A WEIGHTED OSTROWSKI TYPE INEQUALITY FOR L [a,b] 1 AND APPLICATIONS 4 1 A.QAYYUM1,2,S.S.DRAGOMIR1,2,ANDM.SHOAIB 0 2 Abstract. TheaimofthispaperistoobtainsomegeneralizedweightedOs- n trowskiinequalitiesfordifferentiablemappings. Somewellknowninequalities a can be derivedas special cases ofthe inequalities obtained here. Inaddition, J perturbed mid-point inequality and perturbed trapezoid inequality are also 5 obtained. The inequalities obtained here have direct applications in Numeri- 2 calIntegration,ProbabilityTheory,InformationTheoryandIntegralOperator Theory. Someoftheseapplications arediscussed. ] A C . 1. Introduction h t a Inequalities appearinmostofthe domains ofMathematics andhasapplications m in numerical integration, probability theory, information theory and integral oper- [ ator theory. Inequalities as a field came into promince with the publication of a book by Hardy, Littlewood and Polya [6] in 1934. In 1938, Ostrowski [8] discov- 1 ered a useful inequality, which is known after his name as Ostrowski inequality. v 7 In many practical investigation, it is necessary to bound one quantity by another. 0 This classicalOstrowskiinequality is veryuseful for this purpose. Beckenbachand 0 Bellman [2] and Mitrinovi´c [12] highlighted the importance of inequalities in their 7 respective publications. . 1 0 More recently new inequalities of Ostrowski type were presented Dragomir and 4 Wang [5] in 1997 and Dragomir and Rassias [4] in 2002. The weighted version of 1 Ostrowskiinequality was first presented in 1983 by Pecari´cand Savi´c [9]. In 2003, : v Roumeliotis[11]didsomeimprovementinthe weightedversionofOstrowski-Gru¨ss i type inequalities. In [10] and [7], Qayyum and Hussain discussed the weighted X versionof Ostrowski-Gru¨sstype inequalities. The tools that are used in this paper r a are weighted Peano kernel approach which is the classical and extensively used approach in developing Ostrowski integral inequalities. The results presented in this paper are very general in nature. The inequalities proved by Dragomir et al [5], Barnett et al [1] and Cerone et al [3] are special cases of the inequalities developed here. Ostrowski[8]provedtheclassicalintegralinequalitywhichisstatedherewithout proof. Theorem 1. Let f : [a,b] R be continuous on [a,b] and differentiable on (a,b), whose derivative f : →(a,b) R is bounded on (a,b), i.e. f = ′ ′ → k k ∞ Date:Today. 2000 Mathematics Subject Classification. Primary65D30;Secondary65D32. Key words and phrases. Ostrowskiinequality,Weightfunction,NumericalIntegration. Thispaperisinfinalformandnoversionofitwillbesubmittedforpublicationelsewhere. 1 2 A.QAYYUM1,2,S.S.DRAGOMIR1,2,ANDM.SHOAIB sup f (t) < then t∈[a,b]| ′ | ∞ 1 b 1 x a+b 2 (1.1) (cid:12)(cid:12)f(x)− b−aZa f(t)dt(cid:12)(cid:12)≤"4 + (cid:0)(b−−a2)2(cid:1) #(b−a)kf′k∞ (cid:12) (cid:12) for all x [(cid:12)a,b]. The constant 1 is(cid:12)sharp in the sense that it can not be replaced by ∈ (cid:12) 4 (cid:12) a smaller one. Dragomir and Wang [5] proved (1.1) for f L [a,b] , as follows: ′ 1 ∈ Theorem2. Let f :I R R beadifferentiablemappinginI anda,b I with ◦ ◦ ⊆ → ∈ a<b. If f L [a,b], then the inequality holds ′ 1 ∈ 1 b 1 x a+b (1.2) (cid:12)(cid:12) f(x)− b−aZa f(t)dt(cid:12)(cid:12)≤"2 + (cid:12)(cid:12) b−−a2 (cid:12)(cid:12)#kf′k1 (cid:12) (cid:12) for all x [a,b].(cid:12) (cid:12) ∈ (cid:12) (cid:12) Theyalsopointedoutsomeapplicationsof(1.2)inNumericalIntegrationaswell as for special means. Barnett et,al, [1] proved out an inequality of Ostrowski type for twice differen- tiable mappings which is in terms of the . normofthe secondderivative f and kk1 ′′ apply it in numerical integration and for some special means. The following inequality of Ostrowski’s type for mappings which are twice dif- ferentiable, holds [3]. Theorem 3. Let f : [a,b] R be continuous on [a,b] and twice differentiable → in (a,b) and f L (a,b). Then the inequality obtained ′′ 1 ∈ 1 b a+b (1.1) f(x) f(t)dt x f (x) ′ (cid:12)(cid:12) − b−aZa −(cid:18) − 2 (cid:19) (cid:12)(cid:12) (cid:12) 2 (cid:12) (cid:12) 1 a+b 1 (cid:12) ≤ (cid:12)2(b a) x− 2 + 2(b−a) kf′′k1 (cid:12) − (cid:18)(cid:12) (cid:12) (cid:19) (cid:12) (cid:12) for all x [a,b]. (cid:12) (cid:12) (cid:12) (cid:12) ∈ J.Roumeliotis[4],presentedproductinequalitiesandweightedquadrature. The weighted inequlity was also obtained in Lebesgue spaces involving first derivative of the function, which is given by 1 b w(t)f(t)dt m(a,b)f(x) (cid:12)(cid:12) b−aZa − (cid:12)(cid:12) (cid:12)1 (cid:12) (1.2) (cid:12) [m(a,b)+ m(a,x) m(x,b)] f (cid:12) ≤ (cid:12)2 | − | k ′′k(cid:12)1 Motivated and inspired by the work of the above mentioned renowned mathemati- cians, we will establish a new inequality by using weight function , which will be better and generalized than those developed in [1 4]. Some other interesting in- − equalities arealsopresentedas specialcases. In the last,we presentedapplications for some special means and in numerical integration. WEIGHTED OSTROWSKI TYPE INEQUALITY 3 2. Main Results In order to prove our main result we first give the following essential definition. We assume that the weight function (or density) w : (a,b) [0, ) to be −→ ∞ non-negative and integrable over its entire domain and b w(t)dt< . ∞ Za Thedomainof w maybe finite orinfinite andmayvanishatthe boundarypoint. We denote the moment b m(a,b)= w(t)dt. Za We now give our main result. Theorem 4. Let f :[a,b] R be continuous on [a,b] and differentiable on (a,b) → and satisfy the condition θ f Φ , x (a,b). Then we have the inequality ′ ≤ ≤ ∈ 1 a+b 1 b f(x) w(x)(b a) x f ′(x) f(t)w(t)dt (cid:12)(cid:12) − m(a,b) − (cid:18) − 2 (cid:19) − m(a,b)Za (cid:12)(cid:12) (cid:12) 2 (cid:12) (cid:12) 1 1 a+b (cid:12) (cid:12) w(x) (b a)2+2 x (cid:12) ≤ 2m2(a,b) 2 − (cid:18) − 2 (cid:19) ! 1 a+b (2.1) × 2(b−a)+ x− 2 kf ′′kw,1 (cid:18) (cid:12) (cid:12)(cid:19) (cid:12) (cid:12) for all x [a,b]. (cid:12) (cid:12) ∈ (cid:12) (cid:12) Proof. Let us define the mapping P(.,.):[a,b] R given by −→ t w(u)du if t [a,x] P(x,t)= at ∈ ( Rb w(u)du if t∈(x,b]. Integrating by parts, we have R b b (2.2) P(x,t)f (t)dt= f(x)m(a,b) f(t)w(t)dt. ′ − Za Za Applying the identity (2.2) for f (.), we get ′ 1 b 1 b f (t)= P(t,s)f (s)ds+ f (s)w(s)ds. ′ ′′ ′ m(a,b) m(a,b) Za Za Substituting f (t) in the right membership of (2.2),we have ′ 1 b b f(x) = P(x,t)P(t,s)f (s)dsdt ′′ m2(a,b) Za Za 1 b b 1 b (2.2) + P(x,t)dt f (s)w(s)dsdt+ f(t)w(t)dt. ′ m2(a,b) m(a,b) Za Za Za Since b a+b P(x,t)dt=w(x)(b a) x − − 2 Za (cid:18) (cid:19) and b f (s)w(s)ds =f (x)m(a,b). ′ ′ Za 4 A.QAYYUM1,2,S.S.DRAGOMIR1,2,ANDM.SHOAIB From (2.3) therefore we obtain 1 a+b 1 b f(x) = w(x)(b a) x f ′(x)+ f(t)w(t)dt m(a,b) − − 2 m(a,b) (cid:18) (cid:19) Za 1 b b (2.3) + P(x,t)P(t,s)f (s)dsdt. ′′ m2(a,b) Za Za Now b 1 P(t,s) ds= w(t) (t a)2+(t b)2 , | | 2 − − Za h i b w(t) P(x,t) (t a)2+(b t)2 f (s) ds dt ′′ | | 2 − − | | Za (cid:20) (cid:16) (cid:17) (cid:21) 1 w(x) (x a)2+(b x)2 max t a,b t f . ≤ 2 − − { − − }k ′′kw,1 (cid:16) (cid:17) From (2.4), we have 1 a+b 1 b f(x) w(x)(b a) x f (x) f(t)w(t)dt ′ (cid:12)(cid:12) − m(a,b) − (cid:18) − 2 (cid:19) − m(a,b)Za (cid:12)(cid:12) (cid:12) 1 (cid:12) (2.4) (cid:12) w(x) (x a)2+(b x)2 max t a,b t f . (cid:12) ≤ (cid:12)2m2(a,b) − − { − − }k ′′kw,1 (cid:12) (cid:16) (cid:17) Using 1 a+b max t a,b t = (b a)+ x { − − } 2 − − 2 (cid:12) (cid:12) in (2.5), we get our desired result. (cid:12)(cid:12) (cid:12)(cid:12) (cid:3) (cid:12) (cid:12) Remark 1. For w(t)=1, the inequality (2.1) gives b a+b 1 f(x) x f ′(x) f(t)dt (cid:12) − − 2 − (b a) (cid:12) (cid:12)(cid:12) (cid:18) (cid:19) − Za (cid:12)(cid:12) (cid:12) 2 (cid:12) (cid:12) 1 1 a+b (cid:12) (cid:12) (b a)2+2 x (cid:12) ≤ 2(b−a)2 2 − (cid:18) − 2 (cid:19) ! 1 a+b (2.5) (b a)+ x f × 2 − − 2 k ′′k1 (cid:18) (cid:12) (cid:12)(cid:19) (cid:12) (cid:12) which is similar to Barnett’s result p(cid:12)roved in [1(cid:12)]. (cid:12) (cid:12) Corollary 1. Under the assumptions of Theorem 4 and choosing x = a+b , we 2 have the perturbed midpoint inequality a+b 1 b f f(t)w(t)dt (cid:12)(cid:12) (cid:18) 2 (cid:19)− m(a,b)Za (cid:12)(cid:12) (cid:12) 1 (cid:12) (2.6) (cid:12) w(x)(b a)3 f . (cid:12) ≤ (cid:12)8m2(a,b) − k ′′kw,1 (cid:12) Proof. This follows from inequality (2.1). (cid:3) WEIGHTED OSTROWSKI TYPE INEQUALITY 5 Corollary 2. Under the assumptions of Theorem 4, we have the perturbed trape- zoidal inequality f(a)+f(b) 1 b 1 (b a)2 f(t)w(t)dt+ − (w(a)f ′(a) w(b)f ′(b)) (cid:12)(cid:12) 2 − m(a,b)Za m(a,b) 4 − (cid:12)(cid:12) (cid:12) 1 (cid:12) ((cid:12)2.7) (b a)3[w(a)+w(b)] f . (cid:12) ≤ (cid:12)4m2(a,b) − k ′′kw,1 (cid:12) Proof. Put x = a and x = b in (2.1), summing up the obtained inequalities, using the triangle inequality and dividing by 2, we get the required inequality. (cid:3) Remark 2. The result given in (2.8) is different from the comparable results avail- able in [4]. Remark 3. We can get the best estimation from the inequality (2.1) , only when x= a+b, thisyields theinequality(2.7).Itshows thatmidpoint estimationis better 2 than the trapezoid estimation. 3. Application for some special means Wemaynowapplyinequality(2.1),todeducesomeinequalitiesforspecialmeans by the use of particular mappings as follows: Remark 4. Consider f(x)=√x lnx ,x [a,b] (0, ) ∈ ⊂ ∞ and 1 w(x)= , √x The inequality (2.1) therefore gives √xlnx 1 (b a)(x A) 1+ 1lnx − 2(√b √a)x − − 2 (cid:12)(cid:12) −2(√−b1−√a)(b−a)lnI(a,b(cid:0)) (cid:1) (cid:12)(cid:12) (cid:12) (cid:12) (cid:12)(cid:12) 1 1 1(b a)2+2(x A)2 (cid:12)(cid:12) ≤ 2√x 2 − − 8 √b √a (cid:18) (cid:19) − (cid:16) 1 (cid:17) (b a) lnba lnab (3.1) (b a)+ x A − 1 − . × 2 − | − | 4ab − b a (cid:18) (cid:19) (cid:18) − (cid:19) Choosing x=A in (3.1), we get 1 √AlnA (b a)lnI(a,b) (cid:12)(cid:12) − 2 √b √a − (cid:12)(cid:12) (cid:12) − (cid:12) (3.2) (cid:12)(cid:12)(cid:12) 1 (cid:16) 1 ((cid:17)b a)4 1 lnba(cid:12)(cid:12)(cid:12)−lnab . ≤ 2√x − − b a 128ab √b √a (cid:18) − (cid:19) − (cid:16) (cid:17) Remark 5. Consider f(x)= 1√x ,x [a,b] [1, ) x ∈ ⊂ ∞ and 1 w(x)= √x 6 A.QAYYUM1,2,S.S.DRAGOMIR1,2,ANDM.SHOAIB The inequality (2.1) therefore gives 1 √x + 1 1 (b a)(x A) x 4(√b √a)x2 − − (cid:12)(cid:12)(cid:12) −2(√b1−−√a)(b−a)L−−11 (cid:12)(cid:12)(cid:12) (cid:12) 1 1 1 (cid:12) (cid:12)(cid:12) (b a)2+2(x A(cid:12)(cid:12))2 ≤ 2√x 2 − − 8 √b √a (cid:18) (cid:19) − (cid:16)3 1 (cid:17) b2 a2 (3.3) (b a)+ x A − . ×8 2 − | − | a2b2 (cid:18) (cid:19)(cid:18) (cid:19) Choosing x=A in (3.3), we get 1 1 (cid:12)(cid:12)A √A − 2 √b √a (b−a)L−−11(cid:12)(cid:12) (cid:12) − (cid:12) (3.4) (cid:12)(cid:12)(cid:12) 3 (cid:16) 1 (b(cid:17) a)3 b2−a(cid:12)(cid:12)(cid:12)2 . ≤ 256 √b √a 2√A − (cid:18) a2b2 (cid:19) − Remark 6. Consider f((cid:16)x) = xp√(cid:17)x , x [a,b] f : (0, ) R , where p ∈ ∞ → ∈ R/ 1,0 then for a<b, {− } and 1 w(x)= √x The inequality (2.1) therefore gives xp√x 1 1 (b a)(x A) p+ 1 xp − 2(√b √a)x − − 2 (cid:12)(cid:12) −−2(√b1−√a)(b−a)Lpp. (cid:0) (cid:1) (cid:12)(cid:12) (cid:12) (cid:12) (cid:12)(cid:12) 1 1 1(b a)2+2(x A)2 (cid:12)(cid:12) ≤ 2√x 2 − − 8 √b √a (cid:18) (cid:19) − (cid:16) 1 (cid:17) p2 1 (3.5) (b a)+ x A − 4 bp 1 ap 1 . − − × 2 − | − | p 1 − (cid:18) (cid:19)(cid:18) − (cid:19) Choosing x=A in (3.5), we get (cid:0) (cid:1) 1 Ap√A (b a)Lp (cid:12)(cid:12) − 2 √b √a − p(cid:12)(cid:12) (cid:12) − (cid:12) (cid:12)(cid:12) 1 (cid:16) 1 (cid:17) p2 (cid:12)(cid:12)1 (3.6) (cid:12) (b a)3 −(cid:12)4 bp 1 ap 1 . ≤ 32 √b √a 2√A − (cid:18) p−1 (cid:19) − − − − (cid:0) (cid:1) (cid:16) (cid:17) 4. An Application to Numerical integration Let I : a = x < x < x < .... < x < x = b be a division of the interval n 0 1 2 n 1 n − [a,b] and ξ =(ξ ,ξ ,......,ξ ), a sequence of intermediate points ξ [x ,x ] 0 1 n 1 i i i+1 − ∈ (i=0,1,.....,n 1). Consider the perturbed Riemann sum defined by − n 1 n 1 (4.1) A= − m(x ,x )f(ξ ) − w(ξ )h ξ xi+xi+1 f´(ξ ) i i+1 i i i i i − − 2 i=0 i=0 (cid:18) (cid:19) X X WEIGHTED OSTROWSKI TYPE INEQUALITY 7 Theorem 5. Let f :[a,b] R be continuous on [a,b] and differentiable on (a,b), such that f´: (a,b) R is b→ounded on (a,b) and assume that γ f´ Γ for all x (a,b). f :(a,b)→ R belongs to L (a,b), i.e. ≤ ≤ ′′ 1 ∈ −→ b f := w(t)f(t) dt< . k ′′kw,1 | | ∞ Za we have b (4.2) f(t)w(t)dt =A(f,I,w,ξ)+R(f,I,w,ξ), Za where the remainder R satisfies the estimation R(f,I,w,ξ) | | kf ′′kw,1 w(ξ ) 1(h )2+2 ξ xi+xi+1 2 i i i ≤ 2m(xi,xi+1) 2 (cid:18) − 2 (cid:19) ! 1 x +x i i+1 (4.1) (h )+ ξ i i × 2 − 2 (cid:18) (cid:12) (cid:12)(cid:19) (cid:12) (cid:12) for any choice ξ of the interme(cid:12)diate points. (cid:12) (cid:12) (cid:12) Proof. Apply Theorem 4 on the interval [x ,x ], ξ [x ,x ], where h = i i+1 i i i+1 i ∈ x x (i=1,2,3....n 1), to get i+1 i − − xi+1 x +x m(x ,x ) f(ξ ) f(t)w(t)dt (ξ i i+1)w(ξ )(x x )f´(ξ ) (cid:12) i i+1 i − − i− 2 i i+1− i i (cid:12) (cid:12) Z (cid:12) (cid:12) xi (cid:12) (cid:12) (cid:12) (cid:12)(cid:12) kf ′′kw,1 w(ξ ) 1(x x )2+2 ξ xi+xi+1 2 (cid:12)(cid:12) i i+1 i i ≤ 2m(xi,xi+1) 2 − (cid:18) − 2 (cid:19) ! 1 x +x i i+1 (x x )+ ξ i+1 i i × 2 − − 2 (cid:18) (cid:12) (cid:12)(cid:19) (cid:12) (cid:12) forallξ [x ,x ]and(cid:12)i (0,1,....n (cid:12) 1).Summingtheabovetwoinequalities i ∈ i i+1 (cid:12) ∈ −(cid:12) over i from 0 to n 1 and using the generalized triangular inequality, we get the desired estimation.− (cid:3) 5. Conclusions We established weighted Ostrowski type inequality for bounded differentiable mappings which generalizes the previous inequalities developed and discussed in [1],[3],[5] and [8]. Perturbed midpoint and trapezoid inequalities are obtained. Some closely new results are also given. This inequality is extended to account for applications in some special means and numerical integration to show his ap- plicability towards obtaining direct relationship of these means. These generalized inequalitieswillalsobeusefulfortheresearchersworkinginthefieldoftheapprox- imation theory, applied mathematics, probability theory, stochastic and numerical analysis to solve their problems in engineering and in practical life. 8 A.QAYYUM1,2,S.S.DRAGOMIR1,2,ANDM.SHOAIB References [1] N.S. Barnett, P. Cerone, S.S. Dragomir, J. Roumeliotis, A. Sofo, A survey on Ostrowski type inequalities for twice differentiable mappings and applications, Inequality Theory and Applications1(2001) 24–30. [2] E.F.Beckenbach andR.Bellman,Springer-Verlag,Berlin-Gottinggon-Heidelberg,1961. 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[9] J.E.Pecari´candB.Savi´c,Onovompostupkurazvijanjafunkcijauredinekimprimjenama, Zb.Rad.VAKoV(Beograd)9(1983), 171-202. [10] Ather Qayyum and Sabir Hussain, A new generalized Ostrowski Gru¨ss type inequality and applications,AppliedMathematics Letters 25(2012)1875–1880. [11] J.Roumeliotis,ImprovedweightedOstrowskiGru¨sstypeinequalities.InequalityTheoryand Applications(Eds.Y.J.Cho,Y.K.KimandS.S.Dragomir)Vol.3,153-160,NovaSci.Publ., Hauppauge, NY,2003. [12] D.S.Mitrinovi´c,AnalyticInequalities,SpringerVerlag,1970. 1DepartmentofMathematics,UniversititeknologiPatronas,Malaysia. 2Department ofMathematics,University of Hail,Hail2440,SaudiArabia E-mail address: [email protected] 1Mathematics,CollegeofEngineering&Science,VictoriaUniversity,POBox14428, Melbourne City, MC 8001, Australia., 2School of Computational and Applied Mathe- matics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa. E-mail address: [email protected] Departmentof Mathematics,University ofHail, Hail2440,SaudiArabia E-mail address: [email protected]