8 J. Roumeliotis, P. Cerone, S.S. Drgomir In this pper, we extend the ove result nd develop n Ostrowski-type inequlity for weighted integrls. Applicti

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1 J. KSIAM Vol.3, No., 7-9, 999 An Ostrowski Type Inequlity for Weighted Mppings with Bounded Second Derivtives J. Roumeliotis, P. Cerone, S.S. Drgomir Astrct A weighted integrl inequlity of Ostrowski type for mppings whose second derivtives re ounded is proved. The inequlity is extended to ccount for pplictions in numericl integrtion. Introduction In 938, Ostrowski (see for exmple Mitrinovic et l. (994, p. 468)) proved the following inequlity THEOREM.. Let f : I R! R e dierentile mpping in I o (I o is the interior of I), nd let I o with <. If f :( )! R is ounded on ( ), i.e., kf k := sup jf (t)j <, then we hve the inequlity: t( ) Z " ; f(t) dt ; f(x) ; x ; + # 4 + ( ; )kf k (.) ( ; ) for ll x ( ). The constnt 4 is shrp in the sense tht it cnnot e replced y smller one. A similr result for twice dierentile mppings (Cerone et l. elow. 998) is given THEOREM.. Let f :[ ]! R e twice dierentile mpping such tht f : ( )! R is ounded on (,), i.e. kf k := sup jf (t)j <. Then we hve the t( ) inequlity ; for ll x [ ]. f(t) dt ; f(x)+ x ; + " f (x) ; x ; + # 4 + ( ; ) kf k (.) ( ; ) Key Words nd Phrses: Ostrowski's inequlity, weighted integrls, numericl integrtion 7

2 8 J. Roumeliotis, P. Cerone, S.S. Drgomir In this pper, we extend the ove result nd develop n Ostrowski-type inequlity for weighted integrls. Applictions to specil weight functions nd numericl integrtion re investigted. Preliminries In the next section weighted (or product) integrl inequlities re constructed. The weight function (or density) is ssumed to e non-negtive nd integrle over its entire domin. The following generic quntittive mesures of the weight re dened. R Denition.. Let w :( )! [ ) enintegrle function, i.e. w(t) dt <, then dene s the i th moment of w. m i ( ) = t i w(t) dt i = ::: (.) Denition.. Dene the men of the intervl [ ] with respect to the density w s nd the vrince y 3 The Results 3. -point inequlity ( ) = m ( ) m ( ) (.) ( ) = m ( ) m ( ) ; ( ): (.3) THEOREM 3.. Let f w :( )! R e two mppings on ( ) with the following properties:. sup jf (t)j <, t( ). w(t) 8t ( ), R 3. w(t) dt <, then the following inequlities hold for ll x [ ]. m ( ) ; w(t)f(t) dt;f(x)+ x ; ( ) f (x) h; x ; ( ) + ( )i kf k kf k x ; + (3.) + ; (3.)

3 Ostrowski Type Inequlity 9 Proof. Dene the mpping K( ) :[ ]! R y Integrting y prts gives K(x t) := (R t R t (t ; u)w(u) du t x (t ; u)w(u) du x<t : K(x t)f (t) dt = Z x Z t = f (x) = ; (t ; u)w(u)f (t) dudt + Z x Z t (x ; u)w(u) du (t ; u)w(u)f (t) dudt ; w(t)f(t) dt + f (x) Z t x Z t x (t ; u)w(u)f (t) dudt (t ; u)w(u)f (t) dudt (x ; u)w(u) du ; f(x) w(u) du providing the identity K(x t)f (t) dt = tht is vlid for ll x [ ]. Now tking the modulus of (3.3) we hve, w(t)f(t) dt ; m ( )f(x)+m ( ) ; x ; ( ) f (x) (3.3) ; w(t)f(t) dt ; m ( )f(x)+m ( ) x ; ( ) f (x) = Z K(x t)f (t) dt Z kf k jk(x t)j dt = kf k Z x = kf k Z t (t ; u)w(u) dudt + Z t x (t ; u)w(u) dudt (x ; t) w(t) dt: (3.4) The lst line eing computed y reversing the order of integrtion nd evluting the inner integrls. To otin the desired result (3.) oserve tht (x ; t) w(t) dt = m ( )h; i x ; ( ) + ( ) :

4 J. Roumeliotis, P. Cerone, S.S. Drgomir To otin (3.) note tht (x ; t) dt sup (x ; t) m o ( ) t[ ] =mxf(x ; ) (x ; ) gm ( ) = = ; (x ; ) +(x ; ) + (x ; ) ; (x ; ) m ( ) x ; + + ; m ( ) which upon susitution into (3.4) furnishes the result. Note lso tht the inequlity (3.) is vlid even for unounded w or intervl [ ]. This is not the cse with (.). COROLLARY 3.. The inequlity (3.) is minimized t x = ( ) producing the generlized \mid-point" inequlity m ( ) w(t)f(t) dt ; f(( )) kf ( ) k : (3.5) Proof. Sustituting ( ) for x in (3.) produces the desired result. Note tht x = ( ) not only minimizes the ound of the inequlity (3.), ut lso cuses the derivtive term to vnish. The optiml point (.) cn e interpreted in mny wys. In physicl context, ( ) represents the centre of mss of one dimensionl rod with mss density w. Equivlently, this point cne viewed s tht which minimizes the error vrince for the proility density w (see Brnett et l. (995) for n ppliction). Finlly (.) is lso the Guss node point for one-point rule (Stroud nd Secrest 966). The ound in (3.5) is directly proportionl to the vrince of the density w. So tht the tightest ound is chieved y smpling t the men point ofthe intervl ( ), while its vlue is given y the vrince. 3. -point inequlity Here two point nlogy of (3.) is developed where the result is extended to crete n inequlity with two independent prmeters x nd x. This is minly used (Section 5) to nd n optiml grid for composite weighted-qudrture rules. THEOREM 3.3. Let the conditions of Theorem 3. hold, then the following -point

5 Ostrowski Type Inequlity inequlity is otined ; w(t)f(t) dt ; m ( )f(x )+m ( ) x ; ( ) f (x ) ; ; m ( )f(x )+m ( ) x ; ( ) f (x ) kf k i m ( )h; x ; ( ) + ( ) i + m ( )h; x ; ( ) + ( ) (3.6) for ll x <<x. Proof. Dene the mpping K( ) :[ ] 4! R y K(x x t):= 8 >< >: R t R t R t (t ; u)w(u) du (t ; u)w(u) du x t x <t <x (t ; u)w(u) du x t : With this kernel, the proof is lmost identicl to tht of Theorem 3.. Integrting y prts produces the integrl identity K(x x t)f (t) dt = w(t)f(t) dt ; m ( )f(x )+m ( ) ; x ; ( ) f (x ) ; m ( )f(x )+m ( ) ; x ; ( ) f (x ): (3.7) Re-rrnging nd tking ounds produces the result (3.6). COROLLARY 3.4. The optiml loction of the points x x nd stisfy x = ( ) x = ( ) = ( )+( ) Proof. By inspection of the right hnd side of (3.6) it is ovious tht choosing (3.8) x = ( ) nd x = ( ) (3.9) minimizes this quntity. To nd the optiml vlue for write the expression in rces in (3.6) s i jk(x x t)j dt = m ( )h; x ; ( ) + ( ) = i + m ( )h; x ; ( ) + ( ) Z (x ; t) w(t) dt + (x ; t) w(t) dt: (3.)

6 J. Roumeliotis, P. Cerone, S.S. Drgomir Sustituting (3.9) into the right hnd side of (3.) nd dierentiting with respect to gives Z d ; ( )+( ) jk(( ) ( ) t)j dt = ( ) ; ( ) ; w(): d Assuming w() 6=, then this eqution possesses only one root. A minimum exists t this root since (3.) is convex, nd so the corollry is proved. Eqution (3.8) shows not only where smpling should occur within ech suintervl (i.e. x nd x ), ut how the domin should e divided to mke up these suintervls (). 4 Some Weighted Integrl Inequlities Integrtion with weight functions re used in countless mthemticl prolems. Two min res re: (i) pproximtion theory nd spectrl nlysis nd (ii) sttisticl nlysis nd the theory of distriutions. In this section (3.) is evluted for the more populr weight functions. In ech cse (.) cnnot e used since the weight w(t) or the intervl ( ; ) is unounded. The optiml point (.) is esily identied. 4. Uniform (Legendre) Sustituting w(t) = into (.) nd (.3) gives R ( ) = R tdt = + dt R t dt R (4.) nd + ( ; ) ( ) = ; = dt respectively. Sustituting into (3.) produces (.). Note tht the intervl men is simply the midpoint (4.). 4. Logrithm This weight is present in mny physicl prolems the min ody of which exhiit some xil symmetry. Specil logrithmic rules re used extensively in the Boundry Element Method populrized y Brei (see for exmple Brei nd Dominguez (989)). Some pplictions of which include ule cvittion (Blke nd Gison 987) nd viscous drop deformtion (Rllison nd Acrivos (978) nd more recently y Roumeliotis et l. (997)). With w(t) =ln(=t), =, =, (.) nd (.3) re ( ) = t ln(=t) dt ln(=t) dt = 4

7 Ostrowski Type Inequlity 3 nd ( ) = respectively. Sustituting into (3.) gives Z The optiml point ln(=t)f(t) dt ; f(x)+ t ln(=t) dt ln(=t) ; dt x ; 4 = f (x) kf k 7 x 44 + ;! : 4 x = ( ) = 4 is closer to the origin thn the midpoint (4.) reecting the strength of the log singulrity. 4.3 Jcoi Sustituting w(t) == p t, =, =into (.) nd (.3) gives nd ( ) = ( ) = p tdt =p tdt = 3 tp tdt =p ; tdt = respectively. Hence, the inequlity for Jcoi weight is Z The optiml point f(t) p t dt ; f(x)+ x ; 3 f (x) kf k 4 x 45 + ;! : 3 x = ( ) = 3 is gin shifted to the left of the mid-point due to the t ;= singulrity t the origin. 4.4 Cheyshev The men nd vrince for the Cheyshev weight w(t) == p ; t, = ; =re (; ) = ; t=p ; t dt ; =p ; t dt = nd (; ) = ; tp ; t dt ; =p ; t dt ; =

8 4 J. Roumeliotis, P. Cerone, S.S. Drgomir respectively. Hence, the inequlity corresponding to the Cheyshev weight is Z ; The optiml point f(t) p dt ; f(x)+xf (x) ; t kf k x = (; ) = + x : is t the mid-point of the intervl reecting the symmetry of the Cheyshev weight over its intervl. 4.5 Lguerre The conditions in Theorem 3. re not violted if the integrl domin is innite. The Lguerre weight w(t) = e ;t is dened for positive vlues, t [ ). The men nd vrince of the Lguerre weight re nd ( ) = ( ) = respectively. The pproprite inequlity is Z te ;t dt e ;t dt = t e ;t dt ; = e ;t dt e ;t f(t) dt ; f(x)+(x ; )f (x) kf k from which the optiml smple point of x = my e deduced. 4.6 Hermite ; +(x ; ) Finlly, the Hermite weight isw(t) =e ;t nd vrince for this weight re dened over the entire rel line. The men (; ) = ; te;t dt ; e;t dt = nd (; ) = ; t e ;t dt ; e;t dt ; = respectively. The inequlity from Theorem 3. with the Hermite weight function is thus Z p e ;t f(t) dt ; f(x)+xf (x) kf k ; which results in n optiml smpling point of x =. + x

9 Ostrowski Type Inequlity 5 5 Appliction in Numericl Integrtion Dene grid I n : = < < < n; < n = on the intervl [,], with x i [ i i+ ] for i = ::: n;. The following qudrture formule for weighted integrls re otined. THEOREM 5.. Let the conditions in Theorem 3. hold. qudrture rule holds The following weighted w(t)f(t) dt = A(f x)+r(f x) (5.) where nd A(f x) = n; X; hi f(x i ) ; h i (x i ; i )f (x i ) i= jr(f x)j kf k n; X i= (xi ; i ) + i hi : (5.) The prmeters h i, i nd i re given y respectively. h i = m ( i i+ ) i = ( i i+ ) nd i = ( i i+ ) Proof. Apply Theorem 3. over the intervl [ i i+ ]withx = x i to otin Z i+ i w(t)f(t) dt ; h i f(x i )+h i (x i ; i )f (x i ) kf k ; h i (xi ; i ) + i : Summing over i from to n ; nd using the tringle inequlity produces the desired result. COROLLARY 5.. The optiml loction of the points x i, i = ::: n;, nd grid distriution I n stisfy x i = i i = ::: n; nd (5.3) i = i; + i i = ::: n; (5.4) producing the composite generlized mid-point rule for weighted integrls w(t)f(t) dt = n; X i= h i f(x i )+R(f n) (5.5)

10 6 J. Roumeliotis, P. Cerone, S.S. Drgomir where the reminder is ounded y jr(f n)j kf k n; X i= h i i (5.6) Proof. The proof follows tht of Corollry 3.4 where it is oserved tht the minimum ound (5.) will occur t x i = i. Dierentiting the right hnd side of (5.) gives d d i n; X j= (xj ; j ) + j hj =w( i )(x i ; x i;) i ; x i; + x i Inspection of the second derivtive t the root revels tht the sttionry point is minimum nd hence the result is proved. :

11 6 Numericl Results Ostrowski Type Inequlity 7 In this section, for illustrtrtion, the qudrture rule of Section 5 is used on the integrl Z t ln(=t) cos(4t) dt = ;: (6.) This is evluted using the following three rules: () the composite mid-point rule, where the grid hs uniform step-size nd the node is simply the mid-point of ech su-intervl, () the composite generlized mid-point rule (5.). The grid, I n, is uniform nd the nodes re the men point of ech su-intervl (5.3), (3) eqution (5.5) where the grid is distriuted ccording to (5.4) nd the nodes re the su-intervl mens (5.3). Tle shows the numericl error of ech method for n incresing numer of smple points. For uniform grid, it cn e seen tht chnging the loction of the smpling point from the midpoint [method ()] to the men point [method ()] roughly doules the ccurcy. Chnging the grid distriution s well s the node point [method (3)] from the composite mid-point rule [method ()] increses the ccurcy y pproximtely n order of mgnitude. It is importnt to note tht the nodes nd weights for method (3) cn e esily clculted numericlly using n itertive scheme. For exmple on Pentium-9 personl computer, with n = 64, clculting (5.3) nd (5.4) took close to 37 seconds. Note tht equtions (5.3) nd (5.4) re quite generl in nture nd only rely on the weight insofr s knowledge of the rst two moments is required. This contrsts with Gussin qudrture where for n n point rule, the rst n + moments re needed (or equivlently the n + coecients of the continued frction expnsion (Rutishuser 96 Rutishuser 96)) to construct the pproprite orthogonl polynomil nd then root-nding procedure is clled to nd the scisse (Atkinson 989). This n Error () Error () Error (3) Error rtio (3) Bound rtio (3) 4.97().38().48() { { 8 3.4(-).93(-).35(-) (-) 5.68(-).6(-) (-).3(-) 4.34(-3) (-3) 3. (-3) 9.34(-4) (-3) 7.94(-4).3(-4) (-4).98(-4) 5.5(-5) Tle : The error in evluting (6.) under dierent qudrture rules. The prmeter n is the numer of smple points.

12 8 J. Roumeliotis, P. Cerone, S.S. Drgomir procedure, of course, cn e gretly simplied for the more well known weight functions (Gutschi 994). The second lst column of Tle shows the rtio of the numericl errors for method (3) nd the lst column the rtio of the theoreticl error ound (5.5) Bound rtio (3) = jr(f n=)j : (6.) jr(f n)j As n increses the numericl rtio pproches the theoreticl one. The theoreticl rtio is consistently close to 4. This vlue suggests n symptotic form of the error ound jr(f n)j O n (6.3) for the log weight. Similir results hve een otined for the other weights of Section 4. This is consistent with mid-point type rules nd it is nticipted tht developing other product rules, for exmple generlized trpezoidl or Simpsons rule, will yield more ccurte results. REFERENCES Atkinson, K. E. (989). An Introduction to Numericl Anlysis. John Wiley. Brnett, N. S., I. S. Gomm, nd L. Armour (995). Loction of the optiml smpling point for the qulity ssessment of continuous strems. Austrl. J. Sttist. 37 (), 45{5. Blke, J. R. nd D. C. Gison (987). Cvittion ules ner oundries. Ann. Rev. Fluid Mech. 9, 99{3. Brei, C. H. nd J. Dominguez (989). Boundry elements: n introductory course. Southmpton: Computtionl Mechnics. Cerone, P., S. S. Drgomir, nd J. Roumeliotis (998). An inequlity of Ostrowski type for mppings whose derivtives re ounded nd pplictions. sumitted to Est Asin Mthemticl Journl. Gutschi, W. (994). Algorithm 76: ORTHPOL { pckge of routines for generting orthogonl polynomils nd Guss-type qudrture rules. ACM Trns. Mth. Softwre, {6. Mitrinovic, D. S., J. E. Pecric, nd A. M. Fink (994). Inequlities for functions nd their integrls nd derivtives. Dordrecht: Kluwer Acdemic. Rllison, J. M. nd A. Acrivos (978). A numericl study of the deformtion nd urst of viscous drop in n extensionl ow. J. Fluid Mech. 89, 9{. Roumeliotis, J., G. R. Fulford, nd A. Kucer (997). Boundry integrl eqution pplied to free surfce creeping ow. In B. J. Noye, M. D. Teuner, nd A. W. Gill (Eds.), Computtionl Techniques nd Applictions: CTAC97, Singpore, pp. 599{67. World Scientic.

13 Ostrowski Type Inequlity 9 Rutishuser, H. (96). Alogorithm 5: WEIGHTCOEFF. CACM 5 (), 5{5. Rutishuser, H. (96). On modiction of the QD-lgorithm with Gree-type convergence. In Proceedings of the IFIPS Congress, Munich. Stroud, A. H. nd D. Secrest (966). Gussin qudrture formuls. Prentice Hll. School of Communictions nd Informtics, Victori University of Technology, PO Box 448, MCMC, Melourne, Victori, 8, Austrli JohnRoumeliotis@vut.edu.u pc@mtild.vut.edu.u sever@mtild.vut.edu.u

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