Lower and upper bounds of the Čebyšev functional for the Riemann-

Stieltjes integral are given. Applications for the three point quadrature rules

of functions that are n-time differentiable are also provided

Cerone, Pietro

Dragomir, Sever S

English

Victoria University Eprints Repository

BOUNDING THE µCEBYEV FUNCTIONAL FOR THERIEMANN-STIELTJES INTEGRAL VIA A BEESACKINEQUALITY AND APPLICATIONSP. CERONE AND S.S. DRAGOMIRAbstract. Lower and upper bounds of the µCebyev functional for the Riemann-Stieltjes integral are given. Applications for the three point quadrature rulesof functions that are n time di¤erentiable are also provided.1. IntroductionIn 1975, P.R. Beesack [1] showed that, if y; v; w are real valued functions de-ned on a compact interval [a; b] ; where w is of bounded variation with totalvariationWba (w) ; and such that the Riemann-Stieltjes integralsR bay (t) dv (t) andR baw (t) y (t) dv (t) both exist, thenmZ bay (t) dv (t) +b_a(w)  infa<b"Z y (t) dv (t)#(1.1)Z baw (t) y (t) dv (t) mZ bay (t) dv (t) +b_a(w)  supa<b"Z y (t) dv (t)#;where m := inft2[a;b] fw (t)g :The second of the inequalities above extends a result of R. Darst and H. Pollard[5] who dealt with the case y (t) = 1; t 2 [a; b] and v (t) continuous on [a; b] :In [6], S.S. Dragomir has introduced the following µCebyev functional for theRiemann-Stieltjes integral :(1.2) T (f; g;u) :=1u (b)  u (a)Z baf (t) g (t) du (t)  1u (b)  u (a)Z baf (t) du (t)  1u (b)  u (a)Z bag (t) du (t) ;provided u (b) 6= u (a) and the involved Riemann-Stieltjes integrals exist.It has been shown in [6] that, if f; g are continuous, m  f (t) M for each t 2[a; b] and u is of bounded variation, then the error in approximating the Riemann-Stieltjes integral of the product in terms of the product of integrals, as describedDate : 30 April, 2007.2000 Mathematics Subject Classication. Primary 26D15, 41A55.Key words and phrases. Riemann-Stieltjes integral, µCebyev functional, Integral inequalities,Quadrature rules.12 P. CERONE AND S.S. DRAGOMIRin the denition of the µCebyev functional (1.2), satises the inequality:(1.3) jT (f; g;u)j 12(M  m)  1ju (b)  u (a)jg   1u (b)  u (a)Z bag (s) du (s)1b_a(u) ;where the constant 12 is best possible and kk1 is the sup-norm.Moreover, if f; g are continuous, m  f (t) M for t 2 [a; b] and u is monotonicnondecreasing on [a; b] ; then:(1.4) jT (f; g;u)j 12(M  m) 1ju (b)  u (a)j Z bag (t)  1u (b)  u (a)Z bag (s) du (s) du (t)and the constant 12 here is also sharp.Finally, if f; g are Riemann integrable and u is Lipschitzian with the constantL > 0 then also(1.5) jT (f; g;u)j 12(M  m) Lju (b)  u (a)j Z bag (t)  1u (b)  u (a)Z bag (s) du (s) dt:The constant 12 is also best possible in (1.5) (see [7] and [8]).The main aim of the present paper is to provide other bounds for the µCebyevfunctional T (f; g;u) by utilising the Beesack inequality (1.1). Applications for threepoint quadrature rules of functions that are (n  1) di¤erentiable (n  1) with thederivative f (n 1) absolutely continuous are given as well.2. The ResultsThe following result may be stated.Theorem 1. Let f; g; u : [a; b] ! R be such that f is of bounded variation andthe Riemann-Stieltjes integralsR baf (t) g (t) du (t) ;R baf (t) du (t) andR bag (t) du (t)exist. Thenb_a(f)  infa<b"Z g (t) du (t)  u ()  u ()u (b)  u (a) Z bag (s) du (s)#(2.1)Z baf (t) g (t) du (t)  1u (b)  u (a) Z baf (t) du (t) Z bag (t) du (t)b_a(f)  supa<b"Z g (t) du (t)  u ()  u ()u (b)  u (a) Z bag (s) du (s)#;provided u (b) 6= u (a) :BOUNDING THE µCEBYEV FUNCTIONAL 3Proof. We observe that the following identity holds true (see also [6])(2.2) [u (b)  u (a)]T (f; g;u)=Z baf (t)"g (t)  1u (b)  u (a)Z bag (s) du (s)#du (t) :Since f is of bounded variation, it follows that f is bounded below and if we denoteby m the inmum of f on [a; b] ; then on applying the Beesack inequality for thechoicesw (t) = f (t) ; y (t) = g (t)  1u (b)  u (a)Z bag (s) du (s)and v (t) = u (t) ; t 2 [a; b] ; we can write that:mZ ba"g (t)  1u (b)  u (a)Z bag (s) du (s)#du (t)(2.3)+b_a(f)  infa<b(Z "g (t)  1u (b)  u (a)Z bag (s) du (s)#du (t)) [u (b)  u (a)]T (f; g;u) mZ ba"g (t)  1u (b)  u (a)Z bag (s) du (s)#du (t)+b_a(f)  supa<b(Z "g (t)  1u (b)  u (a)Z bag (s) du (s)#du (t)):Since Z ba"g (t)  1u (b)  u (a)Z bag (s) du (s)#du (t) = 0andZ "g (t)  1u (b)  u (a)Z bag (s) du (s)#du (t)=Z g (t) du (t)  u ()  u ()u (b)  u (a) Z bag (s) du (s) ;hence, by (2.3), we deduce the desired result (2.1). The following corollary for weighted integrals may be stated:Corollary 1. Let f; g; w : [a; b] ! R be such that f is of bounded variation andthe Riemann integralsR baf (t) g (t)w (t) dt;R baf (t)w (t) dt andR bag (t)w (t) dt exist.4 P. CERONE AND S.S. DRAGOMIRThenb_a(f)  infa<b"Z g (t)w (t) dt R w (s) dsR baw (s) dsZ bag (t)w (t) dt#(2.4)Z baf (t) g (t)w (t) dt  1R baw (s) dsZ baf (t)w (t) dt Z bag (t)w (t) dtb_a(f)  supa<b"Z g (t)w (t) dt R w (s) dsR baw (s) dsZ bag (t)w (t) dt#;providedR baw (s) ds 6= 0:Remark 1. For the particular case when w (t) = 1; t 2 [a; b] ; then we get from(2.4) the following inequality:b_a(f)  infa<b"Z g (t) dt     b  a Z bag (t) dt#(2.5)Z baf (t) g (t) dt  1b  aZ baf (t) dt Z bag (t) dtb_a(f)  supa<b"Z g (t) dt     b  a Z bag (t) dt#;provided f is of bounded variation and the involved Riemann integrals exist.3. Applications for Three Point QuadraturesRecall that in [4] (see also [9, p. 223]) P. Cerone and S.S. Dragomir estab-lished the following identity concerning a three point quadrature rule for n timedi¤erentiable functions f : [a; b]! R:(3.1)Z baf (t) dt =nXk=11k!n(1  )kh(b  x)k + ( 1)k 1 (x  a)kif (k 1) (x)+ kh(x  a)k f (k 1) (a) + ( 1)k 1 (b  x)k f (k 1) (b)io+ ( 1)nZ baCn (x; t) f(n) (t) dt;where the Peano kernel is given by:(3.2) Cn (x; t) :=8>>><>>>:[t  (x+ (1  ) a)]nn!if t 2 [a; x] ;[t  (x+ (1  ) b)]nn!if t 2 (x; b];and  2 [0; 1] ; x 2 [a; b] :We note that the above representation generalised the interior point quadraturerule obtained in 1999 by Cerone et al. in [2] for  = 0 and the trapezoid type ruleobtained in 2000 by Cerone et al. in [3] for  = 1:BOUNDING THE µCEBYEV FUNCTIONAL 5The function Cn (x; ) is of bounded variation for each xed x 2 [a; b] and asimple calculation reveals that(3.3)b_a(( 1)nCn (x; ))=Z xadCn (x; t)dt dt+ Z bxdCn (x; t)dt dt=Z xajt  (x+ (1  ) a)jn 1(n  1)! dt+Z bxjx+ (1  ) b  tjn 1(n  1)! dt=1n!(x  a)n [n + (1  )n] + 1n!(b  x)n [n + (1  )n]=1n![n + (1  )n] [(b  x)n + (x  a)n]for any x 2 [a; b] :Also,Z baCn (x; t) dt(3.4)=1n!Z xa[t  (x+ (1  ) a)]n dt+ 1n!Z bx[t  (x+ (1  ) b)]n dt=1(n+ 1)!n[x  (x+ (1  ) a)]n+1   [a  (x+ (1  ) a)]n+1+ [b  (x+ (1  ) b)]n+1   [x  (x+ (1  ) b)]n+1o=1(n+ 1)!n(1  )n+1 (x  a)n+1   ( 1)n+1 n+1 (x  a)n+1+ n+1 (b  x)n+1   ( 1)n+1 (1  )n+1 (b  x)n+1o=1(n+ 1)!n(b  x)n+1hn+1 + ( 1)n (1  )n+1i+ ( 1)nhn+1 + ( 1)n (1  )n+1i(x  a)n+1o=1(n+ 1)!h(b  x)n+1 + ( 1)n (x  a)n+1i hn+1 + ( 1)n (1  )n+1ifor any x 2 [a; b] :We can state the following result:6 P. CERONE AND S.S. DRAGOMIRTheorem 2. Let f : [a; b]! R be an (n  1) di¤erentiable function (n  1) withthe derivative f (n 1) absolutely continuous on [a; b] : Then we have(3.5)Z baf (t) dt =nXk=11k!n(1  )kh(b  x)k + ( 1)k 1 (x  a)kif (k 1) (x)+ kh(x  a)k f (k 1) (a) + ( 1)k 1 (b  x)k f (k 1) (b)io+1(n+ 1)!f (n 1) (b)  f (n 1) (a)b  a h(b  x)n+1 + ( 1)n (x  a)n+1ih( 1)n n+1 + (1  )n+1i+ En (f; x; ; a; b) ;where the remainder En (f; x; ; a; b) (which is dened implicitly by (3.5))satisesthe bounds:(3.6)1n![n + (1  )n] [(b  x)n + (x  a)n] infa<b[n (f ;; )] En (f; x; ; a; b) 1n![n + (1  )n] [(b  x)n + (x  a)n] supa<b[n (f ;; )]and(3.7) n (f ;; ) = f (n 1) ()  f (n 1) ()     b  ahf (n 1) (b)  f (n 1) (a)i;where  2 [0; 1] and x 2 [a; b] :Proof. Apply the inequality (2.5) for the functions f = ( 1)n Cn (x; ) and g = f (n)to get1n![n + (1  )n] [(b  x)n + (x  a)n] infa<b[n (f ;; )](3.8) ( 1)nZ baCn (x; t) f(n) (t) dt  1b  a ( 1)nZ baCn (x; t) dt Z baf (n) (t) dt 1n![n + (1  )n] [(b  x)n + (x  a)n] supa<b[n (f ;; )] :Since, by (3.3)b_a(( 1)n Cn (x; )) = 1n![n + (1  )n] [(b  x)n + (x  a)n]and by (3.4)( 1)nZ baCn (x; t) dt=1(n+ 1)!h(b  x)n+1 + ( 1)n (x  a)n+1i h( 1)n n+1 + (1  )n+1i;then, on utilising the inequality (3.8), we haveBOUNDING THE µCEBYEV FUNCTIONAL 7(3.9)1n![n + (1  )n] [(b  x)n + (x  a)n] infa<b[n (f ;; )] ( 1)nZ baCn (x; t) f(n) (t) dt  1(n+ 1)!h(b  x)n+1 + ( 1)n (x  a)n+1i h( 1)n n+1 + (1  )n+1if (n 1) (b)  f (n 1) (a)b  a 1n![n + (1  )n] [(b  x)n + (x  a)n] supa<b[n (f ;; )] :Now, due to the fact that, by the representation (3.1) we have(3.10) ( 1)nZ baCn (x; t) f(n) (t) dt=Z baf (t) dt nXk=11k!n(1  )kh(b  x)k + ( 1)k 1 (x  a)kif (k 1) (x)+ kh(x  a)k f (k 1) (a) + ( 1)k 1 (b  x)k f (k 1) (b)iothen, on making use of remainders representation En (f; x; ; a; b) (which is denedimplicitly by (3.5)), we deduce from (3.9) the desired result (3.6). Remark 2. For  = 0; we get from Theorem 2:(3.11)Z baf (t) dt =nXk=11k!h(b  x)k + ( 1)k 1 (x  a)kif (k 1) (x)+1(n+ 1)!f (n 1) (b)  f (n 1) (a)b  a h(b  x)n+1 + ( 1)n (x  a)n+1i+ Fn (f; x; a; b) ;where the remainder satises the bounds(3.12)1n![(b  x)n + (x  a)n] infa<b[n (f ;; )] Fn (f; x; a; b) 1n![(b  x)n + (x  a)n] supa<b[n (f ;; )]for x 2 [a; b] :8 P. CERONE AND S.S. DRAGOMIRFor  = 12 ; we get from Theorem 2 that:(3.13)Z baf (t) dt =nXk=112kk!nh(b  x)k + ( 1)k 1 (x  a)kif (k 1) (x)+h(x  a)k f (k 1) (a) + ( 1)k 1 (b  x)k f (k 1) (b)io+[1 + ( 1)n]2n+1 (n+ 1)!f (n 1) (b)  f (n 1) (a)b  a h(b  x)n+1 + ( 1)n (x  a)n+1i+Gn (f; x; a; b) ;where the remainder satises the inequality:(3.14)12n 1n![(b  x)n + (x  a)n] infa<b[n (f ;; )] Gn (f; x; a; b) 12n 1n![(b  x)n + (x  a)n] supa<b[n (f ;; )] ;for x 2 [a; b] :Finally, for  = 1; we obtain from Theorem 2 that:(3.15)Z baf (t) dt =nXk=11k!h(x  a)k f (k 1) (a) + ( 1)k 1 (b  x)k f (k 1) (b)i+( 1)n(n+ 1)!f (n 1) (b)  f (n 1) (a)b  a h(b  x)n+1 + ( 1)n (x  a)n+1i+Hn (f; x; a; b)where the remainder Hn (f; x; a; b) satises the bounds:(3.16)1n![(b  x)n + (x  a)n] infa<b[n (f ;; )] Hn (f; x; a; b) 1n![(b  x)n + (x  a)n] supa<b[n (f ;; )]for x 2 [a; b] :The following particular case may be useful in applications:If n = 1 and f : [a; b]! R is an absolutely continuous function on [a; b] then wehave the representation:(3.17)Z baf (t) dt = (1  ) (b  a) f (x) +  [(x  a) f (a) + (b  x) f (b)]+ [f (b)  f (a)]a+ b2  x(1  2) + E (f; x; ; a; b)and the remainder E (f; x; ; a; b) satises the bounds(3.18) (b  a) infa<b[ (f ;; )]  E (f; x; ; a; b)  (b  a) supa<b[ (f ;; )]BOUNDING THE µCEBYEV FUNCTIONAL 9where (f ;; ) := f ()  f ()     b  a [f (b)  f (a)] ;and x 2 [a; b] while  2 [0; 1] :One must observe that for n = 1 the bounds for the error are independent of xand . However, this quality is not inherited for the quadrature rules with n  2:References[1] P.R. BEESACK, Bounds for Riemann-Stieltjes integrals, Rocky Mountain J. Math., 5(1)(1975), 75-78.[2] P. CERONE, S.S. DRAGOMIR and J. ROUMELIOTIS, Some Ostrowski type inequalities forn time di¤erentiable mappings and applications, Demonstratio Math., 32(2) (1999), 697-712.[3] P. CERONE, S.S. DRAGOMIR, J. ROUMELIOTIS and J. S ¼UNDE, A new generalisationof the trapezoid formula for n time di¤erentiable mappings and applications, DemonstratioMath., 33(4) (2000), 719-736.[4] P. CERONE and S.S. DRAGOMIR, Three point identities and inequalities for n time di¤er-entiable functions, SUT J. of Math. (Japan), 36(2) (2000), 351-383.[5] R. DARST and H. POLLARD, An inequality for the Riemann-Stieltjes integral, Proc. Amer.Math. Soc., 25 (1970), 912-913.[6] S.S. DRAGOMIR, Sharp bounds of µCebyev functional for Stieltjes integrals and applications,Bull. Austral. Math. Soc., 67 (2003), 257-266.[7] S.S. DRAGOMIR, New estimates of the µCebyev functional for Stieltjes integrals and appli-cations, J. Korean Math. Soc., 41(2) (2004), 249-264.[8] S.S. DRAGOMIR, Inequalities of Grüss type for the Stieltjes integral and applications, Kragu-jevac J. Math., 26 (2004), 89-122.[9] S.S. DRAGOMIR and Th.M. RASSIAS, Ostrowski Type Inequalities and Applications in Nu-merical Integration, Kluwer Academic Publishers, Dordrecht, 2002.School of Computer Science and Mathematics, Victoria University, PO Box 14428,Melbourne City, VIC 8001, Australia.E-mail address : pietro.cerone@vu.edu.auURL: http://rgmia.vu.edu.au/ceroneE-mail address : sever.dragomir@vu.edu.auURL: http://rgmia.vu.edu.au/dragomir

Bounding the Čebyšev Functional for the Riemann-Stieltjes Integral via a Beesack Inequality and Applications

Lower and upper bounds of the Čebyšev functional for the Riemann-\ud
Stieltjes integral are given. Applications for the three point quadrature rules\ud
of functions that are n-time differentiable are also provided

Bounding the Čebyšev Functional for the Riemann-Stieltjes Integral via a Beesack Inequality and Applications

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