On Hilbert type inequalities

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R E S E A R C H

Open Access

On Hilbert type inequalities

Chang-Jian Zhao

1*

_{and Wing-Sum Cheung}

2

*_{Correspondence:}

[email protected]; [email protected]

1_{Department of Mathematics, China}

Jiliang University, Hangzhou 310018, P.R. China

Full list of author information is available at the end of the article

Abstract

In the present paper we establish new inequalities similar to the extensions of Hilbert’s double-series inequality and also give their integral analogues. Our results provide some new estimates to these types of inequalities.

MSC: 26D15

Keywords: Hilbert’s inequality; Pachpatte’s inequality; Hölder’s inequality

1 Introduction

In recent years several authors have given considerable attention to Hilbert’s double-series inequality together with its integral version, inverse version, and various generalizations (see [–]). In this paper, we establish multivariable sum inequalities for the extensions of Hilbert’s inequality and also obtain their integral forms. Our results provide some new estimates to these types of inequalities.

The well-known classical extension of Hilbert’s double-series theorem can be stated as follows [, p.].

Theorem A If p,p> are real numbers such that _p_+_p_ ≥and <λ=  –_p_ –_p_ = 

q+ 

q ≤, where, as usual, qand qare the conjugate exponents of pand prespectively, then

∞

m=

∞

n= ambn (m+n)λ≤K

_∞

m= ap_m

/p_∞

n= bq_n

/p

, (.)

where K=K(p,p)depends on pand ponly.

In , Pachpatte [] established a new inequality similar to inequality (.) as follows:

Theorem A Let p, q, a(s), b(t), a(), b(),∇a(s)and∇b(t)be as in [], then

m

s=

n

t=

|a(s)||b(t)|

qsp–₊_ptq–≤



pqm

(p–)/p_n(q–)/q

_m

s=

(m–s+ )∇a(s)p

/p

×

_n

t=

(n–t+ )∇b(t)q

/q .

(.)

The integral analogue of inequality (.) is as follows [, p.].

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Theorem B Let p, q, p, qandλbe as in Theorem A. If f ∈Lp_(,_∞₎_{and g}_∈_Lq_(,_∞₎_{, then}

∞



∞



f(x)g(x)

(x+y)λ dx dy≤K

∞



fp(x)dx

/p ∞



gq(y)dy

/q

, (.)

where K=K(p,q)depends on p and q only.

In [], Pachpatte also established a similar version of inequality (.) as follows.

Theorem B Let p, q, f(s), g(t), f(), g(), f(s)and g(t)be as in [], then

x



y



|f(s)||g(t)|

qsp–₊_ptq–dt ds

≤ 

pqx

(p–)/p_y(q–)/q

x



(x–s)f(s)pds

/p y



(y–t)g(t)qdt

/q .

(.)

In the present paper we establish some new inequalities similar to Theorems A, A, B and B. Our results provide some new estimates to these types of inequalities.

2 Statement of results

Our main results are given in the following theorems.

Theorem . Let pi > be constants and _p

i + 

qi = . Let ai(si, . . . ,sni) be real-valued functions deﬁned for sji= , , . . . ,mji, where mji (i,j= , , . . . ,n) are natural numbers.

For convenience, we write ai(, . . . , ) =  and ai(,si, . . . ,sni) = ai(si, ,si, . . . ,sni) = · · ·=ai(si, . . . ,sn–,i, ) = . Deﬁne the operators ∇i by ∇iai(si, . . . ,sni) =ai(si, . . . ,sni) –

ai(si, . . . ,si–,i,sii– ,si+,i, . . . ,sni)for any function ai(si, . . . ,sni). Then

m

s=

· · · mn

sn=

m

s=

· · · mn

sn=

· · · mn

sn=

· · · mnn

snn=

n

i=|ai(si, . . . ,sni)| (n_i_=(si· · ·sni)/qi)

n i=/qi

≤M

n

i=

_m

ni

sni=

· · · mi

si=

n

j=

(mji–sji+ )∇n· · · ∇ai(si, . . . ,sni) pi

/pi

,

(.)

where

M=M(mi, . . . ,mni) =

n– n

i=

/pi

n

i=/pi–n n

i=

(mi· · ·mni)/qi.

Remark . Letai(si, . . . ,sni) change toai(si) in Theorem . and in view ofai() =  and ∇ai(si) =ai(si) –ai(si– ) for any functionai(si),i= , , . . . ,n, then

m

s=

m

s=

· · · mn

sn=

n i=|ai(si)| (n_i_=si/qi)

n

i=/qi ≤ ¯M

n

i=

_m_i

si=

(mi–si+ )∇ai(si) pi

/pi

, (.)

where

¯

M=M¯(m, . . . ,mn) =

n– n

i=



pi

n i=/pi–n

· n

i= m/qi

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Remark . Taking forn=  in Remark .. Ifp,p>  satisfy _p_ +_p_ ≥ and  <λ=

 –_p

– 

p = 

q + 

q ≤, then inequality (.) reduces to

m

s=

m

s=

|a(s)||a(s)|

(qs+qs)λ

≤ 

(λqq)λ m/q

 m

/q 

_m



s=

(m–s+ )∇a(s)

p

/p

(.)

×

_m



s=

(m–s+ )∇a(s)

p

/p

,

which is an interesting variation of inequality (.).

On the other hand, ifλ= , then 

p+ 

p = 

q+ 

q =  and sop=q,p=q. In this case

inequality (.) reduces to

m

s=

m

s=

|a(s)||a(s)| ps+qs

≤ 

pq

m(p–)/p

 m

(q–)/q 

_m_

s=

(m–s+ )∇a(s)

p

/p

×

_m_

s=

(m–s+ )∇a(s)

q

/q

.

This is just a similar version of inequality (.) in Theorem A.

Theorem . Let pi> be constants and _p_i+_q_i = . Let fi(τi, . . . ,τni)be real-valued nth

diﬀerentiable functions deﬁned on[,xi)×· · ·×[,xni), where≤xji≤tji, tji∈(,∞)and

i,j= , , . . . ,n. Suppose

fi(xi, . . . ,xni) =

xi



· · ·

xni



∂n ∂τi· · ·∂τni

fi(τi, . . . ,τni)dτi· · ·dτni,

then

t

 · · ·

tn



t

 · · ·

tn

 · · ·

tn

 · · ·

tnn



n i=(

xi  · · ·

xni  |

∂n

∂τi···∂τnifi(τi, . . . ,τni)|

pi_d_τ

i· · ·dτni)/pi (n_i_=(xi· · ·xni)/qi)

n i=/qi

dx· · ·dxndx· · ·dxn· · ·dxn· · ·dxnn (.)

≤N

n

i=

ti



· · ·

tni



n

j=

(tji–xji)

× ∂n

∂xi· · ·∂xni

fi(xi, . . . ,xni)

pidxi· · ·dxni

/pi

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where

N=N(ti, . . . ,tni) =

n– n

i=



pi

n i=/pi–n

· n

i=

(ti· · ·tni)/qi.

Remark . Letfi(xi, . . . ,xni) change tofi(si) in Theorem . and in view offi() = ,i=

, , . . . ,n, then

x



. . .

xn



n i=|f(si)| (n_i_=si/qi)

n

i=/qidsn· · ·ds

≤ ¯N

n

i=

xi



(xi–si)fi(si) pi

dsi

/pi

,

(.)

where

¯

N=N¯(x, . . . ,xn) =

n– n

i=



pi

n i=/pi–n

· n

i= x/qi

i .

Remark . Taking forn=  in Remark ., ifp,p>  are such that _p_ + _p_ ≥ and

 <λ=  –_p

– 

p = 

q+ 

q ≤, inequality (.) reduces to

x



x



|f(s)||f(s)|

(qs+qs)λ dsds

≤ 

(λqq)λ x/q

 x

/q 

x



(x–s)f(s)

p ds

/p

(.)

×

x



(x–s)f(s)pds

/p

,

which is an interesting variation of inequality (.).

On the other hand, ifλ= , then 

p+ 

p = 

q+ 

q =  and sop=q,p=q. In this case

inequality (.) reduces to

x



x



|f(s)||f(s)| ps+qs

≤hh pq

x(p–)/p

 x

(q–)/q 

x



(x–s)f(s)pds

/p

×

x



(x–s)f(s)

q ds

/q

.

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3 Proofs of results

Proof of Theorem . From the hypothesesai(, . . . , ) =ai(,si, . . . ,sni) =ai(si, ,si, . . . ,

sni) =· · ·=ai(si, . . . ,sn–,i, ) = , we have

ai(si, . . . ,sni)≤ sni

τni=

· · · si

τi=

_∇_n_{· · · ∇}_ai(τi, . . . ,τni). (.)

From the hypotheses of Theorem . and in view of Hölder’s inequality (see []) and inequality for mean [], we obtain

n

i=

ai(si, . . . ,sni)≤ n

i=

sni

τni=

· · · si

τi=

_∇_n_{· · · ∇}_ai(τi, . . . ,τni)

≤ n

i=

(si· · ·sni)/qi

_s_ni

τni=

· · · si

τi=

_∇_n_{· · · ∇}_ai(τi, . . . ,τni)pi

/pi

≤(

n

i=(si· · ·sni)/qi) n

i=/qi

(n–n_i_=/pi)n– n

i=/pi

× n

i=

_s_ni

τni=

· · · si

τi=

_∇_n_{· · · ∇}__a_i(τi, . . . ,τni) pi

/pi

.

(.)

Dividing both sides of (.) by (n_i_=(si· · ·sni)/qi) n

i=/qi _{and then taking sums over}_s

ji from  tomji (i,j= , , . . . ,n), respectively and then using again Hölder’s inequality, we obtain

m

s=

· · · mn

sn=

m

s=

· · · mn

sn=

· · · mn

sn=

· · · mnn

snn=

n

i=|∇n· · · ∇ai(si, . . . ,sni)| (n_i_=(si· · ·sni)/qi)

n i=/qi

≤

n– n

i=

/pi

n i=/pi–n

× n

i=

_m_ni

sni=

· · · mi

si=

_s_ni

τni=

· · · si

τi=

_∇_n_{· · · ∇}_ai(τi, . . . ,τni) pi

/pi

≤

n– n

i=

/pi

n i=/pi–n

× n

i=

(mi· · ·mni)/qi

_m_ni

sni=

· · · mi

si=

_s_ni

τni=

· · · si

τi=

_∇_n_{· · · ∇}_ai(τi, . . . ,τni)pi

/pi

=M

n

i=

_m_ni

τni=

· · · mi

τi=

n

j=

(mji–τji+ )∇n· · · ∇ai(τi, . . . ,τni) pi

/pi

=M

n

i=

_m

ni

sni=

· · · mi

si=

n

j=

(mji–sji+ )∇n· · · ∇ai(si, . . . ,sni) pi

/pi

.

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Proof of Theorem . From the hypotheses of Theorem ., we have

fi(xi, . . . ,xni)≤

xi



· · ·

xni



∂n

∂τi· · ·∂τni

fi(τi, . . . ,τni)

dτi· · ·dτni. (.)

On the other hand, by using Hölder’s integral inequality (see []) and the following inequality for mean [],

_n

i= λ/qi

i

/ni=/qi

≤n i=/qi

n

i=

λi/qi, λi> ,

we obtain

n

i=

_f_i₍_x__i_{, . . . ,}_x_ni₎

≤ n

i=

xi



· · · xni



_∂τ__i_{· · ·}∂n_∂τ_nifi(τi, . . . ,τni)

dτi· · ·dτni

≤ n

i=

(xi· · ·xni)/qi

×

xi



· · ·

xni



∂n

∂τi· · ·∂τni

fi(τi, . . . ,τni)

pidτi· · ·dτni

/pi

≤(

n

i=(xi· · ·xni)/qi) n

i=/qi

(n–n_i_=/pi)n– n

i=/pi

× n

i=

xi



· · ·

xni



_∂τ__i_{· · ·}∂n_∂τ_nifi(τi, . . . ,τni)

/pi

.

(.)

Dividing both sides of (.) by (n_i_=(xi· · ·xni)/qi) n

i=/qi_{and then integrating the result}

inequality overxjifrom  totji(i,j= , , . . . ,n), respectively and then using again Hölder’s integral inequality, we obtain

t



· · ·

tn



t



· · ·

tn



· · ·

tn



· · ·

tnn



n i=(

xi  · · ·

xni 

∂n

∂τi···∂τnifi(τi, . . . ,τni)

pidτi· · ·dτni)/pi (n_i_=(xi· · ·xni)/qi)

n i=/qi

dx· · ·dxndx· · ·dxn· · ·dxn· · ·dxnn

≤

n– n

i=

/pi

n i=/pi–n

× n

i=

ti



· · ·

tni



xi



· · ·

xni



_∂τ__i_{· · ·}∂n_∂τ_nifi(τi, . . . ,τni)

/pi

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≤

n– n

i=

/pi

n

i=/pi–n n

i=

(ti· · ·tni)/qi

×

ti



· · ·

tni



xi



· · ·

xni



_∂τ__i_{· · ·}∂n_∂τ_nifi(τi, . . . ,τni)

pidτi· · ·dτni

dxi· · ·dxni

/pi

=N

n

i=

ti



· · · tni



n

j=

(tji–xji)

_∂_x__i_{· · ·}∂n_∂_x_nifi(xi, . . . ,xni)

pidxi· · ·dxni

/pi

.

This concludes the proof.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

C-JZ and W-SC jointly contributed to the main results Theorems 2.1 and 2.2. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, China Jiliang University, Hangzhou 310018, P.R. China.}2_{Department of Mathematics, The}

University of Hong Kong, Pokfulam Road, Hong Kong, P.R. China.

Acknowledgement

CJZ is supported by National Natural Science Foundation of China (10971205). WSC is partially supported by a HKU URG grant. The authors express their grateful thanks to the referees for their many very valuable suggestions and comments.

Received: 12 January 2012 Accepted: 6 June 2012 Published: 22 June 2012 References

1. Pachpatte, BG: On some new inequalities similar to Hilbert’s inequality. J. Math. Anal. Appl.226, 166-179 (1998) 2. Handley, GD, Koliha, JJ, Peˇcari´c, JE: New Hilbert-Pachpatte type integral inequalities. J. Math. Anal. Appl.257, 238-250

(2001)

3. Gao, MZ, Yang, BC: On the extended Hilbert’s inequality. Proc. Am. Math. Soc.126, 751-759 (1998) 4. Kuang, JC: On new extensions of Hilbert’s integral inequality. J. Math. Anal. Appl.235, 608-614 (1999) 5. Yang, BC: On new generalizations of Hilbert’s inequality. J. Math. Anal. Appl.248, 29-40 (2000)

6. Zhao, CJ: On inverses of disperse and continuous Pachpatte’s inequalities. Acta Math. Sin.46, 1111-1116 (2003) 7. Zhao, CJ: Generalizations on two new Hilbert type inequalities. J. Math. (Wuhan)20, 413-416 (2000)

8. Zhao, CJ, Debnath, L: Some new inverse type Hilbert integral inequalities. J. Math. Anal. Appl.262, 411-418 (2001) 9. Handley, GD, Koliha, JJ, Peˇcari´c, JE: A Hilbert type inequality. Tamkang J. Math.31, 311-315 (2000)

10. Hardy, GH, Littlewood, JE, Pólya, G: Inequalities. Cambridge University Press, Cambridge (1934)

11. Pachpatte, BG: Inequalities similar to certain extensions of Hilbert’s inequality. J. Math. Anal. Appl.243, 217-227 (2000)

doi:10.1186/1029-242X-2012-145

Cite this article as:Zhao and Cheung:On Hilbert type inequalities.Journal of Inequalities and Applications2012