A good and scientific analysis starts with a closer look at the conceptualisation at hand. The definition of CIP is not easy because of its wide range. This paper examines infrastructures that are critical and need protection. Each word entails a specific connotation and is characterized by several components

Cools, Marc

Pashley, Veerle

Journal of Inequalities and Applications

English

Ghent University Academic Bibliography

The power of creative thinking in situations of uncertainties: the almost impossible task of protecting critical infrastructures

We establish a newinequality of Hilbert type for a finite double number ofnonnegative sequences of real numbers and some interrelatedresults, which are inverse and general forms of Pachpatte's andHandley's results. An integral version and some interrelatedresults are also obtained. These results provide some newestimates on such types of inequalities.published_or_final_versio

Zhao, CJ

Cheung, WS

HKU Scholars Hub

Title Inverses of new Hilbert-Pachpatte-type inequalitiesAuthor(s) Zhao, CJ; Cheung, WSCitation Journal Of Inequalities And Applications, 2006, v. 2006, articleno. 97860Issued Date 2006URL http://hdl.handle.net/10722/156173Rights Creative Commons: Attribution 3.0 Hong Kong LicenseINVERSES OF NEW HILBERT-PACHPATTE-TYPEINEQUALITIESCHANG-JIAN ZHAO AND WING-SUM CHEUNGReceived 7 February 2006; Revised 3 June 2006; Accepted 5 June 2006We establish a new inequality of Hilbert type for a finite double number of nonnegativesequences of real numbers and some interrelated results, which are inverse and generalforms of Pachpatte’s and Handley’s results. An integral version and some interrelatedresults are also obtained. These results provide some new estimates on such types of in-equalities.Copyright © 2006 C.-J. Zhao and W.-S. Cheung. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, dis-tribution, and reproduction in any medium, provided the original work is properly cited.1. IntroductionThe various generalizations and sharpenings of Hilbert’s double series inequality and itsintegral version are obtained by Pachpatte, Handley et al., Gao et al. (see [1, 3, 5, 7, 10,12, 16–18]). An elegant survey on this kind of inequalities was provided by Yang andRassias (see [13]). Moreover, Pachpatte [9] established a new Hilbert-type inequality andits integral version as follows.Theorem 1.1. Let {am}, {bn} be two nonnegative sequences of real numbers defined form=1,2, . . . ,k and n= 1,2, . . . ,r with a0 = b0 = 0, and let {pm}, {qn} be two positive sequencesof real numbers defined for m = 1,2, . . . ,k, n = 1,2, . . . ,r, where k, r are natural numbers.Define Pm =∑ms=1 ps andQn =∑nt=1 qt. Let φ and ψ be two real-valued nonnegative, convex,and submultiplicative functions defined on R+ = [0,∞). Thenk∑m=1r∑n=1φ(am)ψ(bn)m+n≤M(k,r)( k∑m=1(k−m+1)(pmφ(∇ampm))2)1/2×( r∑n=1(r−n+1)(qnψ(∇bnqn))2)1/2,(1.1)Hindawi Publishing CorporationJournal of Inequalities and ApplicationsVolume 2006, Article ID 97860, Pages 1–11DOI 10.1155/JIA/2006/978602 Inverses of new Hilbert-Pachpatte-type inequalitieswhereM(k,r)= 12( k∑m=1(φ(Pm)Pm)2)1/2( r∑n=1(φ(Qn)Qn)2)1/2, (1.2)and∇am = am− am−1,∇bn = bn− bn−1.Theorem 1.2. Let f ∈ C1[[0,x),R+], g ∈ C1[[0, y),R+] with f (0) = g(0) = 0 and letp(σ), q(τ) be two positive functions defined for σ ∈ [0,x) and τ ∈ [0, y). Let P(s) =∫ s0 p(σ)dσ and Q(t) =∫ t0 q(τ)dτ for s ∈ [0,x) and t ∈ [0, y), where x, y are positive realnumbers. Let φ, and ψ be as in Theorem 1.1. Then∫ x0∫ y0φ(f (s))ψ(g(t))s+ tdsdt ≤ L(x, y)(∫ x0(x− s)(p(s)φ(f ′(s)p(s)))2ds)1/2×(∫ y0(y− t)(q(y)φ(g′(t)q(t)))2dt)1/2,(1.3)whereL(x, y)= 12(∫ x0(φ(P(s))P(s))2ds)1/2(∫ y0(φ(Q(t))Q(t))2dt)1/2, (1.4)and ′ denotes the derivative of a function.In [4], Handley et al. gave general versions of inequalities (1.1) and (1.3) as follows.Theorem 1.3. Let {ai,mi} (i= 1,2, . . . ,n) be n sequences of nonnegative real numbers definedfor mi = 1,2, . . . ,ki with a1,0 = a2,0 = ··· = an,0 = 0, and let {pi,mi} be n sequences of posi-tive real numbers defined for mi = 1,2, . . . ,ki, where ki (i= 1,2, . . . ,n) are natural numbers.Set Pi,mi =∑misi=1 pi,si . Let φi (i = 1,2, . . . ,n) be n real-valued nonnegative convex and sub-multiplicative functions defined on R+. Let αi ∈ (0,1), and set α′i = 1− αi (i = 1,2, . . . ,n),α=∑ni=1αi, and α′ =∑ni=1 α′i = n−α. Thenk1∑m1=1···kn∑mn=1∏ni=1φi(ai,mi)(∑ni=1α′imi)α′≤M(k1, . . . ,kn) n∏i=1( ki∑mi=1(ki−mi +1)(pi,miφi(∇ai,mipi,mi))1/αi)αi,(1.5)whereM(k1, . . . ,kn)= 1(α′)α′n∏i=1( ki∑mi=1(φi(Pi,mi)Pi,mi)1/α′i)α′i, (1.6)and∇ai,mi = ai,mi − ai,mi−1 (i= 1,2, . . . ,n).C.-J. Zhao and W.-S. Cheung 3Theorem 1.4. Let fi ∈ C1[[0,ki],R+], i = 1,2, . . . ,n, with fi(0) = 0 (i = 1,2, . . . ,n), letpi(σi) be n positive functions defined for σi ∈ [0,xi] (i= 1, . . . ,n). Set Pi(si)=∫ si0 pi(σi)dσi forsi ∈ [0,xi], where xi are positive real numbers. Let φi, αi, α′i , α, and α′ be as in Theorem 1.3.Then∫ x10···∫ xn0∏ni=1φi(fi(si))(∑ni=1α′i si)α′ ds1 ···dsn≤ L(x1, . . . ,xn) n∏i=1(∫ xi0(xi− si)(pi(si)φi(f ′i(si)pi(si)))1/αidsi)αi,(1.7)whereL(x1, . . . ,xn)= 1(α′)α′n∏i=1(∫ xi0(φi(Pi(si))Pi(si))1/α′idsi)α′i. (1.8)The main purpose of the present paper is to establish their reverse versions, whichare more extensive results for this type of inequalities. Our main results are given in thefollowing theorems.Theorem 1.5. Let {asi,ti,msi ,mti } (i= 1,2, . . . ,n) be n sequences of nonnegative numbers de-fined for msi = 1,2, . . . ,ksi , mti = 1,2, . . . ,kti , with asi,ti,0,mti = asi,ti,msi ,0 = 0 (i = 1,2, . . . ,n),where ksi and kti (i = 1,2, . . . ,n) are natural numbers. Let {psi,ti,msi ,mti } be n sequences ofpositive real numbers defined for msi , mti . SetPsi,ti,msi ,mti =msi∑mξi=1mti∑mηi=1psi,ti,mξi ,mηi (i= 1,2, . . . ,n). (1.9)Let φi (i = 1,2, . . . ,n) be n real-valued nonnegative concave and supermultiplicative func-tions defined on R+. Let αi ∈ (1,∞). Set α′i = 1− αi (i = 1,2, . . . ,n), α =∑ni=1αi, and α′ =∑ni=1 α′i = n−α. Define operators∇1asi,ti,msi ,mti = asi,ti,msi ,mti − asi,ti,msi−1,mti ,∇2asi,ti,msi ,mti = asi,ti,msi ,mti − asi,ti,msi ,mti−1,∇2∇1asi,ti,msi ,mti =∇2(∇1asi,ti,msi ,mti).(1.10)Thenks1∑ms1=1kt1∑mt=1···ksn∑msn=1ktn∑mtn=1∏ni=1φi(asi,ti,msi ,mti)((1/α′)∑ni=1α′imsimti)α′≥ C(ks1kt1 , . . . ,ksnktn)×n∏i=1( ksi∑msi=1kti∑mti=1(ksi−msi+1)(kti−mti+1)(psi,ti,msi ,mti φi(∇2∇1asi,ti,msi ,mtipsi,ti,msi ,mti))1/αi)αi,(1.11)4 Inverses of new Hilbert-Pachpatte-type inequalitieswhereC(ks1kt1 , . . . ,ksnktn)=n∏i=1( ksi∑msi=1kti∑mti=1(φi(Psi,ti,msi ,mti)Psi,ti,msi ,mti)1/α′i)α′i. (1.12)Theorem 1.6. Let fi(si, ti) (i = 1,2, . . . ,n) be real-valued continuous functions defined on[0,xi)×[0, yi), where xi ∈ (0,∞), yi ∈ (0,∞), and with fi(0, ti)= fi(si,0)= 0 (i=1,2, . . . ,n).Let pi(σi) and qi(τi) be positive continuous functions defined for σi ∈ (0,si), τi ∈ (0, ti). SetPi(si, ti)=∫ ti0∫ si0pi(σi)qi(τi)dσidτi. (1.13)For the function fi(si, ti), denote the partial derivatives (∂/∂si) fi(si, ti), (∂/∂ti) fi(si, ti), and(∂2/∂si∂ti) fi(si, ti) by D1 fi(si, ti), D2 fi(si, ti), and D2D1 fi(si, ti)=D1D2 fi(si, ti), respectively.Let φi, αi, α′i , α, and α′ be as in Theorem 1.5. Then∫ x10∫ y10···∫ xn0∫ yn0n∏i=1φi(fi(si, ti))((1/α′)∑ni=1α′i siti)α′ ds1dt1 ···dsndtn=G(x1y1, . . . ,xnyn)×n∏i=1(∫ xi0∫ yi0(xi− si)(yi− ti)(pi(si)qi(ti) ·φi(D2D1 fi(si, ti)pi(si)qi(ti)))1/αidsi dti)αi,(1.14)whereG(x1y1, . . . ,xnyn)=n∏i=1(∫ xi0∫ yi0(φi(Pi(si, ti))Pi(si, ti))1/α′idsi dti)α′i. (1.15)These results provide new estimates on these types of inequalities.2. Proofs and remarksProof of Theorem 1.5. From the operators defined in Theorem 1.5, we obtainasi,ti,msi ,mti =msi∑mξi=1{asi,ti,mξi ,mti − asi,ti,mξi−1,mti}=msi∑mξi=1mti∑mηi=1{asi,ti,mξi ,mηi − asi,ti,mξi−1,mηi − asi,ti,mξi ,mηi−1 + asi,ti,mξi−1,mηi−1}=msi∑mξi=1mti∑mηi=1∇2∇1asi,ti,mξi ,mηi .(2.1)C.-J. Zhao and W.-S. Cheung 5From (2.1) and the hypotheses of Theorem 1.5, and in view of Jensen’s inequality andinverse Ho¨lder’s inequality [6], we obtainφi(asi,ti,msi ,mti)= φi(Psi,ti,msi ,mti∑msimξi=1∑mtimηi=1 psi,ti,mξi ,mηi(∇2∇1asi,ti,mξi ,mηi/psi,ti,mξi ,mηi)∑msimξi=1∑mtimηi=1 psi,ti,mξi ,mηi)≥ φi(Psi,ti,msi ,mti) ·φi(∑msimξi=1∑mtimηi=1 psi,ti,mξi ,mηi(∇2∇1asi,ti,mξi ,mηi/psi,ti,mξi ,mηi)∑msimξi=1∑mtimηi=1 psi,ti,mξi ,mηi)≥ φi(Psi,ti,msi ,mti)Psi,ti,msi ,mti( msi∑mξi=1mti∑mηi=1psi,ti,mξi ,mηi φi(∇2∇1asi,ti,mξi ,mηipsi,ti,mξi ,mηi))≥ φi(Psi,ti,msi ,mti)Psi,ti,msi ,mti(msimti)α′i ×( msi∑mξi=1mti∑mηi=1(psi,ti,mξi ,mηi φi(∇2∇1asi,ti,mξi ,mηipsi,ti,mξi ,mηi))1/αi)αi.(2.2)On the other hand, noticing α′i < 0 (i= 1,2, . . . ,n), λi > 0 (i= 1,2, . . . ,n), and applying thewell-known means inequality, we haven∏i=1λα′ii ≥(1α′n∑i=1α′i λi)α′. (2.3)Hence∏ni=1φi(asi,ti,msi ,mti)((1/α′)∑ni=1α′imsimti)α′≥n∏i=1φi(Psi,ti,msi ,mti)Psi,ti,msi ,mti( msi∑mξi=1mti∑mηi=1(psi,ti,mξi ,mηi φi(∇2∇1asi,ti,mξi ,mηipsi,ti,mξi ,mηi))1/αi)αi.(2.4)Taking the sum of both sides of (2.4) overmsi ,mti (i= 1,2, . . . ,n) from 1 to ksi , ksi , respec-tively, and in view of inverse Ho¨lder’s inequality, we getks1∑ms1=1kt1∑mt=1···ksn∑msn=1ktn∑mtn=1∏ni=1φi(asi,ti,msi ,mti)((1/α′)∑ni=1α′imsimti)α′≥n∏i=1ksi∑msi=1kti∑mti=1(φi(Psi,ti,msi ,mti)Psi,ti,msi ,mti( msi∑mξi=1mti∑mηi=1(psi,ti,mξi ,mηi φi(∇2∇1asi,ti,mξi ,mηipsi,ti,mξi ,mηi))1/αi)αi)6 Inverses of new Hilbert-Pachpatte-type inequalities≥n∏i=1( ksi∑msi=1kti∑mti=1(φi(Psi,ti,msi ,mti)Psi,ti,msi ,mti)1/α′i)α′i×( ksi∑msi=1kti∑mti=1( msi∑mξi=1mti∑mηi=1(psi,ti,mξi ,mηi φi(∇2∇1asi,ti,mξi ,mηipsi,ti,mξi ,mηi))1/αi))αi= C(ks1kt1 , . . . ,ksnktn)×n∏i=1( ksi∑mξi=1kti∑mηi=1(ksi−mξi+1)(kti−mηi+1)(psi,ti,mξi ,mηi φi(∇2∇1asi,ti,mξi ,mηipsi,ti,mξi ,mηi))1/αi)αi= C(ks1kt1 , . . . ,ksnktn)×n∏i=1( ksi∑msi=1kti∑mti=1(ksi−msi+1)(kti −mti+1)(psi,ti,msi ,mti φi(∇2∇1asi,ti,msi ,mtipsi,ti,msi ,mti))1/αi)αi.(2.5)This completes the proof. Remark 2.1. Let asi,ti,msi ,mti ,msi ,mti , ksi , kti , asi,ti,0,mti , asi,ti,msi ,0, Psi,ti,msi ,mti ,∇1, and∇2 be asin Theorem 1.5. Let φi, αi, α′i , α, and α′ (i = 1,2, . . . ,n) be as in Theorem 1.3. Similar tothe proof of Theorem 1.5, we obtainks1∑ms1=1kt1∑mt=1···ksn∑msn=1ktn∑mtn=1∏ni=1φi(asi,ti,msi ,mti)(∑ni=1α′imsimti)α′≤ C(ks1kt1 , . . . ,ksnktn)×n∏i=1( ksi∑msi=1kti∑mti=1(ksi−msi+1)(kti −mti+1)(psi,ti,msi ,mti φi(∇2∇1asi,ti,msi ,mtipsi,ti,msi ,mti))1/αi)αi,(2.6)whereC(ks1kt1 , . . . ,ksnktn)= 1(α′)α′n∏i=1( ksi∑msi=1kti∑mti=1(φi(Psi,ti,msi ,mti)Psi,ti,msi ,mti)1/α′i)α′i. (2.7)This is just a general form of inequality (1.5) and the inverse form of inequality (1.11).C.-J. Zhao and W.-S. Cheung 7Moreover, let {ai,mi},mi, ki, ai,0, pi,mi ,∇, and Pi,mi (i= 1,2, . . . ,n) be as in Theorem 1.3.Let φi, αi, α′i , α, and α′ (i = 1,2, . . . ,n) be as in Theorem 1.5. Similar to the proof ofTheorem 1.5, we obtaink1∑m1=1···kn∑mn=1∏ni=1φi(ai,mi)((1/α′)∑ni=1α′imi)α′≥M(k1, . . . ,kn) n∏i=1( ki∑mi=1(ki−mi +1)(pi,miφi(∇ai,mipi,mi))1/αi)αi,(2.8)whereM(k1, . . . ,kn)=n∏i=1( ki∑mi=1(φi(Pi,mi)Pi,mi)1/α′i)α′i. (2.9)This is just an inverse form of inequality (1.5) which was given by Handley et al. [4].Let n= 2, α1 = α2 = 2, then α′1 = α′2 =−1, putting them in inequality (2.8), we havek1∑m1=1k2∑m2=1φ1(a1,m1)φ2(a2,m2)(m1 +m2)−2≥M(k1,k2)( k1∑m1=1(k1−m1 + 1)(p1,miφ1(∇a1,m1p1,m1))1/2)2×( k2∑m2=1(k2−m2 + 1)(p2,m2φ2(∇a2,m2p2,m2))1/2)2,(2.10)whereM(k1,k2)= 4( k1∑m1=1(φ1(P1,m1)P1,m1)−1)−1( k2∑m2=1(φ2(P2,m2)P2,m2)−1)−1. (2.11)This is just an inverse form of inequality (1.1) which was given by Pachpatte [9].Proof of Theorem 1.6. From the hypotheses of Theorem 1.6, we obtainfi(si, ti)=∫ si0∫ ti0D2D1 fi(σi,τi)dσidτi. (2.12)8 Inverses of new Hilbert-Pachpatte-type inequalitiesFrom (2.12) and using Jensen’s integral inequality and Ho¨lder’s integral inequality [6],we obtainφi(fi(si, ti))= φi(Pi(si, ti)∫ ti0∫ si0 pi(σi)qi(τi)(D2D1 fi(σi,τi)/pi(σi)qi(τi))dσi dτi∫ ti0∫ si0 pi(σi)qi(τi)dσi dτi)≥ φi(Pi(si, ti)) ·φi(∫ ti0∫ si0 pi(σi)qi(τi)(D2D1 fi(σi,τi)/pi(σi)qi(τi))dσi dτi∫ ti0∫ si0 pi(σi)qi(τi)dσi dτi)≥ φi(Pi(si, ti))Pi(si, ti)∫ ti0∫ si0pi(σi)qi(τi) ·φi(D2D1 fi(σi,τi)pi(σi)qi(τi))dσi dτi≥ φi(Pi(si, ti))Pi(si, ti)(siti)α′i(∫ ti0∫ si0(pi(σi)qi(τi)·φi(D2D1 fi(σi,τi)pi(σi)qi(τi)))1/αidσi dτi)αi.(2.13)From the well-known inequality for means, we haven∏i=1φi(fi(si, ti))((1/α′)∑ni=1α′i siti)α′≥n∏i=1φi(Pi(si, ti))Pi(si, ti)(∫ ti0∫ si0(pi(σi)qi(τi) ·φi(D2D1 fi(σi,τi)pi(σi)qi(τi)))1/αidσi dτi)αi.(2.14)Integrating both sides of (2.14) over si, ti from 1 to xi, yi (i = 1,2, . . . ,n) and in view ofHo¨lder’s integral inequality and Fubini’s theorem, we observe that∫ x10∫ y10···∫ xn0∫ yn0n∏i=1φi(fi(si, ti))((1/α′)∑ni=1α′i siti)α′ ds1dt1 ···dsndtn≥n∏i=1∫ xi0∫ yi0φi(Pi(si, ti))Pi(si, ti)×(∫ ti0∫ si0(pi(σi)qi(τi) ·φi(D2D1 fi(σi,τi)pi(σi)qi(τi)))1/αidσi dτi)αidsi dti≥n∏i=1(∫ xi0∫ yi0(φi(Pi(si, ti))Pi(si, ti))1/α′idsi dti)α′i×(∫ xi0∫ yi0(∫ ti0∫ si0(pi(σi)qi(τi) ·φi(D2D1 fi(σi,τi)pi(σi)qi(τi)))1/αidσidτi)dsi dti)αiC.-J. Zhao and W.-S. Cheung 9=G(x1y1, . . . ,xnyn)×n∏i=1(∫ xi0∫ yi0(xi− si)(yi− ti)(pi(si)qi(ti) ·φi(D2D1 fi(si, ti)pi(si)qi(ti)))1/αidsi dti)αi.(2.15)The proof is complete. Remark 2.2. Let fi(si, ti), xi, yi, fi(0, ti), fi(si,0),D1 fi(si, ti),D2 fi(si, ti),D2D1 fi(si, ti), pi(σi),qi(τi), and Pi(si, ti) be as in Theorem 1.6. Let φi, αi, α′i , α, and α′ be as in Theorem 1.3.Similar to the proof of Theorem 1.6, we have∫ x10∫ y10···∫ xn0∫ yn0n∏i=1φi(fi(si, ti))(∑ni=1α′i siti)α′ ds1dt1 ···dsndtn≤G(x1y1, . . . ,xnyn)×n∏i=1(∫ xi0∫ yi0(xi− si)(yi− ti)(pi(si)qi(ti) ·φi(D2D1 fi(si, ti)pi(si)qi(ti)))1/αidsi dti)αi,(2.16)whereG(x1y1, . . . ,xnyn)= 1(α′)α′n∏i=1(∫ xi0∫ yi0(φi(Pi(si, ti))Pi(si, ti))1/α′idsi dti)α′i. (2.17)This is just a general form of inequality (1.7) and an inverse form of the inequality(1.14).On the other hand, let fi, fi(0), f ′i (si), pi(σi), σi, ki, si, xi, Pi(si) be as in Theorem 1.4,and let φi, αi, α′i , α, α′ be as in Theorem 1.5. Then∫ x10···∫ xn0∏ni=1φi(fi(si))(1/α′)(∑ni=1α′i si)α′ ds1 ···dsn≥ L(x1, . . . ,xn) n∏i=1(∫ xi0(xi− si)(pi(si)φi(f ′i(si)pi(si)))1/αidsi)αi,(2.18)whereL(x1, . . . ,xn)=n∏i=1(∫ xi0(φi(Pi(si))Pi(si))1/α′idsi)α′i. (2.19)This is just an inverse form of inequality (1.7) which was given by Handley et al. [4].10 Inverses of new Hilbert-Pachpatte-type inequalitiesLet n= 2, α1 = α2 = 2, then α′1 = α′2 =−1, taking them in inequality (2.18), we have∫ x10∫ x20φ1(f1(s1))φ2(f2(s2))(s1 + s2)−2 ds1ds2≥ L(x1,x2)(∫ x10(x1− s1)(p1(s1)φ1(f ′2(s1)p1(s1)))1/2ds1)2×(∫ x20(x2− s2)(p2(s2)φ2(f ′2(s2)p2(s2)))1/2ds2)2,(2.20)whereL(x1,x2)= 4(∫ x10(φ1(P1(s1))P1(s1))−1ds1)−1(∫ x20(φ2(P2(s2))P2(s2))−1ds2)−1. (2.21)This is just an inverse form of inequality (1.3) which was given by Pachpatte [9].For interrelated research on similar inequalities, one is referred to [2, 8, 11, 14, 15],and the references cited therein.AcknowledgmentsResearch of the first author is supported by Zhejiang Provincial Natural Science Foun-dation of China (Y605065), National Natural Sciences Foundation of China (10271071),Foundation of the Education Department of Zhejiang Province of China (20050392),and the Academic Mainstay of Middle-age and Youth Foundation of Shandong Provinceof China (200203). Research of the second author is partially supported by the ResearchGrants Council of the Hong Kong SAR, China (Project no. HKU7040/03P, and HKU7017/05P).References[1] W.-S. Cheung, Zˇ. Hanjsˇ, and J. Pecˇaric´, Some Hardy-type inequalities, Journal of MathematicalAnalysis and Applications 250 (2000), no. 2, 621–634.[2] W.-S. Cheung and Q.-H. Ma, On certain new Gronwall-Ou-Iang type integral inequalities in twovariables and their applications, Journal of Inequalities and Applications 2005 (2005), no. 4, 347–361.[3] M. Gao and B. Yang, On the extended Hilbert’s inequality, Proceedings of the American Mathe-matical Society 126 (1998), no. 3, 751–759.[4] G. D. Handley, J. J. Koliha, and J. E. Pecˇaric´, A Hilbert type inequality, Tamkang Journal of Math-ematics 31 (2000), no. 4, 311–315.[5] , New Hilbert-Pachpatte type integral inequalities, Journal of Mathematical Analysis andApplications 257 (2001), no. 1, 238–250.[6] G. H. Hardy, J. E. Littlewood, and G. Po´lya, Inequalities, Cambridge University Press, Cam-bridge, 1934.[7] K. Jichang, On new extensions of Hilbert’s integral inequality, Journal of Mathematical Analysisand Applications 235 (1999), no. 2, 608–614.C.-J. Zhao and W.-S. Cheung 11[8] C. A. Okpoti, L.-E. Persson, and A. Wedestig, Weight characterizations for the discrete Hardyinequality with kernel, Journal of Inequalities and Applications 2006 (2006), Article ID 18030,14 pages.[9] B. G. Pachpatte, A note on Hilbert type inequality, Tamkang Journal of Mathematics 29 (1998),no. 4, 293–298.[10] , On some new inequalities similar to Hilbert’s inequality, Journal of Mathematical Anal-ysis and Applications 226 (1998), no. 1, 166–179.[11] P. Rˇeha´k, Hardy inequality on time scales and its application to half-linear dynamic equations,Journal of Inequalities and Applications 2005 (2005), no. 5, 495–507.[12] B. Yang, On new generalizations of Hilbert’s inequality, Journal of Mathematical Analysis andApplications 248 (2000), no. 1, 29–40.[13] B. Yang and T. M. Rassias, On the way of weight coefficient and research for the Hilbert-type in-equalities, Mathematical Inequalities & Applications 6 (2003), no. 4, 625–658.[14] S. Yaotian and C. Zhihui, General Hardy inequalities with optimal constants and remainder terms,Journal of Inequalities and Applications 2005 (2005), no. 3, 207–219.[15] J. Yuan and G. S. Leng, Inequalities for dual affine quermassintegrals, Journal of Inequalities andApplications 2006 (2006), Article ID 50181, 7 pages.[16] C.-J. Zhao, Generalization on two new Hilbert type inequalities, Journal of Mathematics 20(2000), no. 4, 413–416.[17] , Inverses of disperse and continuous Pachpatte’s inequalities, Acta Mathematica Sinica 46(2003), no. 6, 1111–1116.[18] C.-J. Zhao and L. Debnath, Some new inverse type Hilbert integral inequalities, Journal of Math-ematical Analysis and Applications 262 (2001), no. 1, 411–418.Chang-Jian Zhao: Department of Information and Mathematics Sciences, College of Science,China Jiliang University, Hangzhou 310018, ChinaE-mail address: chjzhao@cjlu.edu.cnWing-Sum Cheung: Department of Mathematics, The University of Hong Kong, Pokfulam Road,Hong KongE-mail address: wscheung@hku.hk

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