A unified generalization of perturbed mid-point and trapezoid inequalities and asymptotic expressions for its error term

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1 An. Ştiinţ. Univ. Al. I. Cuz Işi. Mt. (N.S. Tomul LXIII, 07, f. A unified generliztion of perturbed mid-point nd trpezoid inequlities nd symptotic expressions for its error term Wenjun Liu Received: 7.XI.0 / Lst revision: 4.III.03 / Accepted: 6.IX.03 Abstrct In this pper, we first estblish new unified proof of perturbed mid-point nd trpezoid inequlities nd give n ppliction of it in numericl integrtion. This result in specil cses yield the known results. We then derive some symptotic expressions for error terms of this unified inequlity, which not only unify the known results, but lso give some new symptotic expressions for reminder terms of other perturbed qudrture rules s specil cses. Finlly, corresponding formuls with finite sums re given. Keywords unified generliztions perturbed mid-point nd trpezoid inequlities numericl integrtion reminder terms symptotic expressions Mthemtics Subject Clssifiction (00 6D5 65D30 4A55 4A80 Introduction Error nlysis for known nd new qudrture rules hs been extensively studied in recent yers. The pproch from n inequlities point of view to estimte the error terms hs been used in these studies (see -5 nd the references therein. In 6, Ujević nd Billć considered the bove mentioned topic in wy of deriving symptotic expressions for error terms of the mid-point, trpezoid nd Simpson s rules. Precisely, bsed on the Assumption : Let f C, b nd sup n N f (n (c f (n (c M < for some rbitrry but fixed c, b, they proved the following theorems: Wenjun Liu College of Mthemtics nd Sttistics, Nnjing University of Informtion Science nd Technology, Nnjing, 0044, Chin E-mil: wjliu@nuist.edu.cn

2 Wenjun Liu Theorem. Let Assumption holds with c =, we hve ( + b f(tdt = f f(tdt = (b + k=3 f( + f(b (b + f(b f(tdt = f( + 4f ( +b 6 3 k k k f (k ((b k. (. k=3 k + (k 6 k3 k f (k ((b k. Theorem. Let Assumption holds with c = b, we hve ( + b f(tdt = f f(tdt = (b k=3 f( + f(b (b + + f(b f(tdt = f( + 4f ( +b k f (k ((b k, (. (b (.3 k k k f (k (b( b k, (.4 k=3 k + (k 6 k3 k f (k (b( b k. k f (k (b( b k, (.5 (b (.6 In 6,8,8,9,5, the perturbed mid-point nd trpezoid inequlities re considered. In 5, Ujević obtined the perturbed mid-point nd trpezoid inequlities ( + b (b f(tdt f (b f (b f ( 4 (S γ(b 3, (.7 f( + f(b (b f(tdt (b + f (b f ( (S γ(b 3, (.8 where f :, b R is twice differentible function nd there exist constnts γ, Γ R, with γ f (t Γ, t, b, S = f (bf ( b. In 7, Liu et l. derived symptotic expressions for error terms of these perturbed mid-point nd trpezoid rules.

3 A unified generliztion 3 Theorem.3 Let Assumption holds with c =, we hve ( + b (b f(tdt = f (b + f (b f ( 4 + k k(k f (k ((b k, (.9 k 4 f( + f(b f(tdt = (b + (b f (b f ( (k 3(k 4 f (k ((b k. (.0 Theorem.4 Let Assumption holds with c = b, we hve ( + b (b f(tdt = f (b + f (b f ( 4 k k(k f (k (b( b k, (. k 4 f( + f(b f(tdt = (b (b f (b f ( (k 3(k 4 f (k (b( b k. (. In 7, Chen et l. obtined unified generliztion of perturbed trpezoid nd mid-point inequlities. Theorem.5 Let I R be n open intervl,, b I, < b. If f : I R is twice differentible function such tht f is integrble nd there exist constnts γ, Γ R, with γ f (t Γ, t, b, 0 λ. Then f(tdt ( λf 3λ 4 ( λ λ + 3 ( λ λ ( λ λ ( + b (b f (b f ( f( + f(b (b ( λ( λ (Γ γ(b 3, λ 0, 4 3, (Γ γ(b 3, λ ( 3, 3, 3 ( λ(λ 4 (Γ γ(b 3, λ ( 3,.

4 4 Wenjun Liu In this pper, we first estblish new unified proof of perturbed mid-point inequlity (.7 nd perturbed trpezoid inequlity (.8 by using unified p(t s in (. below nd give n ppliction of it in numericl integrtion (Section. This result in specil cses yield Theorem 4 nd Corollry in 5. We then derive some symptotic expressions for error terms of this unified inequlity (Section 3, which not only unify the bove Theorems.3 nd.4, but lso give some new symptotic expressions for reminder terms of other perturbed qudrture rules s specil cses. Corresponding formuls with finite sums will lso be given. A new unified proof of perturbed mid-point nd trpezoid inequlities nd ppliction In this section, we first estblish new unified proof of perturbed mid-point nd trpezoid inequlities. Theorem. Let I R be n open intervl,, b I, < b. If f : I R is twice differentible function such tht f is integrble nd there exist constnts γ, Γ R, with γ f (t Γ, t, b, 0 λ. Then where S = f (bf ( b. ( + b f( + f(b f(tdt ( λf (b 3λ (b f (b f ( 4 (. 3λ (S γ(b 3, 0 λ 4, 3λ (S γ(b 3, 4 < λ, Proof. Let p :, b R be given by Integrting by prts, we hve + b (t t ( λ λb, t,, p(t = + b (b tλ + ( λb t, t (, b. (. p(tf (tdt = f(tdt (b ( + b ( λf f( + f(b. (.3

5 If C is constnt, then We lso hve = p(t b p(t b p(sds f (t Cdt A unified generliztion 5 p(sds f (tdt. (.4 f (tdt = f (b f ( (.5 p(tdt = 3λ (b 3. (.6 4 From (.3-(.6 it follows = p(t b f(tdt (b p(sds f (t Cdt ( λf 3λ (b f (b f (. 4 ( + b f( + f(b (.7 On the other hnd, if we set C = γ, then we hve p(t b p(sds f (t γdt mx t,b p(t p(sds b f (t γ dt = (Sγ(b mx (t λ(b(t 3λ (b (.8 t, +b 3λ (S γ(b 3, 0 λ 4, = 3λ (S γ(b 3, 4 < λ. From (.7 nd (.8 we see tht (. holds. Remrk. We note tht in the specil cses, if we tke λ = 0 nd λ = in Theorem. respectively, we get Theorem 4 nd Corollry in 5 respectively.

6 6 Wenjun Liu To verify the correctness of Theorem., we give severl specific exmples shown s the following Tble, in which we set λ = 3, λ = 3, G (λ = 3λ 4 (S γ(b 3, G (λ = 3λ 4 (S γ(b 3, nd ( F (λ := + b f( + f(b f(tdt ( λf (b 3λ (b f (b f ( 4. We find tht F (λ G (λ nd F (λ G (λ. f(x, b F (λ G F (λ G cos x x 0, π e x 0, , x e x sin x, Corollry. Under the ssumptions of Theorem. nd with λ =, we hve the perturbed verged mid-point-trpezoid type inequlity f(tdt ( + b f + 48 (b f (b f ( (b f( + f(b (b 48 (S γ(b 3. (.9 Corollry.3 Under the ssumptions of Theorem. nd with λ = 3, we hve the Simpson inequlity f(tdt b 6 f( + 4f ( + b + f(b 4 (S γ(b 3. (.0 Now, we give n ppliction of Theorem. in numericl integrtion. Theorem.4 Let the ssumptions of Theorem. hold. If D = { = x 0 < x < < x n = b is given division of the intervl, b then we hve f(tdt = A MT (f, D + R MT (f, D, where n ( xi + x i+ A MT (f, D = h i ( λf f(x i + f(x i+ i=0 + 3λ n h 3 i f (x i+ f (x i, 4 i=0 n 3λ (S i γh 3 i, 0 λ 4, i=0 R MT (f, D n 3λ (S i γh 3 i, 4 < λ, i=0

7 nd h i = x i+ x i, S i = f (x i+f (x i h i, i = 0,,,, n. A unified generliztion 7 Proof. Apply Theorem. to the intervl x i, x i+, i = 0,,,, n nd sum. Then use the tringle inequlity to obtin the desired result. 3 Some symptotic expressions for error term of the unified inequlity In this section, we derive some symptotic expressions for error term of the bove unified inequlity (.. Theorem 3. Let Assumption holds with c =, we hve ( + b f( + f(b f(tdt = ( λf (b + 3λ (b f (b f ( 4 { + ( λ k k(k k 4 +λ (k 3(k 4 Proof. We define the function R(x = x f(tdt ( λf f (k ((b k, λ 0,. ( + x 3λ (x f (x f (, 4 f( + f(x (x for ll λ 0,. Obviously, R( = 0. We hve ( + x R (x = f(x ( λ f + ( + x (x f f( + f(x λ + (x f (x 3λ (x f (x f ( 3λ (x f (x 4 such tht R ( = 0. We lso hve ( + x R (x = (λ f (xf λ ( + x (x f 4 3λ f (xf ( 6 (xf (x 3λ (x f (x 4 (3.

8 8 Wenjun Liu nd R ( = 0. Further, 3 R (x = ( λ 4 f (x 3 ( + x 4 f 4 (x f (x ( + x 8 (x f 4 (x f (4 (x +λ (x f (4 (x, R ( = 0, R (4 (x = ( λ f (x f ( + x 3 (x f (4 (x ( (4 + x (x f 6 4 (x f (5 (x 6 (x f (4 (x + (x f (5 (x, R (4 ( = 0, R (5 (x = ( λ 6 f (4 (x 5 ( + x 6 f (4 5 (x f (5 (x ( (5 + x (x f 3 4 (x f (6 (x 6 f (4 (x + 3 (x f (5 (x + (x f (6 (x, R (5 ( = 5λ 7 f (4 (, 48 Generlly, by induction, we cn get { R (k k(k (x =( λ f (k (x 4 k ( (k + x f k k (x f (k (x ( + x k (x f (k 4 (x f (k+ (x { (k 3(k 4 f (k (x + k 3 6 (x f (k (x (3. + (x f (k+ (x, k 5, nd so { R (k (= (λ k k(k +λ (k3(k4 f (k (, k 5. k 4

9 In fct, suppose tht (3. holds for k = m (m 5, then we hve { R (m+ (x = ( λ m(m 4 m f (m (x m (x f (m+ (x A unified generliztion 9 f (m (x m m f (m ( + x ( + x m f (m ( (m+ + x (x f m+ (x f (m+ (x { (m 3(m 4 4 (x f (m+ (x f (m (x + m 3 f (m (x 6 + m 3 (x f (m+ (x (x f (m+ (x + (x f (m+ (x { m(m + = ( λ f (m (x m + ( + x 4 m f (m m + (x f (m+ (x ( (m+ + x (x f m+ 4 (x f (m+ (x { (m (m 3 f (m (x + m (x f (m+ (x 6 + (x f (m+ (x, m 5, which implies tht (3. holds for k = m +. By using of the Tylor series R(x = R (k ( k=0 (x k with the bove dt, we hve { R(x = ( λ k k(k k 4 +λ f (k ((x k. (k 3(k 4 If we substitute x = b in the bove series then we get formul (3.. We use Lemm. of 6 to show tht the series in (3. converges. Theorem 3. Let Assumption holds with c = b, we hve ( + b f( + f(b f(tdt = ( λf (b + 3λ (b f (b f ( 4 { ( λ k k(k k 4 (3.3

10 0 Wenjun Liu +λ (k 3(k 4 f (k (b( b k. Proof. We define the function ( x + b R(x = f(tdt ( λf x 3λ (b x f (b f (x 4 { x ( b + x = f(tdt ( λf b 3λ (x b f (x f (b. 4 f(x + f(b (b x f(b + f(x (x b Now we cn use the results of Theorem 3.. We simply substitute b in the bove reltion nd get R(x = +λ { ( λ (k 3(k 4 k k(k k 4 f (k (b(x b k. The reltion (3.3 follows if we now set x =. We use Lemm. of 6 to show tht the series in (3.3 converges. Corollry 3.3 Under the ssumptions of Theorems 3. nd 3. we hve ( + b f( + f(b f(tdt = ( λf (b + 3λ (b f (b f ( 4 + { ( λ k k(k (k 3(k 4 +λ k 4 f (k ( ( k f (k (b (b k. Proof. We sum (3. nd (3.3. Remrk 3. We note tht in the specil cses, if we tke λ = 0 nd in Theorem 3. nd 3., respectively, we get Theorem.3 nd.4, respectively. If we tke λ = 3 in Theorem 3. nd 3., respectively, we get (.3 in Theorem. nd (.6 in Theorem., respectively. Now, we cn give some new symptotic expressions for reminder terms of other perturbed qudrture rules s specil cses.

11 A unified generliztion Corollry 3.4 Under the ssumptions of Theorem 3. nd 3. with λ =, we hve the symptotic expressions for reminder terms of perturbed verged mid-point-trpezoid type rule f(tdt = ( + b f( + f(b f + (b 48 (b f (b f ( + { k k(k k 4 (k 3(k 4 + f (k ((b k, f(tdt = ( + b f( + f(b f + (b { k k(k k 4 + (k 3(k 4 f (k (b( b k. 48 (b f (b f ( Corollry 3.5 Under the ssumptions of Theorem 3. nd 3. with λ = 3, we hve the symptotic expressions for reminder terms of perturbed verged 3-point type rule f(tdt = 3 f( + f + 3 { ( + b (k 3(k 4 + f(b (b 4 (b f (b f ( k k(k k 4 f (k ((b k, (k 3(k 4 + f(tdt = ( + b f( + f + f(b (b 3 4 (b f (b f ( { k k(k 3 k 4 + f (k (b( b k. Finlly, we derive the corresponding formuls with finite sums.

12 Wenjun Liu Theorem 3.6 Let f C n+, b. Then ( + b f( + f(b f(tdt = ( λf (b + 3λ (b f (b f ( 4 n { + ( λ k k(k k 4 (k 3(k 4 +λ f (k ((b k where + n! { R (m (t =( λ m (m f m (m (x f m R (n+ (t(b t n dt, m(m f (m (t 4 ( + t m (t f (m (t ( + t 4 (t f (m+ (t (3.4 (3.5 nd { (m 3(m 4 f (m (t + m 3 (t f (m (t 6 + (t f (m+ (t f(tdt = ( λf ( + b f( + f(b (b + 3λ (b f (b f ( (3.6 4 n { ( λ k k(k k 4 (k 3(k 4 +λ f (k (b( b k + n! R (n+ (t( t n dt, where in this cse the derivtives R (m (t re equl to (3.5 with the substitution = b.

13 A unified generliztion 3 Proof. Let R(x be defined in the proof of Theorem 3.. From Lemm. of 6 with g=r, c= we get R(x= n R (k ( k=0 (x k + x n! R(n+ (t(x t n dt. If we substitute the vlues from the bove mentioned proof in the bove reltion then we obtin x ( + x f( + f(x f(tdt ( λf (x 3λ (x f (x f ( = 4 n x + n! { (λ k k(k k 4 R (n+ (t(x t n dt +λ (k3(k 4 f (k ((x k nd this is equivlent to (3.4 with the substitution x = b. The formul (3.5 cn be proved by induction. If R(x is defined in the proof of Theorem 3. nd we substitution x = then, in similr wy s bove, we get (3.6. Remrk 3. We note tht in the specil cses, if we tke λ = 0 nd in Theorem 3.6, we get Theorems 5 nd 8 of 7, respectively. If we tke λ = 3 in Theorem 3.6, we get Theorem.9 of 6. We cn lso give some corresponding formuls with finite sums for reminder terms of other qudrture rules s specil cses. For exmples, we cn set λ = nd λ = 3 to get corresponding formuls with finite sums for reminder terms of perturbed verged mid-point-trpezoid type rule nd perturbed verged 3-point type rule, respectively. Acknowledgements The uthor wish to thnk the nonymous referees for their vluble comments. This work ws prtly supported by the Qing Ln Project of Jingsu Province, the Ntionl Nturl Science Foundtion of Chin (Grnt No nd the Teching Reserch Project of NUIST (Grnt No. JY05. References. Brnett, N.S.; Drgomir, S.S. Applictions of Ostrowski s version of the Grüss inequlity for trpezoid type rules, Tmkng J. Mth., 37 (006, Cerone, P. On perturbed trpezoidl nd midpoint rules, Koren J. Comput. Appl. Mth., 9 (00, Cerone, P. Perturbed rules in numericl integrtion from product brnched Peno kernels, Nonliner Anl. Forum, 9 (004, Cerone, P.; Drgomir, S.S. Trpezoidl-type rules from n inequlities point of view, Hndbook of nlytic-computtionl methods in pplied mthemtics, 65 34, Chpmn & Hll/CRC, Boc Rton, FL, Cerone, P.; Drgomir, S.S. Midpoint-type rules from n inequlities point of view, Hndbook of nlytic-computtionl methods in pplied mthemtics, 35 00, Chpmn & Hll/CRC, Boc Rton, FL, Cerone, P.; Drgomir, S. S.; Roumeliotis, J. An inequlity of Ostrowski-Grüss type for twice differentible mppings nd pplictions in numericl integrtion, Kyungpook Mth. J., 39 (999, Chen, W.B.; Chen, Q.; Liu, W.J. A unified generliztion of perturbed trpezoid nd midpoint inequlities nd pplictions in numericl integrtion, Advnces in Applied Mthemticl Anlysis, 3 (008, 5.

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