R E S E A R C H
Open Access
A strengthened Mulholland-type
inequality with parameters
Aizhen Wang
*, Qiliang Huang and Bicheng Yang
*Correspondence:
sxxmaths876@sina.com Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, P.R. China
Abstract
By means of the way of weight coefficients, technique of real analysis, and Hermite-Hadamard’s inequality, a strengthened version of the Mulholland-type inequality with the best possible constant factor and multi-parameters is given. The equivalent forms, the reverses, the operator expressions and a few particular cases are considered.
MSC: 26D15; 47A07
Keywords: Mulholland-type inequality; weight coefficient; equivalent form; reverse; operator
1 Introduction
Ifp> , p +q = ,am,bn≥,a={am}∞m=∈lp,b={bn}∞
n=∈lq,ap= (
∞
m=a
p m)
p > ,
bq> , then we have the following well-known Hardy-Hilbert’s inequality with the best possible constant factorsin(ππ/p) (cf.[], Theorem ):
∞
m= ∞
n=
ambn m+n<
π
sin(π/p)apbq. ()
Also we have the following Mulholland’s inequality similar to () with the same best
pos-sible constant factor π
sin(π/p)(cf.[] or [], Theorem , replacing
am n,
bn
n byam,bn):
∞
m= ∞
n=
ambn
lnmn<
π
sin(π/p)
∞
m=
apm m–p
p∞
n=
bqn n–q
q
. ()
Inequalities () and () are important in analysis and its applications (cf.[, –]). In , Gao and Yang [] gave a strengthened version of () as follows:
∞
m= ∞
n=
ambn m+n<
∞
m=
π
sin(π/p)– –γ
m/p
apm
p
×
∞
n=
π
sin(π/p)– –γ
n/q
bqn
q
, ()
where –γ = .+(γ is Euler constant).
Suppose thatμi,υj> (i,j∈N={, , . . .}),
Um:= m
i=
μi, Vn:= n
j=
υj (m,n∈N), ()
we have the following Hardy-Hilbert-type inequality (cf.[], Theorem ):
∞
m= ∞
n=
μ/mqυn/pambn Um+Vn <
π
sin(π/p)apbq. ()
Forμi=υj= (i,j∈N), inequality () reduces to (). Replacingμ/mqamandυn/pbnbyam andbnin (), respectively, we obtain the equivalent form of () as follows:
∞
m= ∞
n=
ambn Um+Vn<
π
sin(π p)
∞
m=
apm
μpm–
p∞
n=
bqn
υnq–
q
. ()
In , Yang [] gave an extension of () as follows: For <λ,λ≤,λ+λ=λ, we
have
∞
m= ∞
n=
ambn (Um+Vn)λ
<B(λ,λ)
∞
m=
Up(–λ)– m apm
μpm–
p∞
n=
Vq(–λ–) n bqn
υnq–
q
, ()
whereB(u,v) is the beta function indicated by (cf.[])
B(u,v) :=
∞
tu–
( +t)u+vdt (u,v> ). ()
In this paper, by using the way of weight coefficients, the technique of real analysis, and Hermite-Hadamard’s inequality, a Mulholland-type inequality with the best possible con-stant factor sin(ππ/p) is given as follows: Forμ=υ= ,{μm}∞m=and{υn}∞n=are decreasing,
andU∞=V∞=∞, we have
∞
m= ∞
n=
ambn
lnUmVn
< π
sin(π/p)
∞
m=
Um
μm+
p–
apm
p∞
n=
Vn
υn+
q–
bqn
q
, ()
2 Some lemmas
In the following, we make appointment thatp= , , p+q= , <λ,λ≤,λ+λ=λ, μi,υj> (i,j∈N), withμ=υ= ,UmandVnare defined by (),am,bn≥,ap,λ:=
(∞m=λ(m)apm)
p andb q,λ:= (
∞
n=λ(n)bqn)
q, where
λ(m) :=
(lnUm)p(–λ)– Um–pμpm–+
,
λ(n) :=
(lnVn)q(–λ)– Vn–qυnq+–
m,n∈N\{}.
()
Lemma If a∈R,f(x)is continuous in[a–,a+],f(x)is strictly increasing in(a–,a) and(a,a+),respectively,and
lim
x→a–f
(x) =f(a– )≤f(a+ ) = lim
x→a+f (x),
then we have the following Hermite-Hadamard’s inequality(cf.[]):
f(a) < a+
a–
f(x)dx. ()
Proof Sincef(a– ) (≤f(a+ )) is finite, we set a functiong(x) as follows:
g(x) :=f(a– )(x–a) +f(a), x∈
a– ,a+
.
In view off(x) being strictly increasing in (a–,a), then forx∈(a–,a), (f(x) –g(x))= f(x) –f(a– ) < . Sincef(a) –g(a) = , it follows thatf(x) –g(x) > ,x∈(a–,a). In the same way, we can obtainf(x) –g(x) > ,x∈(a,a+). Hence, we find
a+
a–
f(x)dx> a+
a–
g(x)dx=f(a),
namely () follows.
Example If{μm}∞m= and{υn}∞n= are also decreasing, we setμ(t) :=μm,t∈(m– ,m] (m∈N);υ(t) :=υn,t∈(n– ,n] (n∈N),
U(x) := x
μ(t)dt (x≥), V(y) :=
y
υ(t)dt (y≥). ()
Then it follows thatU(m) =Um,V(n) =Vn(m,n∈N),U(∞) =U∞,V(∞) =V∞and
U(x) =μ(x) =μm
x∈(m– ,m),
V(y) =υ(y) =υn
y∈(n– ,n).
For fixedm,n∈N\{}, we also set a functionf(x) as follows:
f(x) = ln
λ–V(x)
V(x)(lnUm+lnV(x))λ, x∈
n– ,n+
Thenf(x) in continuous in [n–,n+]. Forx∈(n–,n) (n∈N\{}), we find
f(x) = –
lnλ–V(x) V(x) +
λlnλ–V(x)
lnUm+lnV(x)
+ –λ V–λ(x)
× υn
V(x)(lnUm+lnV(x))λ .
Since –λ≥, it follows thatf(x) (< ) is strictly increasing in (n–,n) and
lim
x→n–f
(x) =f(n– )
= –
lnλ–V n Vn +
λlnλ–V n
lnUm+lnVn+ –λ
V–λ n
× υn
Vn(lnUm+lnVn)λ.
In the same way, forx∈(n,n+), we find
f(x) = –
lnλ–V(x) V(x) +
λlnλ–V(x)
lnUm+lnV(x)+ –λ
V–λ(x)
× υn+
V(x)(lnUm+lnV(x))λ,
f(x) (< ) is strictly increasing in (n,n + ). In view of υn+ ≤υn, it follows that
limx→n+f(x) =f(n+ )≥f(n– ). Then by () we have
f(n) < n+
n–
f(x)dx= n+
n–
lnλ–V(x)
V(x)(lnUm+lnV(x))λdx. ()
Definition Define the following weight coefficients:
ω(λ,m) := ∞
n=
lnλ(UmVn)
υn+lnλUm
Vnln–λVn, m∈N\{}, ()
(λ,n) := ∞
m=
lnλ(UmVn)
μm+lnλVn
Umln–λUm, n∈N\{}. ()
Lemma If{μm}∞m= and{υn}∞n= are decreasing,and U∞ =V∞=∞,then we have the
following inequalities:
ω(λ,m) <B(λ,λ)
– θ
lnλU m
<λ≤,λ> ;m∈N\{}
, ()
(λ,n) <B(λ,λ)
– θ
lnλVn
<λ≤,λ> ;n∈N\{}
where
θ:=
B(λ,λ)
lnλ( +υ
/)
λ[ +lnln(+(+μυ/)/)]
λ, ()
θ:=
B(λ,λ)
lnλ( +μ
/)
λ[ +lnln(+(+μυ/)/)]
λ. ()
Proof Since forx∈(n– ,n+
)\{n},υn+≤V(x), by () we find
ω(λ,m) < ∞
n= υn+
n+
n–
lnλUmlnλ–V(x) V(x)(lnUm+lnV(x))λdx
≤ ∞
n=
n+
n–
lnλU
mlnλ–V(x) V(x)(lnUm+lnV(x))λV
(x)dx
=
∞
lnλUmlnλ–V(x) V(x)(lnUm+lnV(x))λ
V(x)dx
=
∞
lnλUmlnλ–V(x) V(x)(lnUm+lnV(x))λV
(x)dx
–
lnλU
mlnλ–V(x) V(x)(lnUm+lnV(x))λV
(x)dx.
Settingt=lnlnVU(x)
m, we obtain V(x)
V(x) dx=lnUmdtand
ω(λ,m) <
∞
( +t)λt
λ–dt–
lnλUm (lnUm+lnV(x))λ
lnλ–V(x) V(x) V
(x)dx
=B(λ,λ)
–θ(m), ()
where
θ(m) := B(λ,λ)
lnλU m (lnUm+lnV(x))λ
lnλ–V(x) V(x) V
(x)dx. ()
We find
θ(m)≥ B(λ,λ)
lnλUm (lnUm+lnV())λ
lnλ–V(x) V(x) V
(x)dx
=
B(λ,λ)
lnλUm
λ(lnUm+lnV())λ
lnλV
=
B(λ,λ)
lnλ( +υ
/)
λ( +ln(+lnUυm/)) λ
lnλU m
≥
B(λ,λ)
lnλ( +υ
/)
λ( +ln(+lnUυ/)) λ
lnλU m
= θ
lnλU m
.
Hence, by (), we have () and (). In the same way, we obtain () and ().
Lemma With the assumptions of Lemma, (i)for m,n∈N\{},we have
B(λ,λ)
–θ(λ,m)
<ω(λ,m) ( <λ≤,λ> ), ()
B(λ,λ)
–ϑ(λ,n)
< (λ,n) ( <λ≤,λ> ), ()
where
θ(λ,m) =
B(λ,λ)
lnλ( +υ
)
λ[ +ln(+lnθU(mm)υ)] λ
lnλU m
=O
lnλUm
∈(, ) θ(m)∈(, ), ()
ϑ(λ,n) =
B(λ,λ)
lnλ( +μ
)
λ[ +ln(+lnϑV(nn)μ)] λ
lnλVn
=O
lnλVn
∈(, ) ϑ(n)∈(, ); ()
(ii)for any a> ,we have
∞
m=
μm+
Umln+aUm =
a
lna( +μ)
+aO()
, ()
∞
n= υn+
Vnln+aVn= a
lna( +υ)
+aO()
. ()
Proof Since by Example ,f(x) is strictly decreasing in [n,n+ ], then we find
ω(λ,m) > ∞
n=
n+
n
υn+
lnλU
mlnλ–V(x) V(x)(lnUm+lnV(x))λdx
=
∞
lnλUmlnλ–V(x) V(x)(lnUm+lnV(x))λ
V(x)dx
=
∞
lnλU
mlnλ–V(x) V(x)(lnUm+lnV(x))λV
(x)dx
–
lnλUmlnλ–V(x) V(x)(lnUm+lnV(x))λV
(x)dx
=B(λ,λ)
–θ(λ,m)
,
where
θ(λ,m) :=
B(λ,λ)
V(x)lnλUmlnλ–V(x)
V(x)(lnUm+lnV(x))λ dx∈(, ).
There existsθ(m)∈(, ) such that
θ(λ,m) =
B(λ,λ)
lnλU m
[lnUm+lnV( +θ(m))]λ
×
lnλ–V(x) V(x) V
= B(λ,λ)
lnλUmlnλ( +υ
) λ[lnUm+lnV( +θ(m))]λ
=
B(λ,λ)
lnλ( +υ
)
λ[ +ln(+lnθU(mm)υ)] λ
lnλUm.
Since we obtain
<θ(λ,m)≤
lnλ( +υ
) λB(λ,λ)
lnλU m
,
namelyθ(λ,m) =O(lnλUm), we have (). In the same way, we obtain (). Fora> , we find
∞
m=
μm+
Umln+aUm ≤
∞
m= μm Umln+aUm
= μ
Uln+aU
+
∞
m= μm Umln+aUm
= μ
Uln+aU
+
∞
m=
m
m–
U(x) Umln+aUm
dx
< μ
Uln+aU
+
∞
m=
m
m–
U(x) U(x)ln+aU(x)dx
= μ
Uln+aU
+
∞
U(x) U(x)ln+aU(x)dx
= μ
( +μ)ln+a( +μ)
+
alna( +μ
)
= a
lna( +μ)
+ aμ
( +μ)ln+a( +μ)
,
∞
m=
μm+
Umln+aUm =
∞
m=
m+
m
U(x)dx Umln+aUm
>
∞
m=
m+
m
U(x) U(x)ln+aU(x)dx
=
∞
U(x)dx U(x)ln+aU(x)=
alna( +μ)
.
Hence we have (). In the same way, we have ().
3 Main results and operator expressions We also set
λ(m) :=ω(λ,m)
(lnUm)p(–λ)–
Um–pμpm–+
,
λ(n) := (λ,n)
(lnVn)q(–λ)– Vn–qυnq+–
m,n∈N\{}.
Theorem (i)For p> ,we have the following equivalent inequalities:
I:=
∞
n= ∞
m=
ambn
lnλ(UmVn)≤ ap,λbq,λ, ()
J:=
∞
n=
υn+lnpλ–Vn
( (λ,n))p–Vn
∞
m=
am
lnλ(UmVn) p p
≤ ap,λ. ()
(ii)For <p< (or p< ),we have the equivalent reverses of()and().
Proof (i) By Hölder’s inequality with weight (cf.[]) and (), we have
∞
m=
am
lnλ(U mVn)
p =
∞
m=
lnλ(U mVn)
Um/q(lnUm)(–λ)/qυn/+p
(lnVn)(–λ)/pμ/q m+
am
×
(lnVn)(–λ)/pμ/q m+
Um/q(lnUm)(–λ)/qυn/+p
p
≤ ∞
m=
lnλ(UmVn)
Ump–(lnUm)(–λ)p/qυn+
(lnVn)–λμp/q m+
apm
×
∞
m=
lnλ(UmVn)
(lnVn)(–λ)(q–)μ m+
Um(lnUm)–λυq– n+
p–
= ( (λ,n)) p–Vn
(lnVn)pλ–υn+
∞
m=
υn+Ump–(lnUm)(–λ)(p–)apm
lnλ(U
mVn)Vn(lnVn)–λμpm–+
. ()
Then by () we find
J≤
∞
n= ∞
m= υn+
lnλ(U mVn)
Ump–(lnUm)(–λ)(p–) Vn(lnVn)–λμp–
m+
apm
p
=
∞
m= ∞
n=
υn+(lnUm)λ
lnλ(UmVn)
Ump–(lnUm)p(–λ)–
Vn(lnVn)–λμp– m+
apm
p
=
∞
m=
ω(λ,m)
(lnUm)p(–λ)–
Um–pμpm–+
apm
p
, ()
and then () follows.
By Hölder’s inequality (cf.[]), we have
I=
∞
n=
(lnVn)λ–pυ/p n+
( (λ,n))
qV
p n
∞
m=
am
lnλ(UmVn)
× (λ,n)
q(lnVn)
p–λ
V –
p n υ
p n+
bn
≤Jbq,λ. ()
On the other hand, assuming that () is valid, we set
bn:=(lnVn) pλ–υ
n+
( (λ,n))p–Vn
∞
m=
am
lnλ(U mVn)
p–
, n∈N\{}. ()
Then we findJp=bq
q,λ. IfJ= , then () is trivially valid; ifJ=∞, then, by (), () takes the form of equality. Suppose that <J<∞. By (), it follows that
bqq,
λ=J
p=I≤ a
p,λbq,λ, ()
bqq–,
λ=J≤ ap,λ, ()
and then () follows, which is equivalent to ().
(ii) For <p< (orp< ), by the reverse Hölder’s inequality with weight (cf.[]) and (), we obtain the reverse of () (or ()), then we have the reverse of (), and then the reverse of () follows. By Hölder’s inequality (cf.[]), we have the reverse of () and then by the reverse of (), the reverse of () follows.
On the other hand, assuming that the reverse of () is valid, we setbnas (). Then we findJp=bq
q,λ. IfJ=∞, then the reverse of () is trivially valid; ifJ= , then, by the reverse of (), () takes the form of equality (= ). Suppose that <J<∞. By the reverse of (), it follows that the reverses of () and () are valid, and then the reverse
of () follows, which is equivalent to the reverse of ().
Setting
λ(m) :=
– θ
lnλU m
(lnUm)p(–λ)– Um–pμpm–+
,
λ(n) :=
– θ
lnλVn
(lnVn)q(–λ)– Vn–qυnq–+
m,n∈N\{},
()
we have the following.
Theorem If p> ,{μm}∞m=and{υn}∞n=are decreasing,U∞=V∞=∞,ap,λ∈R+and
bq,λ∈R+,then we have the following equivalent inequalities:
∞
n= ∞
m=
ambn
lnλ(UmVn)<B(λ,λ)ap,λbq,λ, ()
J:=
∞
n=
υn+lnpλ–Vn ( – θ
lnλVn) p–Vn
∞
m=
am
lnλ(UmVn) p p
<B(λ,λ)ap,λ, ()
where the constant factor B(λ,λ)is the best possible.
Proof Using () and () in () and (), since
ω(λ,m)
p<B(λ
,λ)
p
– θ
lnλU m
p
(p> ),
(λ,n)
q <B(λ
,λ)
q
– θ
lnλVn
q
and
(B(λ,λ))p–( –lnλθVn) p–<
( (λ,n))p–
(p> ),
we obtain equivalent inequalities () and ().
Forε∈(,pλ), we setλ=λ–pε(∈(, )),λ=λ+εp (> ), and
am:=
μm+
Um ln λ–U
m=
μm+
Um ln λ–εp–U
m,
bn=
υn+
Vn ln λ–ε–V
n=
υn+
Vn ln λ–εq–V
n.
()
Then, by (), () and (), we have
ap,λbq,λ
≤ ap,λbq,λ
=
∞
m=
μm+
Umln+εUm
p∞
n= υn+
Vnln+εVn
q
=
ε
lnε( +μ)
+εO()
p
lnε( +υ)
+εO()
q
,
I:=
∞
n= ∞
m=
ambn
lnλ(UmVn)
=
∞
n=
∞
m=
lnλ(U mVn)
μm+lnλVn
Umln–λUm
υn+
Vnlnε+Vn
=
∞
n=
(λ,n) υn+
Vnlnε+Vn
≥B(λ,λ) ∞
n=
–O
lnλVn
υn+
Vnlnε+Vn
=B(λ,λ)
∞
n= υn+
Vnlnε+Vn–
∞
n=
O
υn+
Vn(lnVn)(εq+λ)+
=
εB(λ,λ)
lnε( +υ)
+εO() –O().
If there exists a positive constantK≤B(λ,λ) such that () is valid when replacing
B(λ,λ) byK, then, in particular, we haveεI<εKap,λbq,λ, namely
B(λ,λ)
lnε( +υ)
+εO() –O()
<K
lnε( +μ
)
+εO()
p
lnε( +υ
)
+εO()
q
It follows thatB(λ,λ)≤K (ε→+). Hence,K=B(λ,λ) is the best possible constant
factor of ().
Similarly to (), we still can find that
I≤Jbq,λ. ()
Hence, we can prove that the constant factorB(λ,λ) in () is the best possible.
Other-wise, we would reach a contradiction by () that the constant factor in () is not the best
possible.
Remark (i) It is evident that () and () are strengthened versions of the following equivalent Mulholland-type inequalities:
∞
n= ∞
m=
ambn
lnλ(UmVn)<B(λ,λ)ap,λbq,λ, ()
∞
n= υn+
Vn ln pλ–Vn
∞
m=
am
lnλ(UmVn) p p
<B(λ,λ)ap,λ, ()
where the constant factorB(λ,λ) is still the best possible.
(ii) Forλ= ,λ=q,λ=p,
θ=ϑ:=
psin(π/p)ln/p( +υ/) π[ + ln(+υ/)
ln(+μ/)] ,
θ=ϑ:=
qsin(π/p)ln/q( +μ/) π[ +ln(+μ/)
ln(+υ/)] ,
() reduces to the strengthened version of () as follows:
∞
m= ∞
n=
ambn
lnUmVn <
π
sin(π/p)
∞
m=
– ϑ
ln/pUm
Um
μm+
p–
apm
p
×
∞
n=
– ϑ
ln/qVn
Vn
υn+
q–
bqn
q
. ()
Forμi=υj= (i,j∈N), () reduces to the following strengthened Mulholland’s inequal-ity:
∞
m= ∞
n=
ambn
lnmn<
∞
m=
π
sin(π/p)–
ln√/
ln/pm
apm m–p
p
×
∞
n=
π
sin(π/p)–
ln√/
ln/qm
bqn n–q
q
, ()
Forp> ,λ–p(n) = υn+ Vn (lnVn)
pλ–, we define the following normed spaces:
lp,λ:=
a={am}∞m=;ap,λ<∞
,
lq,λ:=
b={bn}∞n=;bq,λ<∞
,
lp,–p
λ :=
c={cn}∞n=;cp,–p
λ <∞
.
Assuming thata={am}m∞=∈lp,λ, setting
c={cn}∞n=,cn:=
∞
m=
am
lnλ(UmVn), n∈N,
we can rewrite () as follows:
cp,–p
λ
<B(λ,λ)ap,λ<∞,
namelyc∈lp,–p
λ .
Definition Define a Mulholland-type operatorT :lp,λ →lp,–λp as follows: For any a={am}∞m=∈lp,λ, there exists a unique representationTa=c∈lp,–λp. Define the formal inner product ofTaandb={bn}∞n=∈lq,λas follows:
(Ta,b) :=
∞
n=
∞
m=
am
lnλ(UmVn)
bn. ()
Then we can rewrite () and () as follows:
(Ta,b) <B(λ,λ)ap,λbq,λ, ()
Tap,–p
λ <B(
λ,λ)ap,λ. ()
Define the norm of operatorTas follows:
T:= sup
a(=θ)∈lp,λ
Tap,–p
λ
ap,λ .
Then by () we findT ≤B(λ,λ). Since the constant factor in () is the best possible,
we have
T=B(λ,λ). ()
4 Some strengthened versions of the reverses In the following, we also set
λ(m) :=
–θ(λ,m)
(lnUm)p(–λ)–
Um–pμpm–+
,
λ(n) := –ϑ(λ,n)
(lnVn)q(–λ)– Vn–qυnq–+
m,n∈N\{}.
For <p< orp< , we still use the formal symbolsap,λ,bq,λ,ap,λ,bq,λ,
ap,λandbq,λ.
Theorem If <p< ,{μm}m∞=and{υn}∞n=are decreasing,U∞=V∞=∞,ap,λ∈R+
andbq,λ∈R+,then we have the following equivalent inequalities with the best possible
constant factor B(λ,λ):
∞
n= ∞
m=
ambn
lnλ(UmVn)
>B(λ,λ)ap,λbq,λ, ()
∞
n=
υn+lnpλ–Vn
( – θ
lnλVn) p–Vn
∞
m=
am
lnλ(U mVn)
p p
>B(λ,λ)ap,λ. ()
Proof Using () and () in the reverses of () and (), since
ω(λ,m)
p>B(λ
,λ)
p –θ(λ
,m)
p ( <p< ),
(λ,n)
q >B(λ
,λ)
q
– θ
lnλV n
q
(q< )
and
(B(λ,λ))p–( – θ
Vnλ
)p–>
( (λ,n))p–
( <p< ),
we obtain equivalent inequalities () and ().
Forε∈(,pλ), we setλ,λ,amandbnas (). Then, by (), () and (), we find
ap,λbq,λ
≥ ap,λbq,λ
=
∞
m=
–θ(λ,m)
μm+
Umln+εUm
p∞
n= υn+
Vnln+εVn
q
=
∞
m=
μm+
Umln+εUm –
∞
m=
O
μm+
Umln+λ+εUm
p
×
∞
n= υn+
Vnln+εVn
q
=
ε
lnε( +μ
)
+εO() –O()
p
lnε( +υ
)
+εO()
q
,
I:=
∞
n= ∞
m=
ambn
lnλ(UmVn)
=
∞
n=
∞
m=
lnλ(UmVn)
μm+lnλVn
Umln–λUm
υn+
=
∞
n=
(λ,n) υn+
Vnlnε+Vn≤B(λ,λ)
∞
n= υn+
Vnlnε+Vn
=
εB(λ,λ)
lnε( +υ)
+εO()
.
If there exists a positive constantK≥B(λ,λ) such that () is valid when replacing
B(λ,λ) byK, then, in particular, we haveεI>εKap,λbq,λ, namely
B(λ,λ)
lnε( +υ)
+εO()
>K
lnε( +μ)
+εO() –O()
p
lnε( +υ)
+εO()
q
.
It follows thatB(λ,λ)≥K (ε→+). Hence,K=B(λ,λ) is the best possible constant
factor of ().
The constant factorB(λ,λ) in () is still the best possible. Otherwise, we would reach
a contradiction by the reverse of () that the constant factor in () is not the best
possi-ble.
Remark It is evident that () and () are strengthened versions of the following equiv-alent inequalities:
∞
n= ∞
m=
ambn
lnλ(UmVn)>B(λ,λ)ap,λbq,λ, ()
∞
n= υn+
Vn ln pλ–V
n
∞
m=
am
lnλ(UmVn) p p
>B(λ,λ)ap,λ, ()
where the constant factorB(λ,λ) is still the best possible.
Theorem If p< ,{μm}∞m= and{υn}∞n= are decreasing,U∞=V∞=∞,ap,λ∈R+
andbq,λ∈R+,then we have the following equivalent inequalities with the best possible
constant factor B(λ,λ):
∞
n= ∞
m=
ambn
lnλ(U mVn)
>B(λ,λ)ap,λbq,λ, ()
J:=
∞
n=
υn+lnpλ–Vn
( –ϑ(λ,n))p–Vn
∞
m=
am
lnλ(U mVn)
p p
>B(λ,λ)ap,λ. ()
Proof Using () and () in the reverses of () and (), since
ω(λ,m)
p>B(λ
,λ)
p
– θ
Uλ m
p
(p< ),
(λ,n)
q >B(λ
,λ)
q –ϑ(λ
,n)
and
(B(λ,λ))p–( –ϑ(λ,n))p–
p
>
( (λ,n))p–
p
(p< ),
we obtain equivalent inequalities () and ().
Forε∈(,qλ), we setλ=λ+εq(> ),λ=λ–qε(∈(, )), and
am:=μm+ Um
lnλ–ε–Um=μm+ Um
lnλ–εp–Um,
bn=υn+ Vn ln
λ–Vn=υn+ Vn ln
λ–qε–Vn.
Then, by (), () and (), we have
ap,λbq,λ
≥ ap,λbq,λ
=
∞
m=
μm+
Umlnε+Um
p∞
n=
–ϑ(λ,n)
υn+
Vnlnε+Vn
q
=
∞
m=
μm+
Umlnε+Um
p∞
n= υn+
Vnlnε+Vn–
∞
n=
O
υn+
Vnln+(λ+ε)Vn
q
=
ε
lnε( +μ
)
+εO()
p
lnε( +υ
)
+εO() –O()
q
,
I=
∞
m= ∞
n=
ambn
lnλ(UmVn)
=
∞
m=
∞
n=
lnλUm
lnλ(UmVn)
υn+
Vn ln λ–V
n
μm+
Umlnε+Um
=
∞
m=
ω(λ,m)
μm+
Umlnε+Um ≤B(λ,λ)
∞
n=
μm+
Umlnε+Um
=
εB(λ,λ)
lnε( +μ)
+εO()
.
If there exists a positive constantK≥B(λ,λ) such that () is valid when replacing
B(λ,λ) byK, then, in particular, we haveεI>εKap,λbq,λ, namely
B(λ,λ)
lnε( +μ)
+εO()
>K
lnε( +μ
)
+εO()
p
lnε( +υ
)
+εO() –O()
q
.
It follows thatB(λ,λ)≥K (ε→+). Hence,K=B(λ,λ) is the best possible constant
Similarly to the reverse of (), we still find that
I≥Jbq,λ. ()
Hence the constant factorB(λ,λ) in () is still the best possible. Otherwise, we would
reach a contradiction by () that the constant factor in () is not the best possible.
Remark It is evident that () and () are strengthened versions of the following equiv-alent inequalities:
∞
n= ∞
m=
ambn
lnλ(UmVn)
>B(λ,λ)ap,λbq,λ, ()
∞
n=
υn+lnpλ–Vn
( –ϑ(λ,n))p–Vn
∞
m=
am
lnλ(U mVn)
p p
>B(λ,λ)ap,λ, ()
where the constant factorB(λ,λ) is still the best possible.
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. AW and QH participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.
Acknowledgements
The authors wish to express their thanks to the referees for their careful reading of the manuscript and for their valuable suggestions. This work is supported by the National Natural Science Foundation (No. 61370186), and 2013 Knowledge Construction Special Foundation Item of Guangdong Institution of Higher Learning College and University
(No. 2013KJCX0140).
Received: 5 August 2015 Accepted: 30 September 2015
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