p–)(–β(q+λ–))+pαβ _{andw(n) =n}(q–)(–γ(p+λ–))+qαγ_.

Competing interests

The author declares that he has no competing interests.

Acknowledgements

This research was supported by the Sookmyung Women’s University Research Grants 2012.

Received: 10 July 2015 Accepted: 17 August 2015

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