A Generalized Inequality of Ostrowski Type for Twice Differentiable Bounded Mappings and Applications

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1 Applied Mthemticl Sciences, Vol. 8, 04, no. 38, HIKARI Ltd, A Generlized Inequlity of Ostrowski Type for Twice Differentile Bounded Mppings nd Applictions Ather Qyyum Deprtment of Fundmentl nd Applied Sciences Universiti Teknologi Ptrons, Mlysi Muhmmd Shoi Deprtment of Mthemtics, University of Hil Hil 440, Sudi Ari Muhmmd Amer Ltif School of Computtionl nd Applied Mthemtics University of the Witwtersrnd, Privte Bg 3, Wits 050 Johnnesurg, South Afric Copyright c 04 Ather Qyyum, Muhmmd Shoi nd Muhmmd Amer Ltif. This is n open ccess rticle distriuted under the Cretive Commons Attriution License, which permits unrestricted use, distriution, nd reproduction in ny medium, provided the originl work is properly cited. Astrct In this pper, we will improve nd generlize Ostrowski type inequlity for twice differentile mppings in terms of the upper nd lower ounds of the second derivtive. Some well known inequlities cn e derived s specil cses of the inequlities otined here. In ddition, pertured mid-point inequlity nd pertured trpezoid inequlity re lso otined. The otined inequlities hve immedite pplictions in numericl integrtions where new estimtes re otined for the reminder term of the trpezoid nd midpoint formule. Applictions to specil mens re lso investigted. Keywords: Ostrowski inequlity, Specil mens Numericl Integrtion

2 890 Ather Qyyum, Muhmmd Shoi nd Muhmmd Amer Ltif Introduction Inequlities hve proved to e n exlted nd pplicle tool for the development of mny rnches of Mthemtics. It s importnce hs incresed noticely during the pst few decdes nd it is now treted s n independent rnch of Mthemtics. This field is ctive nd experiencing tremendous oost with the pssge of time in theory s well s in pplictions. One element tht prticulrly signifies its importnce is its pplictions in vrious fields. Uptill now, vst numer of reserch ppers nd ooks hve een dedicted to inequlities nd their numerous pplictions. Ostrowski s inequlities ply n importnt role in severl other rnches of mthemtics nd sttistics with reference to its pplictions. In recent yers numer of uthors,3,9 to hve written out generliztions of Ostrowski s inequlity. In 998, Drgomir et l.7 presented new proof to the clssicl Ostrowski s inequlity nd for the first time pplied it to the estimtion of error ounds for some specil mens nd for some numericl qudrture rules. It is with the sme viewpoint, the two monogrphs 8 were written in 00 nd 004 to present some selected results on Ostrowski type inequlities nd their pplictions. The current pper will otin ounds for qudrture rules consisting of, t most, three points for twice differentile functions. These results will e otined with the help of kernels. Ostrowski 3 proved the following clssicl integrl inequlity. Theorem Let f : I R R e differentile mpping on I the interior of I nd let, I with <.If f :, R is ounded on, i.e. f = sup t, f t <, then fx ftdt 4 + x + f. for ll x,. The constnt 4 y smller one. is shrp in the sense tht it cn not e replced Some pplictions of Ostrowski s inequlity to specil mens nd numericl qudrture rules, re given in 6 y Drgomir et l. In 976, Milovnović et l. proved generliztion of Ostrowski s inequlity for n-time differentile mppings from which we would like to mention only the cse of twice differentile mppings, p.470. Theorem Let f :, R e twice differentile mpping such tht f t is ounded on, i.e f = sup t, f t <, then the in-

3 Generlized inequlity of Ostrowski type 89 equlity: fx+ f 4 x f+ x f x + + ftdt for ll x,. In Brnett et l. pointed out n inequlity of Ostrowski s type which ws similr, in sense, to the Milovnović -Pecrić result nd pplied it for specil mens nd in numericl integrtion. Some of the results of the Brnett s pper hs een reported y Cerone et l 4. Theorem 3 Let f :, R e twice differentile on, nd f :, R is ounded i.e f = sup t, f t <, then the inequlity: fx 4 + ftdt x + f x x + f 6 f for ll x,. Motivted nd inspired y the work of the ove mentioned renowned mthemticins, we will estlish new generlized inequlity. Some other interesting inequlities re lso presented s specil cses. In the end, we will give pplictions for some specil mens nd in numericl integrtion. Min Results We now give our min result. Theorem 4 Let f :, R e continuous on, nd twice differentile

4 89 Ather Qyyum, Muhmmd Shoi nd Muhmmd Amer Ltif on,, then h fx h x + f x+ h f +f h h f f 8 h 4 + x + ftdt + h3 f 4 3 h + f 4 3 for ll x + h, h nd h 0,. Proof. Let us define the mpping K :, R 8 y K x, t = t + h,ift, x t h,ift x, Let Kx, tf t dt = x t + h f t dt + x t h f t dt. After some mnipultions, we otined the following identity. ftdt = h fx h x + f x 4 +h f +f h f f + 8 Kx, tf t dt for ll x + h, h. This is prticulr form of the identity given in 8, p.67, Theorem 8.

5 Generlized inequlity of Ostrowski type 893 Using the identity., we hve h fx ftdt + h f +f h f f h x + 8 f x Kx, tf t dt f = f x Now, oserve tht x t + h dt + Kx, t dt 5 t + h dt + x x h t h dt = h + 4 Using.4 in.3, we get our required result given in.. t h dt x + + h Remrk For h =0, in., we otin Brnett s result.3 proved in 3. It shows tht our result contins Brnett s result.3 s specil cse. Remrk For h =in., we otin nother useful inequliy. f+f f f ftdt 8 f 4 Hence, for different vlues of h, we cn otin vriety of results. Corollry 5 If f is s in Theorem 4, then we hve the following pertured midpoint inequlity h f + + h f +f h f f 8 ftdt.5 3 h 4 + f 4

6 894 Ather Qyyum, Muhmmd Shoi nd Muhmmd Amer Ltif giving, + f ftdt for h =0, this recptures the clssicl midpoint inequlity. f 4.6 Remrk 3 The estimtion provided y.5 is etter thn the estimtion provided y the clssicl midpoint inequlity. Corollry 6 Let f e s in Theorem 4, then : h f+f h f f + h f +f 4 h f f ftdt 8 3 h + f 4 7 Proof. Put x = nd x = in., summing up the otined inequlities, using the tringle inequlity nd dividing y, we get the required inequlity. Corollry 7 Let f e s in Theorem 4, then we hve the pertured trpezoidl inequlity: f+f f f ftdt 4 6 f. 8 Proof. Put h =0, in.7. Remrk 4 The estimtion provided y.8, is similr to tht of the clssicl trpezoidl inequlity. 3 Applictions in Numericl integrtion Let I n : = x 0 <x <x <... < x n <x n = e division of the intervl,, ξ i x i + δ h i,x i+ δ h i, i =0,,..., n sequence of intermedite points nd h i = x i+ x i,i =0,,..., n. then we hve the following qudrture rule:

7 Generlized inequlity of Ostrowski type 895 Theorem 8 Let f :, R e twice differentile on, nd f :, R is ounded, i.e f <. Then we hve the following: ftdt = A f,f,i n,ξ,δ + R f,f,i n,ξ,δ 3. where A f,f,i n,ξ,δ = n n δ h i fξ i δ h i ξ i x i + x i+ f ξ i 9 + δ n h i f x i +f x i+ δ n h i f x i+ f x i 8 nd the reminder R f,f,i n,ξ,δ stisfies the estimtion R f,f,i n,ξ,δ n h i δ 3 h i 4 + ξ i x i + x i+ + δ3 h i f 4 3 δ + n h 3 i 4 f 0 where δ 0, nd x i + δ h i ξ i x i+ δ h i. Proof. otin Apply Theorem 4 on the intervl x i,x i+, i =0,,...n, to δ h i fξ i δ h i ξi x i+x i+ f ξ i + δ h i f x i +f x i+ δ 8 h i δ h i h i δ 4 3 δ + h 3 i 4 f f x i+ f x i + x i+ x i ξ i x i + x i+ ftdt + δ3 h i 4 f for ny choice ξ of the intermedite points. Summing over i from 0 to n nd using the generlized tringulr inequlity, we deduce the desired estimtion 3.3.

8 896 Ather Qyyum, Muhmmd Shoi nd Muhmmd Amer Ltif Corollry 9 The following pertured midpoint rule holds: fxdx = M f,f,i n + RM f,f,i n, where M n xi + x i+ f,f,i n = h i f 3.4 nd the reminder term R M f,f,i n stisfies the estimtion: n RM f,f,i n f h 3 i Corollry 0 The following pertured trpezoidl rule holds: fxdx = T f,f,i n + RT f,f,i n 3.6 where T n f x i +f x i+ f,f,i n = h i nd the reminder term n RT f,f,i n f 8 h 3 i n h i f x i+ f x i 3.7 x i+ x i Remrk 5 Note tht the ove mentioned pertured midpoint formul 3.4 nd pertured trpezoid formul 3.7 cn give etter pproximtions of the integrl fxdx for generl clsses of mppings. 4 Appliction for some specil mens Let us recll the following mens: The Arithmetic Men A = A, = +,, 0.

9 Generlized inequlity of Ostrowski type 897 The Geometric Men The Hrmonic Men The Logrithmic Men The Identric Men G = G, =,, 0. H = H, = + L = L, = I = I, =,, 0; if =,, 0; if ; ln ln if =,, > 0; e if ; The P-Logrithmic Men p+ p+ p, if p+ L p = L p, := if =, where p R\{, 0},, > 0. The following simple reltionships re known in this pper. H G L I A It is lso known tht L p is monotoniclly incresing in p R with L 0 = I nd L = L. We my now pply inequlity., to deduce some inequlities for specil mens y the use of prticulr mppings s follows: Remrk 6 Consider the mpping f :0, R, fx =x r,r R\{, 0}, for 0 <<, then ftdt = L r r,,

10 898 Ather Qyyum, Muhmmd Shoi nd Muhmmd Amer Ltif nd f = r r δ r,, where r if r, δ r, = r if r, \{0}. Then the inequlity. gives h x r hx A rx r + h r + r h r r r L r 8 r, h h + 4 x A + h3 r r δ r,. 4 Choosing x = A in 4., we get h Ar + h r + r h r r r L r r 8, 3 h + r r δ r,. 4 4 Choosing h =0in 4., we get A r L r r, 4 r r δ r,. 4.3 Remrk 7 Consider the mpping fx =,x, 0,, then x nd ftdt = L,, f = 3.

11 Generlized inequlity of Ostrowski type 899 Then the inequlity. gives h x + hx A x + h H h 8 h h + 4 x A + h3 4 Choosing x = A in 4.4, we get h A + h H h 8 3 h + 4 Choosing h =0in 4.5, we get A L, L, L, Remrk 8 Consider the mpping fx =lnx,x, 0,, nd ftdt =lni,, f =. Then inequlity. gives h lnx hx A + h ln +ln x + h ln I, 8 h h + x A 4 + h Choosing x = A in 4.7, we get h lna h ln +ln+h ln I, 8 h

12 900 Ather Qyyum, Muhmmd Shoi nd Muhmmd Amer Ltif Choosing h =0in 4.8, we get A I 5 Conclusion: exp We estlished generlized Ostrowski type inequlity for ounded differentile mppings. We hve shown tht the inequlities otined in,,4 nd 7 re specil cses of our inequlities. Pertured midpoint nd trpezoid inequlities re lso otined. Improvements to clssicl trpezoidl nd midpoint inequlities re provided. These generlized inequlities dd up to the literture in the sense tht they hve immedite pplictions in Numericl Integrtion nd Specil Mens. These generlized inequlities will lso e useful for the reserchers working in the field of pproximtion theory, pplied mthemtics, proility theory, stochstic nd numericl nlysis. References G.A. Anstssiou, Ostrowski type inequlities, Proc. Amer. Mth. Soc., 3 995, N.S. Brnett, P. Cerone, S.S. Drgomir, J. Roumeliotis, A. Sofo, A survey on Ostrowski type inequlities for twice differentile mppings nd pplictions, Inequlity Theory nd Applictions N.S. Brnett, S.S. Drgomir nd A. Sofo, Better ounds for n inequlity of the Ostrowski type with pplictions, RGMIA Reserch Report Collection, 3 000, Article. 4 P. Cerone, S.S. Drgomir nd J. Roumeliotis, An Inequlity of Ostrowski Type for Mppings whose Second Derivtives re Bounded nd Applictions. RGMIA Reserch Report Collection, 998, P. Cerone, S.S. Drgomir nd J. Roumeliotis, An Inequlity of Ostrowski type for mppings whose second derivtives elong to L, nd pplictions, RGMIA Reserch Report Collection, 998, S.S. Drgomir nd S. Wng, Applictions of Ostrowski s inequlity to the estimtion of error ounds for some specil mens nd some numericl qudrture rules, Appl. Mth. Lett., 998,

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