In this paper, we firstly establish generalized weighted Montgomery identity for double integrals. Then, some generalized weighted Cebysev and Ostrowski type inequalities for double integrals are given.Publisher's Versio

Budak, Hüseyin

Sarıkaya, Mehmet Zeki

TWMS Journal of Applied and Engineering Mathematics

English

Isik University Academic Open Access

TWMS J. App. Eng. Math. V.7, N.2, 2017, pp. 272-281GENERALIZED WEIGHTED ČEBYSEV AND OSTROWSKI TYPEINEQUALITIES FOR DOUBLE INTEGRALSH. BUDAK1, M. Z. SARIKAYA1, §Abstract. In this paper, we firstly establish generalized weighted Montgomery identityfor double integrals. Then, some generalized weighted Čebysev and Ostrowski typeinequalities for double integrals are given.Keywords: Čebysev type inequalities, Ostrowski type inequalities, weighted integral in-equalities.AMS Subject Classification: 26D07, 26D10, 26D15, 26A331. IntroductionLet f : [a, b]→ R be a differentiable mapping on (a, b) whoose derivative f ′ : (a, b)→ Ris baunded on (a, b) , i.e. ‖f ′‖∞ := supt∈(a,b)|f ′(t)| <∞. Then we have∣∣∣∣∣∣f(x)− 1b− ab∫af(t)dt∣∣∣∣∣∣ ≤[14+(x− a+b2)2(b− a)2](b− a)∥∥f ′∥∥∞ ,for all x ∈ [a, b]. The constant 14 is the best possible [10]. This inequality is well known inthe literature as the Ostrowski inequality. For some results which generalize, improve andextend the inequality (1), see ([2], [5], [18], [19], [21]) and the references therein.In [4], P. L. Čebysev proved the following important integral inequality|T (f, g)| ≤ 112(b− a)2∥∥f ′∥∥∞ ∥∥g′∥∥∞ (1)where f, g : [a, b] → R are absolutely continuous functions whose derivatives f ′, g′ ∈L∞ [a, b] andT (f, g) =1b− ab∫af(x)g(x)dx− 1b− ab∫af(x)dx 1b− ab∫ag(x)dx (2)which is called the Čebysev functional, provided the integrals in (2) exist. In recentyears many researchers have given the generalization of Čebysev type inequalities, we canmention the works ([1], [3], [6], [9], [12], [13], [14], [16], [20]).1 Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce-TURKEY.hsyn.budak@gmail.com, sarikayamz@gmail.com;ORCID: http://orcid.org/0000-0001-8843-955X, http://orcid.org/0000-0002-6165-9242§ Manuscript received: June 06, 2016; accepted: October 30, 2016.TWMS Journal of Applied and Engineering Mathematics, Vol.7, No.2; c© Işık University, Departmentof Mathematics, 2017; all rights reserved.272H. BUDAK, M. Z. SARIKAYA: GENERALIZED WEIGHTED ČEBYSEV AND OSTROWSKI TYPE ... 273Let w1 : [a, b] → [0,∞) be a weight function. We define m1(a, b) =b∫aw1(s)ds andm1(a, t) =t∫aw1(s)ds, so that m1(a, t) = 0 for t < a.In [13], Rafiq et al. proved the following weighted Montgomery’s indentity:Let f : [a, b] → R be absolutely continuous, ϕ1 : R+ → R+ be a differentiable functionon R+ with ϕ1(0) = 0, ϕ1(m1(a, b)) 6= 0 and ϕ′1 is integrable on R+, thenf(x) =1ϕ1(m1(a, b))b∫aw1(t)ϕ′1(m1(a, t))f(t)dt (3)+1ϕ1(m1(a, b))b∫aPw1,ϕ1(x, t)f′(t)dtfor all x ∈ [a, b] , wherePw1,ϕ1(x, t) = ϕ1(m1(a, t)), a ≤ t ≤ xϕ1(m1(a, t))− ϕ1(m1(a, b)), x ≤ t ≤ b.(4)Recently, many authors have studied on Čebysev inequality for double integrals, please see([7], [8], [11] [15]). In [8], Guazene-Lakoud and Aissaoui established a weighted Čebysevtype inequality for double integrals using the probability density function. In this paper, weobtain a generalized weighted Čebysev type inequality similar to this inequality for doubleintegrals using the weighted funtions which are not necessarily the probability densityfunctions. Moreover, we established an Ostrowski type inequality for double integralwhich is the generalization of the inequality given in [17].2. Generalized Weighted Montgomery Identity for Double IntegralsIn order to prove our main theorems, we need to prove following identities:Let w2 : [c, d] → [0,∞) be a weight function. We define m2(c, d) =d∫cw2(u)du andm2(c, s) =s∫cw2(u)du, so that m2(c, s) for s < c. ϕ2 : R+ → R+ be a differentiablefunction on R+ with ϕ2(0) = 0, ϕ2(m2(c, d)) 6= 0 and ϕ′2 is integrable on R+.Theorem 2.1. Let f : ∆ = [a, b]× [c, d]→ R be a partial differentiable function such thatsecond derivative ∂2f(t,s)∂s∂t is integrable on ∆. Then for all (x, y) ∈ ∆ we havef(x, y) =1ϕ1(m1(a, b))ϕ2(m2(c, d))(5)× b∫ad∫cw1(t)w2(s)ϕ′1(m1(a, t))ϕ′2(m2(c, s))f(t, s)dsdt+b∫ad∫cw1(t)ϕ′1(m1(a, t))Qw2,ϕ2(y, s)∂f(t, s)∂sdsdt274 TWMS J. APP. ENG. MATH. V.7, N.2, 2015+b∫ad∫cw2(s)ϕ′2(m2(c, s))Pw1,ϕ1(x, t)∂f(t, s)∂tdsdt+b∫ad∫cPw1,ϕ1(x, t)Qw2,ϕ2(y, s)∂2f(t, s)∂s∂tdsdtwhere Pw1,ϕ1(x, t) is defined as in (4) and Qw2,ϕ2(y, s) defined byQw2,ϕ2(y, s) = ϕ2(m2(c, s)), c ≤ s ≤ yϕ2(m2(c, s))− ϕ2(m2(c, d)), y ≤ s ≤ d.(6)Proof. Aplying the identity (3) for the partial derivative ∂f(t,y)∂t , we havef(x, y) =1ϕ(m(a, b))b∫aw1(t)ϕ′1(m1(a, t))f(t, y)dt (7)+1ϕ1(m1(a, b))b∫aPw1,ϕ1(x, t)∂f(t, y)∂tdtfor all (x, y) ∈ ∆. Similarly, applying the identity (3) for the partial derivative ∂f(t,s)∂s , wegetf(t, y) =1ϕ2(m2(c, d))d∫cw2(s)ϕ′2(m2(c, s))f(t, s)ds (8)+1ϕ2(m2(c, d))d∫cQw2,ϕ2(y, s)∂f(t, s)∂sdsfor all (t, y) ∈ ∆. For partial derivative of (8) according to t, we have∂f(t, y)∂t=1ϕ2(m2(c, d))d∫cw2(s)ϕ′2(m2(c, s))∂f(t, s)∂tds (9)+1ϕ2(m2(c, d))d∫cQw2,ϕ2(y, s)∂2f(t, s)∂s∂tdsfor all (t, y) ∈ ∆. If we subsitute the equalities (8) and (9) in (7), then we obtain therequired result. Remark 2.1. If we choose ϕ1(u) ≡ ϕ2(u) ≡ u in the Theorem 2.1, then the Theorem 2.1reduces the Lemma 2 in [17].H. BUDAK, M. Z. SARIKAYA: GENERALIZED WEIGHTED ČEBYSEV AND OSTROWSKI TYPE ... 275Theorem 2.2. Let f : ∆→ R be a partial differentiable function such that second deriv-ative ∂2f(t,s)∂s∂t is integrable on ∆. Then we have the following generalized weighted Mont-gomery’s identity,f(x, y)− 1ϕ1(m1(a, b))b∫aw1(t)ϕ′1(m1(a, t))f(t, y)dt (10)− 1ϕ2(m2(c, d))d∫cw2(s)ϕ′2(m2(c, s))f(x, s)ds+1ϕ1(m1(a, b))ϕ2(m2(c, d))b∫ad∫cw1(t)w2(s)ϕ′1(m1(a, t))ϕ′2(m2(c, s))f(t, s)dsdt=1ϕ1(m1(a, b))ϕ2(m2(c, d))b∫ad∫cPw1,ϕ1(x, t)Qw2,ϕ2(y, s)∂2f(t, s)∂t∂sdsdtfor all (x, y) ∈ ∆.Proof. Using the integration by parts we have,b∫ad∫cPw1,ϕ1(x, t)Qw2,ϕ2(y, s)∂2f(t, s)∂t∂sdsdt (11)=b∫aPw1,ϕ1(x, t) d∫cϕ2(m2(c, s))∂2f(t, s)∂t∂sds− ϕ2(m2(c, d))d∫c∂2f(t, s)∂t∂sds dt=b∫aPw1,ϕ1(x, t)ϕ2(m2(c, d))∂f(t, y)∂t−d∫cw2(s)ϕ′2(m2(c, s))∂f(t, s)∂tds dt= ϕ2(m2(c, d))b∫aPw1,ϕ1(x, t)∂f(t, y)∂tdt−d∫cb∫aw2(s)ϕ′2(m2(c, s))Pw1,ϕ1(x, t)∂f(t, s)∂tdtds.Similarly, we haveb∫aPw1,ϕ1(x, t)∂f(t, y)∂tdt (12)=b∫aϕ1(m1(a, t))∂f(t, y)∂tdt− ϕ1(m1(a, b))b∫x∂f(t, y)∂tdt= ϕ1(m1(a, b))f(x, y)−b∫aw1(t)ϕ′1(m1(a, t))f(t, y)dt,276 TWMS J. APP. ENG. MATH. V.7, N.2, 2015andd∫cb∫aw2(s)ϕ′2(m2(c, s))Pw1,ϕ1(x, t)∂f(t, s)∂tdtds (13)=d∫cw2(s)ϕ′2(m2(c, s)) b∫aϕ1(m1(a, t))∂f(t, s)∂tdt− ϕ1(m(a, b))b∫x∂f(t, s)∂tdt ds=d∫cw2(s)ϕ′2(m2(c, s))ϕ(m1(a, b))f(x, s)− b∫aw1(t)ϕ′1(m1(a, t))f(t, s)dt ds= ϕ1(m1(a, b))d∫cw2(s)ϕ′2(m2(c, s))f(x, s)ds−b∫ad∫cw1(t)w2(s)ϕ′1(m1(a, t))ϕ′2(m2(c, s))f(t, s)dt.If we subsitute the equalities (12) and (13) in (11), then we obtain the required identity(10). Remark 2.2. If we take w1 and w2 as two probability density functions in (10), then theidentity (10) reduces the identity (6) in [8].3. New Generalized Weighted Ostrowski and Čebysev InequalitiesTheorem 3.1. Let f : ∆ → R have continuous partial derivatives ∂f(t,s)∂t ,∂f(t,s)∂s and∂2f(t,s)∂s∂t on ∆. Then we have the following weighted Ostrowski inequality∣∣∣∣∣∣f(x, y)− 1ϕ1(m1(a, b))ϕ2(m2(c, d))b∫ad∫cw1(t)w2(s)ϕ′1(m1(a, t))ϕ′2(m2(c, s))f(t, s)dsdt∣∣∣∣∣∣≤ 1ϕ1(m1(a, b))ϕ2(m2(c, d))[m1(a, b)∥∥∥∥∂f(t, s)∂s∥∥∥∥∞∥∥ϕ′1∥∥∞H2(y)+m2(c, d)∥∥∥∥∂f(t, s)∂t∥∥∥∥∞∥∥ϕ′2∥∥∞H1(x) + ∥∥∥∥∂f(t, s)∂t∥∥∥∥∞H1(x)H2(y)]whereH1(x) =b∫a|Pw1,ϕ1(x, t)| dt,andH2(y) =d∫c|Qw2,ϕ2(y, s)| ds.H. BUDAK, M. Z. SARIKAYA: GENERALIZED WEIGHTED ČEBYSEV AND OSTROWSKI TYPE ... 277Proof. Taking modulus in Theorem 2.1, we have∣∣∣∣∣∣f(x, y)− 1ϕ1(m1(a, b))ϕ2(m2(c, d))b∫ad∫cw1(t)w2(s)ϕ′1(m1(a, t))ϕ′2(m2(c, s))f(t, s)dsdt∣∣∣∣∣∣≤ 1ϕ1(m1(a, b))ϕ2(m2(c, d)) b∫ad∫cw1(t)∣∣ϕ′1(m1(a, t))∣∣ |Qw2,ϕ2(y, s)| ∣∣∣∣∂f(t, s)∂s∣∣∣∣ dsdt+b∫ad∫cw2(s)∣∣ϕ′2(m2(c, s))∣∣ |Pw1,ϕ1(x, t)| ∣∣∣∣∂f(t, s)∂t∣∣∣∣ dsdt+b∫ad∫c|Pw1,ϕ1(x, t)| |Qw2,ϕ2(y, s)|∣∣∣∣∂2f(t, s)∂s∂t∣∣∣∣ dsdt≤ 1ϕ1(m1(a, b))ϕ2(m2(c, d))∥∥∥∥∂f(t, s)∂s∥∥∥∥∞∥∥ϕ′1∥∥∞b∫ad∫cw1(t) |Qw2,ϕ2(y, s)| dsdt+∥∥∥∥∂f(t, s)∂t∥∥∥∥∞∥∥ϕ′2∥∥∞b∫ad∫cw2(s) |Pw1,ϕ1(x, t)| dsdt+∥∥∥∥∂f(t, s)∂t∥∥∥∥∞b∫ad∫c|Pw1,ϕ1(x, t)| |Qw2,ϕ2(y, s)| dsdt .Here, we have the equalitiesb∫ad∫cw1(t) |Qw2,ϕ2(y, s)| dsdt = b∫aw1(t)dt d∫c|Qw2,ϕ2(y, s)| ds = m1(a, b)H2(y),b∫ad∫cw2(s) |Pw1,ϕ1(x, t)| dsdt = d∫cw2(s)ds b∫a|Pw1,ϕ1(x, t)| dt = m2(c, d)H1(x)andb∫ad∫c|Pw1,ϕ1(x, t)| |Qw2,ϕ2(y, s)| dsdt= b∫a|Pw1,ϕ1(x, t)| dt d∫c|Qw2,ϕ2(y, s)| ds = H1(x)H2(y)which complete the proof. Remark 3.1. If we choose ϕ1(u) ≡ ϕ2(u) ≡ u in the Theorem 3.1, then the Theorem 3.1reduces the Theorem 2 in [17].Theorem 3.2. Let f, g : ∆→ R be partial differentiable functions such that their secondderivatives ∂2f(t,s)∂s∂t and∂2g(t,s)∂s∂t are integrable on ∆. Then we have the weighted Čebysev278 TWMS J. APP. ENG. MATH. V.7, N.2, 2015inequality|T (w1, ϕ1, w2, ϕ2, f, g)| (14)≤ 1ϕ31(m1(a, b))ϕ32(m2(c, d))∥∥∥∥∂2f(t, s)∂t∂s∥∥∥∥∞∥∥∥∥∂2g(t, s)∂t∂s∥∥∥∥∞∥∥ϕ′1∥∥∞ ∥∥ϕ′2∥∥∞×b∫ad∫cw1(x)w2(y)H21 (x)H22 (y)dydxwhere H1(x) and H2(y) are defined as in (3.1) andT (w1, ϕ1, w2, ϕ2, f, g)=1ϕ1(m1(a, b))ϕ2(m2(c, d))× b∫ad∫cw1(x)w2(y)ϕ′1(m1(a, x))ϕ′2(m2(c, y))f(x, y)g(x, y)dydx− 1ϕ2(m2(c, d))b∫ad∫cw1(x)w2(y)ϕ′1(m1(a, x))ϕ′2(m2(c, y))f(x, y)× d∫cw2(s)ϕ′2(m2(c, s))g(x, s)ds dydx− 1ϕ1(m1(a, b))b∫ad∫cw1(x)w2(y)ϕ′1(m1(a, x))ϕ′2(m2(c, y))g(x, y)× b∫aw1(t)ϕ′1(m1(a, t))f(t, y)dt dydx+1ϕ1(m1(a, b))ϕ2(m2(c, d))× b∫ad∫cw1(x)w2(y)ϕ′1(m1(a, x))ϕ′2(m2(c, y))f(x, y)dydx× b∫ad∫cw1(x)w2(y)ϕ′1(m1(a, x))ϕ′2(m2(c, y))g(x, y)dydx .H. BUDAK, M. Z. SARIKAYA: GENERALIZED WEIGHTED ČEBYSEV AND OSTROWSKI TYPE ... 279Proof. From Theorem 2.2, writing again the identity (10) for the function g(x, y), we haveg(x, y)− 1ϕ1(m1(a, b))b∫aw1(t)ϕ′1(m1(a, t))g(t, y)dt (15)− 1ϕ2(m2(c, d))d∫cw2(s)ϕ′2(m2(c, s))g(x, s)ds+1ϕ1(m1(a, b))ϕ2(m2(c, d))b∫ad∫cw1(t)w2(s)ϕ′1(m1(a, t))ϕ′2(m2(c, s))g(t, s)dsdt=1ϕ1(m1(a, b))ϕ2(m2(c, d))b∫ad∫cPw1,ϕ1(x, t)Qw2,ϕ2(y, s)∂2g(t, s)∂t∂sdsdt.After multiplying the identities (10) and (15), multiplying both sides result byw1(x)w2(y)ϕ′1(m1(a,x))ϕ′2(m2(c,y))ϕ1(m1(a,b))ϕ2(m2(c,d))and integrating over ∆, we haveT (w1, ϕ1, w2, ϕ2, f, g) (16)=1ϕ31(m1(a, b))ϕ32(m2(c, d))b∫ad∫c b∫ad∫cPw1,ϕ1(x, t)Qw2,ϕ2(y, s)∂2f(t, s)∂t∂sdsdt× b∫ad∫cPw1,ϕ1(x, t)Qw2,ϕ2(y, s)∂2g(t, s)∂t∂sdsdt dydx.Taking the modulus in (16), we obtain|T (w1, ϕ1, w2, ϕ2, f, g)|≤ 1ϕ31(m1(a, b))ϕ32(m2(c, d))b∫ad∫cw1(x)w2(y)ϕ′1(m1(a, x))ϕ′2(m2(c, y))× b∫ad∫c|Pw1,ϕ1(x, t)Qw2,ϕ2(y, s)|∣∣∣∣∂2f(t, s)∂t∂s∣∣∣∣ dsdt× b∫ad∫c|Pw1,ϕ1(x, t)Qw2,ϕ2(y, s)|∣∣∣∣∂2g(t, s)∂t∂s∣∣∣∣ dsdt dydx≤ 1ϕ31(m1(a, b))ϕ32(m2(c, d))∥∥∥∥∂2f(t, s)∂t∂s∥∥∥∥∞∥∥∥∥∂2g(t, s)∂t∂s∥∥∥∥∞∥∥ϕ′1∥∥∞ ∥∥ϕ′2∥∥∞×b∫ad∫cw1(x)w2(y) b∫ad∫c|Pw1,ϕ1(x, t)Qw2,ϕ2(y, s)| dsdt2 dydx280 TWMS J. APP. ENG. MATH. V.7, N.2, 2015=1ϕ31(m1(a, b))ϕ32(m2(c, d))∥∥∥∥∂2f(t, s)∂t∂s∥∥∥∥∞∥∥∥∥∂2g(t, s)∂t∂s∥∥∥∥∞∥∥ϕ′1∥∥∞ ∥∥ϕ′2∥∥∞×b∫ad∫cw1(x)w2(y)H21 (x)H22 (y)dydx.This completes the proof. Remark 3.2. If we take w1 and w2 as two probability density functions in (14), then theidentity (14) reduces the identity (14) in [8].4. ConclusionsIn this study, we presented some Čebysev and Ostrowski type inequalities generalizedweighted Montgomery identity. A further study could assess similar inequalities by usingdifferent types of generalized weighted Montgomery identity.References[1] Ahmad,F., Barnett,N.S., and Dragomir,S.S., (2009), New weighted Ostrowski and Čebysev type in-equalities, Nonlinear Anal., 71(12), pp.1408-1412.[2] Barnett,N.S. and Dragomir,S.S., (2001), An Ostrowski type inequality for double integrals and appli-cations for cubature formulae, Soochow J. 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SARIKAYA: GENERALIZED WEIGHTED ČEBYSEV AND OSTROWSKI TYPE ... 281[17] Sarikaya,M.Z., (2015), On the generalized weighted integral inequality for double integrals, Annals ofthe Alexandru Ioan Cuza University-Mathematics, LXI, pp.169-179. DOI: 10.2478/aicu-2014-0008.[18] Sarikaya,M.Z., (2012)On the Ostrowski type integral inequality for double integrals, DemonstratioMathematica, XLV(3), pp.533-540.[19] Sarikaya,M.Z., Yaldiz,H., and Erden,S., (2014), On the weigted Ostrowski type inequalities for doubleintegrals, Kragujevac Journal of Mathematics, 38(2), pp.303-314.[20] Set,E., Sarikaya,M.Z., and Ahmad,F., (2011), A generalization of Chebychev type inequalities for firstdifferentiable mappings, Miskolc Mathematical Notes, 12(2), pp.245-253.[21] Set,E., Sarikaya,M.Z., and Ahmad,F., (2015), On weighted Ostrowski type inequalities for double in-tegrals, Pure and Applied Mathematics Letters, Volume 2015, pp.55-58.Hüseyin BUDAK graduated from Kocaeli University, Kocaeli, Turkey in 2010.He received his M.Sc. from Kocaeli University in 2003. Since 2014, he is a Ph.D.student and a research assistant at Duzce University. His research interests focuson functions of bounded variation and theory of inequalities.Mehmet Zeki SARIKAYA received his BSc (Maths), MSc (Maths) and PhD(Maths) degrees from Afyon Kocatepe University, Afyonkarahisar, Turkey in 2000,2002 and 2007 respectively. At present, he is working as a professor in the De-partment of Mathematics at Duzce University (Turkey) and is the head of the de-partment. Moreover, he is the founder and Editor-in-Chief of Konuralp Journal ofMathematics (KJM). He is the author or coauthor of more than 200 papers in thefield of theory of inequalities, potential theory, integral equations and transforms,special functions, time-scales.

Generalized weighted Cebysev and Ostrowski type inequalities for double integrals

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Generalized weighted Cebysev and Ostrowski type inequalities for double integrals

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