In this paper, we establish a new generalized Ostrowski type inequality for double integrals involving functions of two independent variables by using fairly elementary analysis

Mehmet Zeki Sarikaya

Mustafa Kemal Yildiz

International Journal of Analysis and Applications

Directory of Open Access Journals

On the Generalized Ostrowski Type Integral Inequality for Double Integrals

WOS: 000391118800007In this paper, we establish a new generalized Ostrowski type inequality for double integrals involving functions of two independent variables by using fairly elementary analysis

Yıldız, Mustafa Kemal

Sarıkaya, Mehmet Zeki

Duzce University Open Access

ON THE GENERALIZED OSTROWSKI TYPE INTEGRAL INEQUALITY FOR DOUBLE INTEGRALS

International Journal of Analysis and ApplicationsISSN 2291-8639Volume 13, Number 1 (2017), 64-69http://www.etamaths.comON THE GENERALIZED OSTROWSKI TYPE INTEGRAL INEQUALITY FORDOUBLE INTEGRALSMUSTAFA KEMAL YILDIZ1,∗ AND MEHMET ZEKI SARIKAYA2Abstract. In this paper, we establish a new generalized Ostrowski type inequality for double inte-grals involving functions of two independent variables by using fairly elementary analysis.1. IntroductionIn 1938, the classical integral inequality was established by Ostrowski [5] as follows:Theorem 1.1. Let f : [a, b]→ R be a differentiable mapping on (a, b) whose derivative f ′ : (a, b)→ Ris bounded on (a, b), i.e., ‖f ′‖∞ = supt∈(a,b)|f ′(t)| <∞. Then, the inequality holds:∣∣∣∣∣∣f(x)− 1b− ab∫af(t)dt∣∣∣∣∣∣ ≤[14+(x− a+b2)2(b− a)2](b− a) ‖f ′‖∞ (1.1)for all x ∈ [a, b]. The constant 14 is the best possible.In a recent paper [3], Barnett and Dragomir proved the following Ostrowski type inequality fordouble integrals:Theorem 1.2. Let f : [a, b]×[c, d]→ R be continuous on [a, b]×[c, d], f ′′x,y =∂2f∂x∂y exists on (a, b)×(c, d)and is bounded, i.e., ∥∥f ′′x,y∥∥∞ = sup(x,y)∈(a,b)×(c,d)∣∣∣∣∂2f(x, y)∂x∂y∣∣∣∣ <∞.Then, we have the inequality:∣∣∣∣∣∣b∫ad∫cf(s, t)dtds− (d− c)(b− a)f(x, y)−(b− a) d∫cf(x, t)dt + (d− c)b∫af(s, y)ds∣∣∣∣∣∣ (1.2)≤[14(b− a)2 + (x− a + b2)2] [14(d− c)2 + (y − d + c2)2] ∥∥f ′′x,y∥∥∞for all (x, y) ∈ [a, b]× [c, d].In [3], the inequality (1.2) is established by the use of integral identity involving Peano kernels.In [7], Pachpatte obtained an inequality in the view (1.2) by using elementary analysis. The interestedreader is also refered to ( [3], [4], [6]- [13]) for Ostrowski type inequalities in several independentvariables and for recent weighted version of these type inequalities see [1], [2], [9] and [11].Received 22nd July, 2016; accepted 19th September, 2016; published 3rd January, 2017.2010 Mathematics Subject Classification. 26D07, 26D15.Key words and phrases. integral inequality; Ostrowski’s inequality.c©2017 Authors retain the copyrights oftheir papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.64ON THE GENERALIZED OSTROWSKI TYPE INTEGRAL INEQUALITY 65Meanwhile, in [11] Sarikaya and Ogunmez gave the following interesting identity and by using thisindentity they establised some interesting integral inequalities:Lemma 1.1. Let f : [a, b] × [c, d]→ R be an absolutely continuous function such that the partialderivative of order ∂2f(t,s)∂t∂s exists for all (t, s) ∈ [a, b]× [c, d] and the weight function w : [a, b]→ [0,∞)is integrable, nonnegative andm(a, b) =b∫aw(t)dt <∞. (1.3)Then, we havef(x, y) =1m(a, b)b∫aw(t)f(t, y)dt +1m(c, d)d∫cw(s)f(x, s)ds− 1m(a, b)m(c, d) b∫ad∫cw(t)w(s)f(t, s)dsdt−b∫ad∫cp(x, t)q(y, s)∂2f(t, s)∂t∂sdsdt (1.4)wherep(x, t) =p1(a, t) =t∫aw(u)du, a ≤ t < xp2(b, t) =t∫bw(u)du, x ≤ t ≤ bandq(y, s) =q1(c, s) =s∫cw(v)dv, c ≤ s < yq2(d, s) =s∫dw(v)dv, y ≤ s ≤ d.The main aim of this paper is to establish a new generalized Ostrowski type inequality for doubleintegrals involving functions of two independent variables and their partial derivatives.2. Main ResultWe begin with the following important result:Lemma 2.1. Let f : [a, b] × [c, d]→ R be an absolutely continuous function such that the partialderivative of order ∂2f(t,s)∂t∂s exists for all (t, s) ∈ [a, b]× [c, d], and the function p : [a, b]× [c, d]→ [0,∞)is integrable. Then, we have b∫ad∫cp(u, v)dvdu f(x, y)− b∫ad∫cp(t, v)f(t, y)dvdt (2.1)−b∫ad∫cp(u, s)f(x, s)dsdu +b∫ad∫cp(t, s)f(t, s)dsdt=b∫ad∫cP (x, t; y, s)∂2f(t, s)∂t∂sdsdt66 YILDIZ AND SARIKAYAwhereP (x, t; y, s) =t∫as∫cp(u, v)dvdu, a ≤ t < x, c ≤ s < yt∫as∫dp(u, v)dvdu, a ≤ t < x, y ≤ s ≤ dt∫bs∫cp(u, v)dvdu, x ≤ t ≤ b, c ≤ s < yt∫bs∫dp(u, v)dvdu, x ≤ t ≤ b, y ≤ s ≤ d.Proof. By definitions of P (x, t; y, s), we haveb∫ad∫cP (x, t; y, s)∂2f(t, s)∂t∂sdsdt =x∫ay∫c[t∫as∫cp(u, v)dvdu]∂2f(t, s)∂t∂sdsdt+x∫ad∫y[t∫as∫dp(u, v)dvdu]∂2f(t, s)∂t∂sdsdt +b∫xy∫c[t∫bs∫cp(u, v)dvdu]∂2f(t, s)∂t∂sdsdt+b∫xd∫y[t∫bs∫dp(u, v)dvdu]∂2f(t, s)∂t∂sdsdt.Integrating by parts, we can state:x∫ay∫c[t∫as∫cp(u, v)dvdu]∂2f(t, s)∂t∂sdsdt=x∫a[(t∫ay∫cp(u, v)dvdu)∂f(t, y)∂t−y∫c(t∫ap(u, s)du)∂f(t, s)∂tds]dt=(x∫ay∫cp(u, v)dvdu)f(x, y)−x∫a(y∫cp(t, v)dv)f(t, y)dt−y∫c(x∫ap(u, s)du)f(x, s)ds +x∫ay∫cp(t, s)f(t, s)dsdt,(2.2)x∫ad∫y[t∫as∫dp(u, v)dvdu]∂2f(t, s)∂t∂sdsdt= −x∫a[(t∫ay∫dp(u, v)dvdu)∂f(t, y)∂t+d∫y(t∫ap(u, s)du)∂f(t, s)∂tds]dt=(x∫ad∫yp(u, v)dvdu)f(x, y)−x∫a(d∫yp(t, v)dv)f(t, y)dt−d∫y(x∫ap(u, s)du)f(x, s)ds +x∫ad∫yp(t, s)f(t, s)dsdt,(2.3)b∫xy∫c[t∫bs∫cp(u, v)dvdu]∂2f(t, s)∂t∂sdsdt= −b∫x[(b∫ty∫cp(u, v)dvdu)∂f(t, y)∂t−y∫c(b∫tp(u, s)du)∂f(t, s)∂tds]dt=(b∫xy∫cp(u, v)dvdu)f(x, y)−b∫x(y∫cp(t, v)dv)f(t, y)dt−y∫c(b∫xp(u, s)du)f(x, s)ds +b∫xy∫cp(t, s)f(t, s)dsdt,(2.4)ON THE GENERALIZED OSTROWSKI TYPE INTEGRAL INEQUALITY 67b∫xd∫y[t∫bs∫dp(u, v)dvdu]∂2f(t, s)∂t∂sdsdt=b∫x[(b∫td∫yp(u, v)dvdu)∂f(t, y)∂t+d∫y(b∫tp(u, s)du)∂f(t, s)∂tds]dt=(b∫xd∫yp(u, v)dvdu)f(x, y)−b∫x(d∫yp(t, v)dv)f(t, y)dt−d∫y(b∫xp(u, s)du)f(x, s)ds +b∫xd∫yp(t, s)f(t, s)dsdt.(2.5)Adding (2.2)-(2.5) and rewriting, we easily deduce required identity (2.1) which completes the proof.Remark 2.1. If take p(., .) ≡ 1 in Lemma 2.1, we getf(x, y)− 1(b− a)b∫af(t, y)dt− 1(d− c)d∫cf(x, s)dsdu +1(b− a) (d− c)b∫ad∫cf(t, s)dsdt=1(b− a) (d− c)b∫ad∫cP (x, t; y, s)∂2f(t, s)∂t∂sdsdtwhereP (x, t; y, s) =(t− a) (s− c) , a ≤ t < x, c ≤ s < y(t− a) (s− d) , a ≤ t < x, y ≤ s ≤ d(t− b) (s− c) , x ≤ t ≤ b, c ≤ s < y(t− b) (s− d) , x ≤ t ≤ b, y ≤ s ≤ d.which is given by Barnett and Dragomir in [3].Remark 2.2. If take p(u, v) = w(u)w(v) in Lemma 2.1, then the Lemma 2.1 reduces to the Lemma1.1 which is proved by Sarikaya and Ogunmez in [11].Theorem 2.1. Let f : [a, b] × [c, d]→ R be an absolutely continuous function such that the partialderivative of order ∂2f(t,s)∂t∂s exists and is bounded, i.e.,∥∥∥∥∂2f(t, s)∂t∂s∥∥∥∥∞= sup(t,s)∈(a,b)×(c,d)∣∣∣∣∂2f(t, s)∂t∂s∣∣∣∣ <∞for all (t, s) ∈ [a, b]× [c, d], the function p : [a, b]× [c, d]→ [0,∞) is integrable. Then, we have∣∣∣∣∣∣ b∫ad∫cp(u, v)dvdu f(x, y)− b∫ad∫cp(t, v)f(t, y)dvdt−b∫ad∫cp(u, s)f(x, s)dsdu +b∫ad∫cp(t, s)f(t, s)dsdt∣∣∣∣∣∣ (2.6)≤∥∥∥∥∂2f(t, s)∂t∂s∥∥∥∥∞x∫a(x− u)A(u, y)du +b∫x(u− x)A(u, y)duwhereA(u, y) =y∫c(y − v) |p(u, v)| dv +d∫y(v − y) |p(u, v)| dv.68 YILDIZ AND SARIKAYAProof. From Lemma 2.1 and using the properties of modulus, we observe that∣∣∣∣∣∣ b∫ad∫cp(u, v)dvdu f(x, y)− b∫ad∫cp(t, v)f(t, y)dvdt−b∫ad∫cp(u, s)f(x, s)dsdu +b∫ad∫cp(t, s)f(t, s)dsdt∣∣∣∣∣∣≤b∫ad∫c|P (x, t; y, s)|∣∣∣∣∂2f(t, s)∂t∂s∣∣∣∣ dsdt (2.7)≤∥∥∥∥∂2f(t, s)∂t∂s∥∥∥∥∞b∫ad∫c|P (x, t; y, s)| dsdt≤∥∥∥∥∂2f(t, s)∂t∂s∥∥∥∥∞x∫ay∫c t∫as∫c|p(u, v)| dvdu dsdt+x∫ad∫y t∫ad∫s|p(u, v)| dvdu dsdt + b∫xy∫c b∫ts∫c|p(u, v)| dvdu dsdt+b∫xd∫y b∫td∫s|p(u, v)| dvdu dsdt≤∥∥∥∥∂2f(t, s)∂t∂s∥∥∥∥∞{J1 + J2 + J3 + J4} .Now, using the change of order of integration we getJ1 =x∫ay∫c t∫as∫c|p(u, v)| dvdu dsdt=x∫at∫a y∫cs∫c|p(u, v)| dvds dudt=x∫at∫a y∫c(y − v) |p(u, v)| dv dudt=y∫c x∫at∫a(y − v) |p(u, v)| dudt dv (2.8)=x∫ay∫c(x− u) (y − v) |p(u, v)| dvduand similarly,J2 =x∫ad∫y(x− u) (v − y) |p(u, v)| dvdu, (2.9)J3 =b∫xy∫c(u− x) (y − v) |p(u, v)| dvdu, (2.10)ON THE GENERALIZED OSTROWSKI TYPE INTEGRAL INEQUALITY 69J4 =b∫xd∫y(u− x) (v − y) |p(u, v)| dvdu. (2.11)Thus, using (2.8), (2.9), (2.10) and (2.11) in (2.7), we obtain the inequality (2.6) and the proof iscompleted. Remark 2.3. If we choose p(., .) ≡ 1 in Theorem 2.1, then the inequality (2.6) reduces the inequality(1.2) which is proved by Barnett and Dragomir in [3].Remark 2.4. If take p(u, v) = w(u)w(v) in Theorem 2.1, then the inequality (2.6) reduces∣∣∣∣∣∣f(x, y)− 1m(a, b)b∫aw(t)f(t, y)dt− 1m(c, d)d∫cw(s)f(x, s)ds +1m(a, b)m(c, d)b∫ad∫cw(s)w(t)f(t, s)dsdt∣∣∣∣∣∣≤∥∥∥∥∂2f(t, s)∂t∂s∥∥∥∥∞x∫a(x− u)A(u, y)du +b∫x(u− x)A(u, y)duwhereA(u, y) =y∫c(y − v)w(u)w(v)dv +d∫y(v − y)w(u)w(v)dv.which is proved by Sarikaya and Ogunmez in [11].References[1] F. Ahmad, N. S. Barnett and S. S. Dragomir, New Weighted Ostrowski and Cebysev Type Inequalities, NonlinearAnalysis: Theory, Methods & Appl., 71 (12) (2009), 1408-1412.[2] F. Ahmad, A. Rafiq, N. A. Mir, Weighted Ostrowski type inequality for twice differentiable mappings, Global Journalof Research in Pure and Applied Math., 2 (2) (2006), 147-154.[3] N. S. Barnett and S. S. Dragomir, An Ostrowski type inequality for double integrals and applications for cubatureformulae, Soochow J. Math., 27(1) (2001), 109-114.[4] S. S. Dragomir, N. S. Barnett and P. Cerone, An n-dimensional version of Ostrowski’s inequality for mappings ofHölder type, RGMIA Res. Pep. Coll., 2(2) (1999), 169-180.[5] A. M. Ostrowski, Über die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Comment.Math. Helv. 10 (1938), 226-227.[6] B. G. Pachpatte, On an inequality of Ostrowski type in three independent variables, J. Math.Anal. Appl., 249(2000), 583-591.[7] B. G. Pachpatte, On a new Ostrowski type inequality in two independent variables, Tamkang J. Math., 32 (1)(2001), 45-49[8] B. G. Pachpatte, A new Ostrowski type inequality for double integrals, Soochow J. Math., 32 (2) (2006), 317-322.[9] A. Rafiq and F. Ahmad, Another weighted Ostrowski-Grüss type inequality for twice differentiable mappings, Kragu-jevac Journal of Mathematics, 31 (2008), 43-51.[10] M. Z. Sarikaya, On the Ostrowski type integral inequality, Acta Math. Univ. Comenianae, 79 (1) (2010), 129-134.[11] M. Z. Sarikaya and H. Ogunmez, On the weighted Ostrowski type integral inequality for double integrals, TheArabian Journal for Science and Engineering (AJSE)-Mathematics, 36 (2011), 1153-1160.[12] M. Z. Sarikaya and H. Yildirim, New inequalities for local fractional integrals, Iranian Journal of Science andTechnology (Sciences), in press.[13] N. Ujević, Some double integral inequalities and applications, Appl. Math. E-Notes, 7 (2007), 93-101.1Department of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University, Afyon-Turkey2Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce-Turkey∗Corresponding author: myildiz@aku.edu.tr

On the Generalized Ostrowski Type Integral Inequality for Double Integrals

Abstract

Similar works

Full text

Available Versions

Directory of Open Access Journals

Duzce University Open Access

Directory of Open Access Journals