Basing on Taylor’s formula with an integral remaider, an integral is estimated in Lp norm of the (n + 1)-th derivative of its integrand, and the Iyengar’s inequality and many other useful inequalities are generalized

Guo, Bai-Ni

Qi, Feng

English

Victoria University Eprints Repository

ESTIMATES FOR AN INTEGRAL IN Lp NORM OF THE (n+ 1)-THDERIVATIVE OF ITS INTEGRANDBAI-NI GUO AND FENG QIAbstract. Basing on Taylor’s formula with an integral remaider, an integral is estimated in Lpnorm of the (n+ 1)-th derivative of its integrand, and the Iyengar’s inequality and many otheruseful inequalities are generalized.1. IntroductionThe study of Iyengar’s inequality [6] has a rich literature and a long history. For details pleaserefer to references in this article.In [4, 5] and [11]–[15], making use of mean value theorems for derivative (including Rolle’s,Lagarange’s, and Taylor’s), mean value theorem for integral, the Taylor’s formula for functions ofseveral variables, and other technique, the following result was obtained:Theorem 1. Let the function f ∈ Cn([a, b]) have derivative of (n+1)-th order in (a, b) satisfyingN 6 f (n+1)(x) 6M . Denote(1)Sn(u, v, w) =n−1∑k=1(−1)kk!· ukf (k−1)(v) + wn!· (−1)nun,∂kSn∂uk= S(k)n (u, v, w),then, for any t ∈ (a, b), when n is odd we have(2)n+2∑i=0(−1)ii!(S(i)n+2(a, a,N)− S(i)n+2(b, b,N))ti 6∫ baf(x) dx6n+2∑i=0(−1)ii!(S(i)n+2(a, a,M)− S(i)n+2(b, b,M))ti;1991 Mathematics Subject Classification. Primary 26D15, 47A30, 65D30; Secondary 26D10.Key words and phrases. Integral, estimate, derivative, Taylor’s formula, integral remainder, Lp norm.The authors were supported in part by NSF of Henan Province (no. 004051800), SF for Pure Research of theEducation Committee of Henan Province (no. 1999110004), and Doctor Fund of Jiaozuo Institute of Technology,The People’s Republic of China.This paper was typeset using AMS-LATEX.12 BAI-NI GUO AND FENG QIwhen n is even we have(3)n+2∑i=0(−1)ii!(S(i)n+2(a, a,N)− S(i)n+2(b, b,M))ti 6∫ baf(x) dx6n+2∑i=0(−1)ii!(S(i)n+2(a, a,M)− S(i)n+2(b, b,N))ti.In [1]–[3], [8] and [18], similar problems were investigated by using Hayashi’s inequality. Formore information, please see [7], [9] and [10].In this article, by the Taylor’s mean value theorem [17] with an integral remainder, we haveTheorem 2. Let the function f ∈ Cn([a, b]) have derivative of (n + 1)-th order in (a, b), andf (n+1) ∈ Lp([a, b]) for positive numbers p and q satisfying 1p + 1q = 1. Then, for any t ∈ (a, b), wehave(4)∣∣∣∣∣∫ ba f(x) dx− n∑i=0 f (i)(a)(i+ 1)! (t− a)i+1 +n∑i=0f (i)(b)(i+ 1)!(t− b)i+1∣∣∣∣∣6 (t− a)n+1+ 1q + (b− t)n+1+ 1q(n+ 1)! q√nq + q + 1∥∥∥f (n+1)∥∥∥Lp([a,b]).Corollary 1. Let f ∈ Cn([a, b]). If f (i)(a) = f (i)(b) = 0 for 1 6 i 6 n, and f (n+1) ∈ Lp([a, b]) isnot identically zero, then for positive numbers p and q satisfying 1p +1q = 1, we have(5)∣∣∣∣∣∫ ba f(x) dx∣∣∣∣∣ 6 (b− a)1+ 1qq√2(q + 1) ∥∥∥f (n+1)∥∥∥Lp([a,b]) .2. Proof of Theorem 2Let t be a parameter satisfying a < t < b, and write∫ baf(x) dx =∫ taf(x) dx+∫ btf(x) dx.(6)The well-known Taylor’s formula with an integral remainder [17, pp. 4–6] states thatf(x) =n∑i=0f (i)(a)i!(x− a)i + 1n!∫ xa(x− s)nf (n+1)(s) ds,(7)f(x) =n∑i=0f (i)(b)i!(x− b)i + 1n!∫ xb(x− s)nf (n+1)(s) ds,(8)and ∫ xa∫ ta(t− s)nf (n+1)(s) ds dt = 1n+ 1∫ xa(x− s)n+1f (n+1)(s) ds.(9)ESTIMATES FOR AN INTEGRAL IN Lp NORM 3Integrating on both sides of (7) over [a, t], we obtain∫ taf(x) dx =n∑i=0f (i)(a)(i+ 1)!(t− a)i+1 + 1n!∫ ta∫ xa(x− s)nf (n+1)(s) ds dx=n∑i=0f (i)(a)(i+ 1)!(t− a)i+1 + 1(n+ 1)!∫ ta(t− s)n+1f (n+1)(s) ds.(10)Therefore, we have∣∣∣∣∣∫ ta f(x) dx− n∑i=0 f (i)(a)(i+ 1)! (t− a)i+1∣∣∣∣∣6 1(n+ 1)!∫ ta∣∣∣(t− s)n+1f (n+1)(s)∣∣∣ ds6 1(n+ 1)!(∫ ta(t− s)q(n+1) ds)1/q (∫ ta∣∣∣f (n+1)(s)∣∣∣p ds)1/p6 1(n+ 1)!((t− a)nq+q+1nq + q + 1)1/q ∥∥∥f (n+1)∥∥∥Lp[a,b],(11)where 1p +1q = 1.Integrating on both sides of (8) over [t, b], we get∫ btf(x) dx = −n∑i=0f (i)(b)(i+ 1)!(t− b)i+1 + 1n!∫ bt∫ xb(x− s)nf (n+1)(s) ds dx= −n∑i=0f (i)(b)(i+ 1)!(t− b)i+1 + 1(n+ 1)!∫ bt(t− s)n+1f (n+1)(s) ds.(12)Thus, from (12), it follows that∣∣∣∣∣∫ bt f(x) dx+ n∑i=0 f (i)(b)(i+ 1)! (t− b)i+1∣∣∣∣∣6 1(n+ 1)!∫ bt∣∣∣(t− s)n+1f (n+1)(s)∣∣∣ ds6 1(n+ 1)!(∫ bt(s− t)p(n+1) ds)1/p(∫ bt∣∣∣f (n+1)(s)∣∣∣p ds)1/p6 1(n+ 1)!((b− t)np+p+1np+ p+ 1)1/p ∥∥∥f (n+1)∥∥∥Lp[a,b].(13)4 BAI-NI GUO AND FENG QIFrom (6), (11) and (13), we have∣∣∣∣∣∫ ba f(x) dx− n∑i=0 f (i)(a)(i+ 1)! (t− a)i+1 +n∑i=0f (i)(b)(i+ 1)!(t− b)i+1∣∣∣∣∣6 1(n+ 1)!{((t− a)np+p+1np+ p+ 1)1/p+((b− t)np+p+1np+ p+ 1)1/p}·∥∥∥f (n+1)∥∥∥Lp[a,b]=(t− a)n+1+ 1q + (b− t)n+1+ 1q(n+ 1)! q√nq + q + 1∥∥∥f (n+1)∥∥∥Lp([a,b]).(14)The proof is complete.References[1] R. P. Agarwal and S. S. Dragomir, An application of Hayashi’s inequality for differentiable functions, Com-puters Math. Appl. 32 (1996), no. 6, 95–99.[2] P. Cerone and S. S. Dragomir, On a weighted generalization of Iyengar type inequalities in-volving bounded first derivative, RGMIA Research Report Collection 2 (1999), no. 2, 147–157.http://rgmia.vu.edu.au/v2n2.html.[3] P. Cerone and S. S. Dragomir, Lobatto type quadrature rules for functions with bounded derivative, RGMIAResearch Report Collection 2 (1999), no. 2, 133–146. http://rgmia.vu.edu.au/v2n2.html.[4] Li-Hong Cui and Bai-Ni Guo, On proofs of an integral inequality and its generalizations, Journal of ZhengzhouGrain College, 17 (1996), Supplement, 152–154 and 158. (Chinese)[5] Bai-Ni Guo and Feng Qi, Proofs of an integral inequality, Mathematics and Informatics Quarterly, 7 (1997),no. 4, 182–184.[6] K. S. K. Iyengar, Note on an inequality, Math. Student 6 (1938), 75–76.[7] Ji-Chang Kuang, Applied Inequalities (Changyong Budengshi), 2nd edition, Hunan Education Press, Chang-sha, China, 1993, Chapter 8, page 584. (Chinese)[8] G. V. Milovanovic´ and J. E. Pecˇaric´, Some considerations on Iyengar’s inequality and some related applications,Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 544–576 (1976), 166–170.[9] D. S. Mitrinovic´, Analytic Inequalities, Springer-Verlag, 1970, page 297.[10] D. S. Mitrinovic´, J. E. Pecˇaric´, and A. M. Fink, Inequalities Involving Functions and Their Integrals andDerivatives, Kluwer Academic Publishers, 1991, Chapter XV.[11] Feng Qi, Inequalities for an integral, The Mathematical Gazette 80 (1996), no. 488, 376–377.[12] Feng Qi, Further generalizations of inequalities for an integral, Univ. Beograd. Publ. Elektrotehn. Fak. Ser.Mat. 8 (1997), 79–83.[13] Feng Qi, Inequalities for a multiple integral, Acta Mathematica Hungarica 84 (1999), no. 1-2, 19–26.[14] Feng Qi, Inequalities for a weighted integral, RGMIA Research Report Collection 2 (1999), no. 7, Article 2.http://rgmia.vu.edu.au/v2n7.html.[15] Feng Qi, Inequalities for a weighted multiple integral, Journal of Mathematical Analysis and Ap-plications (2000), in the press. RGMIA Research Report Collection 2 (1999), no. 7, Article 4.http://rgmia.vu.edu.au/v2n7.html.ESTIMATES FOR AN INTEGRAL IN Lp NORM 5[16] Feng Qi, Estimates in Lp norms of partial derivatives of the (n+1)-th order for a multiple integral, submitted.[17] Michael E. Taylor, Partial Differential Equations I—Basic Theory, Springer-Verlag, New York, 1996. Reprintedin China by Beijing World Publishing Corporation, Beijing, 1999.[18] P. M. Vasic´ and G. V. Milonanovic´, On an inequality of Iyengar, Univ. Beograd. Publ. Elektrotehn. Fak. Ser.Mat. Fiz. No. 544–576 (1976), 18–24.Department of Mathematics, Jiaozuo Institute of Technology, Jiaozuo City, Henan 454000, ThePeople’s Republic of ChinaDepartment of Mathematics, Jiaozuo Institute of Technology, Jiaozuo City, Henan 454000, ThePeople’s Republic of ChinaE-mail address: qifeng@jzit.edu.cnURL: http://rgmia.vu.edu.au/qi.html or http://rgmia.vu.edu.au/authors/FQi.htm

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