on a half-discrete reverse mulholland-type inequality and an extension

Liu et al. Journal of Inequalities and Applications 2014, 2014:103http://www.journalofinequalitiesandapplications.com/content/2014/1/103

RESEARCH Open Access

On a half-discrete reverse Mulholland-typeinequality and an extensionTuo Liu1, Bicheng Yang2 and Leping He1*

*Correspondence:[email protected] of Mathematics andStatistics, Jishou University, Hunan,Jishou 416000, P.R. ChinaFull list of author information isavailable at the end of the article

AbstractBy using the way of weight functions and the Hermite-Hadamard inequality,a half-discrete reverse Mulholland-type inequality with a best constant factor is given.The extension with multi-parameters, the equivalent forms as well as the relatinghomogeneous inequalities are also considered.MSC: 26D15

Keywords: Mulholland-type inequality; weight function; equivalent form; reverse

1 IntroductionAssuming that f , g ∈ L(R+), ‖f ‖ = {∫ ∞

f (x)dx} > , ‖g‖ > , we have the following

Hilbert integral inequality (cf. []):

∫ ∞

∫ ∞

f (x)g(y)x + y

dxdy < π‖f ‖‖g‖, ()

where the constant factor π is best possible. If a = {an}∞n=,b = {bn}∞n= ∈ l, ‖a‖ ={∑∞

n= an} > , ‖b‖ > , then we still have the following discrete Hilbert inequality:

∞∑m=

∞∑n=

ambnm + n

< π‖a‖‖b‖, ()

with the same best constant factor π . Inequalities () and () are important in analysisand its applications (cf. [–]). Also we have the followingMulholland inequality with thesame best constant factor π (cf. [, ]):

∞∑m=

∞∑n=

ambnlnmn

< π

{ ∞∑m=

mam∞∑n=

nbn

}

. ()

In , by introducing an independent parameter λ ∈ (, ], Yang [] gave an exten-sion of (). For generalizing the results from [], Yang [] gave some best extensions of() and () as follows. If p > ,

p + q = , λ + λ = λ, kλ(x, y) is a non-negative homo-

geneous function of degree –λ, k(λ) =∫ ∞ kλ(t, )tλ– dt ∈ R+, φ(x) = xp(–λ)–, ψ(x) =

xq(–λ)–, f (≥ ) ∈ Lp,φ(R+) = {f |‖f ‖p,φ := {∫ ∞ φ(x)|f (x)|p dx}

p < ∞}, g(≥ ) ∈ Lq,ψ (R+),

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Liu et al. Journal of Inequalities and Applications 2014, 2014:103 of 9http://www.journalofinequalitiesandapplications.com/content/2014/1/103

‖f ‖p,φ ,‖g‖q,ψ > , then

∫ ∞

∫ ∞

kλ(x, y)f (x)g(y)dxdy < k(λ)‖f ‖p,φ‖g‖q,ψ , ()

where the constant factor k(λ) is best possible. Moreover, if kλ(x, y) is finite andkλ(x, y)xλ–(kλ(x, y)yλ–) is strict decreasing for x > (y > ), then for am,bn ≥ , a ={am}∞m= ∈ lp,φ = {a|‖a‖p,φ := {∑∞

n= φ(n)|an|p}p < ∞}, b = {bn}∞n= ∈ lq,ψ , ‖a‖p,φ ,‖b‖q,ψ > ,

we have

∞∑m=

∞∑n=

kλ(m,n)ambn < k(λ)‖a‖p,φ‖b‖q,ψ , ()

with the same best constant factor k(λ). Clearly, for p = q = , λ = , k(x, y) = x+y , λ =

λ = , () reduces to (), while () reduces to ().

Some other results including the reverse Hilbert-type inequalities are provided by [–]. On 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 constantfactors in the inequalities are best possible. However, Yang [] gave a result by introducingan interval variable and proved that the constant factor is best possible. Recently, Yang []gave a half-discrete Hilbert inequality with multi-parameters, and [] gave the followinghalf-discrete reverse Hilbert-type inequality with the best constant factor : For < p < ,p +

q = , we have θ(x) ∈ (, ), and

∫ ∞

f (x)

∞∑n=

min{x,n}an dx

> {∫ ∞

– θ(x)x–

p

f p(x)dx}

p{ ∞∑

n=

aqnn–

q

} q

. ()

In this paper, by using the way of weight functions and the Hermite-Hadamard inequal-ity, a half-discrete reverse Mulholland-type inequality similar to () is given as follows:

∫ ∞

f (x)

∞∑n=

anln e(n +

)xdx

> π

{∫ ∞

– θ(x)x–

p

f p(x)dx}

p{ ∞∑

n=

(n + )

q–aqnln–

q (n +

)

} q

. ()

Moreover, a best extension of () with multi-parameters, the equivalent forms and therelating homogeneous inequalities are considered.

2 Some lemmasLemma If < λ ≤ , α ≥

, setting weight functions ω(n) and (x) as follows:

ω(n) := lnλ (n + α)

∫ ∞

lnλ e(n + α)x

x λ – dx, n ∈ N, ()



(x) := x λ

∞∑n=

lnλ –(n + α)

(n + α) lnλ e(n + α)x, x ∈ (,∞), ()

we have

B(

λ

,λ

)( – θλ(x)

)< (x) < ω(n) = B

(λ

,λ

), ()

where

θλ(x) =

B( λ ,

λ )

∫ x ln(+α)

t(λ/)–

( + t)λdt ∈ (, ),

satisfying θλ(x) =O(x λ ).

Proof Substituting of t = x ln(n + α) in (), by calculation, we have

ω(n) =∫ ∞

( + t)λ

t λ – dt = B

(λ

,λ

).

Since by the conditions and for fixed x >

h(x, y) := lnλ –(y + α)

(y + α) lnλ e(y + α)x=

lnλ –(y + α)

(y + α)[ + x ln(y + α)]λ

is strictly decreasing and strictly convex in y ∈ ( ,∞), then by the Hermite-Hadamardinequality (cf. []), we find

(x) < x λ

∫ ∞

lnλ –(y + α)

(y + α)[ + x ln(y + α)]λdy t=x ln(y+α)=

∫ ∞

x ln( +α)

t λ –

( + t)λdt ≤ B

(λ

,λ

),

(x) > x λ

∫ ∞

lnλ –(y + α)

(y + α)[ + x ln(y + α)]λdy

t=x ln(y+α)=∫ ∞

x ln(+α)

t λ – dt

( + t)λ= B

(λ

,λ

)( – θλ(x)

)> ,

< θλ(x) :=

B( λ ,

λ )

∫ x ln(+α)

t λ –

( + t)λdt <

B( λ

,λ )

∫ x ln(+α)

t λ – dt =

[x ln( + α)] λ

λB( λ ,

λ )

,

that is, () is valid. �

Lemma Let the assumptions of Lemma be fulfilled and, additionally, < p < , p +

q =

, an ≥ , n ∈ N, f (x) is a non-negative measurable function in (,∞). Then we have thefollowing inequalities:

J :={ ∞∑

n=

lnpλ –(n + α)n + α

[∫ ∞

f (x)lnλ e(n + α)x

dx]p

} p

≥[B(

λ

,λ

)] q{∫ ∞

(x)xp(– λ

)–f p(x)dx}

p, ()



L :={∫ ∞

xqλ –

[ (x)]q–

[ ∞∑n=

anlnλ e(n + α)x

]q

dx}

q

≥{B(

λ

,λ

) ∞∑n=

(n + α)q– lnq(–λ )–(n + α)aqn

} q

. ()

Proof By the reverse Hölder inequality (cf. []) and (), it follows that

[∫ ∞

f (x)dxlnλ e(n + α)x

]p

={∫ ∞

lnλ e(n + α)x

[x(– λ

)/q

ln(–λ )/p(n + α)

f (x)

(n + α)p

]

·[

ln(–λ )/p(n + α)x(– λ

)/q(n + α)

p

]dx

}p

≥∫ ∞

lnλ e(n + α)x

x(– λ )(p–)

(n + α) ln–λ (n + α)

f p(x)dx

·{∫ ∞

(n + α)q–

lnλ e(n + α)xln(–

λ )(q–)(n + α)x– λ

dx

}p–

={ω(n)(n + α)q– lnq(–

λ )–(n + α)

}p–·∫ ∞

lnλ e(n + α)x

x(– λ )(p–)

(n + α) ln–λ (n + α)

f p(x)dx

=[B(

λ

,λ

)]p–(n + α) ln–

pλ (n + α)

·∫ ∞

lnλ e(n + α)x

x(– λ )(p–)

(n + α) ln–λ (n + α)

f p(x)dx.

Then, by the Lebesgue term-by-term integration theorem (cf. []), we have

J ≥[B(

λ

,λ

)] q{ ∞∑

n=

∫ ∞

lnλ e(n + α)x

x(– λ )(p–)f p(x)dx

(n + α) ln–λ (n + α)

} p

=[B(

λ

,λ

)] q{∫ ∞

∞∑n=

x λ

lnλ e((n + α))xxp(– λ

)–f p(x)dx(n + α) ln–

λ (n + α)

} p

=[B(

λ

,λ

)] q{∫ ∞

(x)xp(– λ

)–f p(x)dx}

p,

and () follows. Still, by the reverse Hölder inequality, we have

[ ∞∑n=

anlnλ e(n + α)x

]q

={ ∞∑

n=

lnλ e(n + α)x

[x(– λ

)/q

ln(–λ )/p(n + α)

·

(n + α)p

][ln(–

λ )/p(n + α)x(– λ

)/q(n + α)

p an

]}q



≤{ ∞∑

n=

lnλ e(n + α)x

x(– λ )(p–)

(n + α) ln–λ (n + α)

}q–

·∞∑n=

(n + α)q–

lnλ e(n + α)xln(–

λ )(q–)(n + α)x– λ

aqn

=[ (x)]q–

xqλ –

∞∑n=

(n + α)q–

lnλ e(n + α)xx λ

– ln(–λ )(q–)(n + α)aqn.

Then, by the Lebesgue term-by-term integration theorem, we have

L ≥{∫ ∞

∞∑n=

(n + α)q–

lnλ e(n + α)xx λ

– ln(–λ )(q–)(n + α)aqn dx

} q

={ ∞∑

n=

[ln

λ (n + α)

∫ ∞

x λ – dx

lnλ e(n + α)x

](n + α)q– lnq(–

λ )–(n + α)aqn

} q

={ ∞∑

n=ω(n)(n + α)q– lnq(–

λ )–(n + α)aqn

} q

,

and then in view of (), inequality () follows. �

3 Main resultsIn this paper, for < p < (q < ), we still use the normal expressions ‖f ‖p, and ‖a‖q,� .We also introduce two functions

(x) :=( – θλ(x)

)xp(– λ

)– (x > ) and

�(n) := (n + α)q– lnq(–λ )–(n + α) (n ∈N),

wherefrom [(x)]–q = ( – θλ(x))–qxqλ – and [�(n)]–p = ln

pλ –(n+α)n+α

.

Theorem If < λ ≤ , α ≥ , < p < ,

p + q = , f (x),an ≥ , < ‖f ‖p, < ∞ and

< ‖a‖q,� <∞, then we have the following equivalent inequalities:

I :=∞∑n=

an∫ ∞


dx

=∫ ∞

f (x)

∞∑n=

anlnλ e(n + α)x

dx > B(

λ

,λ

)‖f ‖p,‖a‖q,� , ()

J ={ ∞∑

n=

[�(n)

]–p[∫ ∞


dx]p

} p

> B(

λ

,λ

)‖f ‖p,, ()

L :={∫ ∞

[(x)

]–q[ ∞∑n=

anlnλ e(n + α)x

]q

dx}

q

> B(

λ

,λ

)‖a‖q,� , ()

where the constant B( λ ,

λ ) is best possible.



Proof By the Lebesgue term-by-term integration theorem, there are two expressions forI in (). In view of (), for (x) > B( λ

,λ )( – θλ(x)), we have (). By the reverse Hölder

inequality, we have

I =∞∑n=

[�

–q (n)

∫ ∞

lnλ e(n + α)x

f (x)dx][

�q (n)an

] ≥ J‖a‖q,� . ()

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

an :=[�(n)

]–p[∫ ∞

lnλ e(n + α)x

f (x)dx]p–

, n ∈N,

we obtain that Jp– = ‖a‖q,� . By (), we find J > . If J = ∞, then () is trivially valid; ifJ <∞, then by () we have

‖a‖qq,� = Jp = I > B(

λ

,λ

)‖f ‖p,‖a‖q,� , i.e.,

‖a‖q–q,� = J > B(

λ

,λ

)‖f ‖p,,

that is, () is equivalent to (). In view of (), for

[ (x)

]–q > [B(

λ

,λ

)( – θλ(x)

)]–q,

we have (). By the reverse Hölder inequality, we find

I =∫ ∞

[

p (x)f (x)

][

–p (x)

∞∑n=

lnλ e(n + α)x

an

]dx ≥ ‖f ‖p,L. ()

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

f (x) :=[(x)

]–q[ ∞∑n=

lnλ e(n + α)x

an

]q–

, x ∈ (,∞),

we obtain that Lq– = ‖f ‖p,. By (), we find L > . If L = ∞, then () is trivially valid; ifL <∞, then by () we have

‖f ‖pp, = Lq = I > B(

λ

,λ

)‖f ‖p,‖a‖q,� , i.e.,

‖f ‖p–p, = L > B(

λ

,λ

)‖a‖q,� ,

that is, () is equivalent to (). Hence, inequalities (), () and () are equivalent.For < ε < pλ

, set f (x) = xλ +

εp–, x ∈ (, ); f (x) = , x ∈ [,∞), and

an =

n + αln

λ –

εq–(n + α), n ∈N.



If there exists a positive number k (≥ B( λ ,

λ )) such that () is valid when replacing B(

λ ,

λ )

with k, then, in particular, it follows that

I :=∞∑n=

∫ ∞

lnλ e(n + α)x

anf (x)dx > k‖f ‖p,‖a‖q,�

= k{∫

( –O

(x λ

)) dxx–ε+

} p{

( + α) lnε+( + α)

+∞∑n=

(n + α) lnε+(n + α)

} q

> k{ε–O()

} p{

( + α) lnε+( + α)

+∫ ∞

dx(x + α) lnε+(x + α)

} q

=kε

{ – εO()

} p

{ε

( + α) lnε+( + α)+

lnε( + α)

} q, ()

I =∞∑n=

n + α

lnλ –

εq–(n + α)

∫

lnλ e(n + α)x

xλ +

εp– dx

t=x ln(n+α)=∞∑n=

(n + α) lnε+(n + α)

∫ ln(n+α)

(t + )λ

tλ +

εp– dt

≤ B(

λ

+

ε

p,λ

–

ε

p

)[

( + α) lnε+( + α)+

∞∑n=

(n + α) lnε+(n + α)

]

< B(

λ

+

ε

p,λ

–

ε

p

)[

( + α) lnε+( + α)+

∫ ∞

(y + α) lnε+(y + α)

dy]

=εB(

λ

+

ε

p,λ

–

ε

p

)[ε

( + α) lnε+( + α)+

lnε( + α)

]. ()

Hence by () and () it follows that

B(

λ

+

ε

p,λ

–

ε

p

)[ε

( + α) lnε+( + α)+

lnε( + α)

]

> k{ – εO()

} p

{ε

( + α) lnε+( + α)+

lnε( + α)

} q,

and B( λ ,

λ )≥ k(ε → +). Hence k = B( λ

,λ ) is the best value of ().

By the equivalence, the constant factorB( λ ,

λ ) in () and () is best possible. Otherwise

we would reach a contradiction by () and () that the constant factor in () is not bestpossible. �

Remark (i) For λ = , λ = λ = , α =

in (), () and (), we have () and the fol-lowing equivalent inequalities:

∞∑n=

lnp –(n +

)n +

[∫ ∞

f (x)ln e(n +

)xdx

]p

> πp∫ ∞

– θ(x)x–

p

f p(x)dx, ()



∫ ∞

( – θ(x))–q

x–q

[ ∞∑n=

anln e(n +

)x

]q

dx

< πq∞∑n=

(n + )

q–

ln–q (n +

)aqn. ()

(ii) Setting x = ln y , g(y) :=

y (ln y)

λ–f ( ln y ) and

φ(y) :=( – θλ

(

ln y

))yp–(ln y)p(– λ

)–(y ∈ (,∞)

)in (), by simplifications, we find the following inequality with the homogeneous kernel:

∞∑n=

an∫ ∞

g(y)lnλ y(n + α)

dy

=∫ ∞

g(y)

∞∑n=

anlnλ y(n + α)

dx > B(

λ

,λ

)‖g‖p,φ‖a‖q,� . ()

It is evident that () is equivalent to (), and then the constant factor B( λ ,

λ ) in () is

still best possible. In the same way as in () and (), we have the following inequalitiesequivalent to () with the best constant factor B( λ

,λ ):

{ ∞∑n=

[�(n)

]–p[∫ ∞

g(y)lnλ y(n + α)

dy]p

} p

> B(

λ

,λ

)‖g‖p,φ , ()

{∫ ∞

[φ(y)

]–q[ ∞∑n=

anlnλ y(n + α)

]q

dy}

q

> B(

λ

,λ

)‖a‖q,� . ()

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsTL participated in the design of the study and performed the numerical analysis. BY carried out the mathematical studies,participated in the sequence alignment and drafted the manuscript. LH conceived of the study, and participated in itsdesign and coordination. All authors read and approved the final manuscript.

Author details1College of Mathematics and Statistics, Jishou University, Hunan, Jishou 416000, P.R. China. 2Department of Mathematics,Guangdong University of Education, Guangzhou, Guangdong 510303, P.R. China.

AcknowledgementsThis work is supported by the National Natural Science Foundation of China (No. 61370186).

Received: 19 November 2013 Accepted: 6 February 2014 Published: 04 Mar 2014

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10.1186/1029-242X-2014-103Cite this article as: Liu et al.: On a half-discrete reverse Mulholland-type inequality and an extension. Journal ofInequalities and Applications 2014, 2014:103


on a half-discrete reverse mulholland-type inequality and an extension

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